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Connections between symmetries and integrating factors of ODEs Kolokolnikov, Theodore 1999

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C O N N E C T I O N S B E T W E E N S Y M M E T R I E S A N D I N T E G R A T I N G F A C T O R S OF ODES by THEODORE KOLOKOLNIKOV B.Math. University of Waterloo, 1997 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1999 © Theodore Kolokolnikov, 1999 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Date 3Q /Kpn^jL \J3J Abstract In this thesis we examine the connections between conservation laws and symmetries, both for self-adjoint and non self-adjoint ODEs. The goal is to gain a better understanding of how to combine symmetry methods with the method of conservation laws to obtain results not obtainable by either method separately. We review the concepts of symmetries and integrating factors. We present two known meth-ods of obtaining conservation laws without quadrature, using known conservation laws and symmetries. We show that the two methods yield the same result. For self-adjoint systems, we examine Noether's theorem in detail and discuss its generalisation for ODEs admitting more than one variational symmetry. We generalise an example from Sheftel [20] and show how to use r-dimensional Lie Algebra of variational symmetries to obtain more then r reductions of order. We develop an ansatz for finding point variational symmetries. We also develop ansatzes that use a known symmetry to find an integrating factor or another symmetry. These ansatzes are then used to classify ODEs. New solvable cases of Emden-Fowler and Abel ODEs result. ii Acknowledgments I thank George B luman , my supervisor, for his careful readings and suggestions throughout the preparat ion of this thesis. Many thanks to Edgardo S. Cheb-Terrab and A u s t i n D. Roche for useful and l ively discussions and for suggesting what eventually became Section 4.3. F inal ly , I would like to thank Greg Reid for his reading of the f inal draf t . i i i Table of Contents Abstract ii Acknowledgments iii Table of Contents iv List of Tables vi List of Figures vii Preface viii Chapter 1. Symmetries of ODEs 1 1.1 Symmetries of differential equations 2 1.1.1 Continuous transformations groups 2 1.1.2 Inf ini tesimals 4 1.1.3 Change of coordinates; canonical coordinates 5 1.1.4 Invariance under continuous t ransformat ion 6 1.1.5 Point symmetries of ODEs 8 1.1.6 Lie-Backlund symmetries of ODEs 12 1.1.7 T r i v ia l symmetries 12 1.1.8 Evolut ionary fo rm for symmetry generators 13 1.1.9 Lie Algebra of symmetries 15 1.2 Reduct ion of order using symmetries 17 1.2.1 Canonical coordinates 17 1.2.2 Dif ferent ial invariants 20 1.3 Discussion 24 Chapter 2. Conservation laws and integrating factors 26 2.1 Conservation laws and using symmetries to find them 27 2.1.1 Conservation laws 27 2.1.2 Ac t ion of symmetry generators on conservation laws 29 2.1.3 Using symmetries to find conservation laws 30 2.1.4 Using one symmetry to find two conservation laws 34 2.2 In tegrat ing factors and Euler Operator 37 2.2.1 Integrat ing factors 37 2.2.2 Ad jo in t Direct ional Derivative and Euler Operator 38 2.2.3 Euler-Lagrange Equations 40 2.2.4 Kernel of Euler Operator 41 2.3 Relat ionship between integrat ing factors, conservation laws, and symmetries 44 2.4 Discussion 48 iv Chapter 3. Self-adjoint systems 50 3.1 In t roduc t ion 50 3.2 Self-adjoint systems and Noether's theorem 51 3.2.1 Characterisat ion of a self-adjoint system 51 3.2.2 Var iat ional symmetries and Noether's theorem 53 3.3 Obta in ing conservation laws w i thout integrat ion 57 3.3.1 Commuta tor of var iat ional symmetries 58 3.3.2 Characterisat ion of var iat ional point symmetries 62 3.3.3 Using non-variat ional symmetries in conjunct ion w i t h var iat ional symmetries . 64 3.4 Conclusions 66 Chapter 4. Classification of solvable ODEs 68 4.1 Classif ication of the Emden-Fowler Equat ion, y" = Axnymy'1 69 4.1.1 Point symmetry classification of (4.3) 70 4.1.2 Ad jo in t symmetry classification of (4.3) 70 4.2 Using known symmetries or integrat ing factors as ansatzes 74 4.2.1 Using a symmetry to generate an ansatz for an integrat ing factor 74 4.2.2 Using a symmetry to generate an ansatz for a symmetry 78 4.3 Classif ication of y" + f{y)y' + g(y) = 0 79 4.3.1 Symmetry classification of (4.19) 80 4.3.2 Classification of integrat ing factors of (4.19) 85 4.4 Discussion 88 Chapter 5. Conclusions and future work 90 5.1 Conclusions 90 5.2 Future research 94 Bibliography 95 List of Tables 4.1 Symmetry classification of Emden-Fowler Equat ion 71 4.2 Ad jo in t symmetries of Emden-Fowler Equat ion 73 v i List of Figures 1.1 A n evolut ionary symmetry transforms in the vert ical direct ion only 14 2.1 A maximiser and curves nearby 40 v i i Preface One of the most algor i thmic methods of finding exact solutions to dif ferential equations is the method of continuous symmetries developed by Sophius Lie in the lat ter hal f of the 19th century. A m o n g many applications for ODEs, one can use a continuous symmetry to find a change of variables tha t leads to a reduct ion of order. A more direct approach for reducing the order of ODEs is to use conservation laws. Each conservation law leads to a reduct ion of order w i thout change of variables. Th is is unl ike a symmet ry reduct ion which relies on a change of variables. I n this thesis we explore the connections that exist between symmetry reduct ion and reduct ion using conservation laws. I n Chapter 1 we review Lie theory of symmetries, inc luding how to use symmetries to obta in reductions of order of ODEs. I n Chapter 2 we study conservation laws and integrat ing factors. A conservation law is char-acterised uniquely by its integrating factor and one can obta in a conservation law f rom an integrat ing factor using a quadrature. Al ternat ively, any pair ( integrat ing factor, symmetry) and any pair (conservation law, symmetry) are shown to yield a (possibly t r i v ia l ) conservation law without quadrature. I n Chapter 3 we study self-adjoint ODEs whose integrat ing factors correspond to a special class of symmetries, variational symmetries. I n her famous paper [15], E m m y Noether showed how to construct a conservation law f rom a variat ional symmetry. Moreover, the result ing conservation law admits the var iat ional symmetry that was used to find i t . Thus two reductions of order are possible using a single var iat ional symmetry: one reduct ion in the or iginal variables and one symmetry reduct ion. I n general, r var iat ional symmetries do not necessarily y ie ld 2r reductions of order. I n Chapter 3 we w i l l establish a lower bound on how many reductions of order of each type is to be expected. The answer depends on the structure of commutators of admi t ted var iat ional symmetries. For a self-adjoint O D E explici t i n its highest derivative, i t is possible to te l l when a point symmetry is var iat ional w i thou t looking at the O D E itself. We explore this to give an ansatz for seeking var iat ional symmetries of such ODEs. I n Chapter 4 we consider a classification problem: find al l ODEs belonging to a given fami ly of ODEs for which a solut ion can be found. As an example, we classify the Emden-Fowler fami ly of ODEs and find several new solvable cases. We also develop ansatzes tha t use a known symmetry to help in finding an integrat ing factor or another symmetry. These ansatzes are then used to find new solvable cases of Abe l ODEs. v i i i Chapter 1 Symmetries of O D E s I n th is chapter we introduce symmetry methods and show how symmetries can be used to reduce the order of ODEs. Since many physically relevant differential equations admi t symmetries, symmetry methods have become increasingly popular since Lie's fundamental work and its rediscovery i n the lat ter hal f of the 20th century, especially by mathematicians in the former Soviet Un ion. Lie gave a common framework and extended different ad-hoc techniques used to f ind solutions of ODEs. The symmetry methods he developed are highly algor i thmic. For instance, using Lie's a lgor i thm one can systematically f ind point symmetries of dif ferential equations and then f ind a change of variables which leads to a reduct ion of order. Many of these methods have been implemented on computer algebra systems (for review, see Hermann [13]). Some applications of symmetries include: Reduct ion of order of ODEs, f inding special solu-t ions of PDEs, f ind ing conservation laws for self-adjoint systems using Noether's Theorem, and l inearizing differential equations. These and other applications are described in [7]. I n Section 1.1 we describe how the symmetries arise as well as Lie's a lgor i thm to f ind them. For ODEs of second or higher order, f ind ing point symmetries leads to solving an overdetermined system of PDEs w i t h only f in i te ly many solutions. Such systems can often be solved completely and computer programs exist to f ind their solutions. I n Section 1.2 we show how to use symmetries to obta in reduct ion of order of ODEs using the method of canonical coordinates or the method of differential invariants. 1 1.1 Symmetries of differential equations Consider a dif ferential equation y' = f(x), x € » (1.1) I f we make a change of variables y = y + e (1.2) where e is any real number then y w i l l satisfy the same differential equation (1.1). Hence i f y = 4>(x) is a solut ion of (1.1) then so is y = <f>(x) + e. Th is is an example of a continuous symmetry of a differential equation. I n general, a symmetry of a dif ferential equat ion is a t ransformat ion tha t maps the solut ion of an equation into another solut ion of the same equation. Symmetries can be discrete (such as reflection or a ro ta t ion by 30°), or continuous (such as in the above example, or a ro ta t ion by an arb i t rary angle) which depends on a continuous parameter e. The fact tha t continuous symmetries are an uncountably inf ini te fami ly of t ransformations makes them much more useful for applications than discrete symmetries. I t also makes cont in-uous symmetries easier to f ind. I n what follows, we w i l l only discuss continuous symmetries. The mater ia l tha t we shall present in this section is wel l-known. See for example [5], [7], [21]. 1.1.1 Continuous transformations groups Since continuous symmetries are continuous transformations, we f irst study the transformat ions themselves. Consider a fami ly of transformations xe(x) : M x I ->• M indexed by a continuous parameter e £ Z , I is an open interval of containing zero 2 which maps x E M into xe G M, where M is a smooth man i fo ld 1 . E x a m p l e 1.1.1 A ro ta t ion of the two-dimensional plane is a fami ly of t ransformat ions tha t can be parametrised by an angle e G (—7r, IT): X\(XI,X2) = xi cos e — X2 sine £2) = £1 sine + X2 cos e A l ternat ive ly this can be represented as: xe(x) = cos e sine — sine cos e x where x — Xl X2 (1.3) Note tha t in the above example, ro ta t ing by angle e and then by angle 6 is the same as ro ta t ing by angle e + 5; ro ta t ion by zero is the ident i ty t ransformat ion. Thus the fami ly of rotat ions above forms an addit ive group w i t h respect to e. Th is motivates the fol lowing def in i t ion: Definit ion 1.1.2 A flow is a funct ion xe(x) :Xx M -> M w i t h the fol lowing properties: 1. X is an open interval tha t contains zero w i t h e G X and M is a smooth mani fo ld w i t h x G M 2. xs+€(x) = xs(xe(x)) whenever 6, e, S + e G X 3. x°(x) = x 4. xe(x) is analyt ic in e when e is near zero, for every x G M. A flow forms a one parameter continuous group of transformations on M, since x€(x) is an addit ive group w i t h respect to e. To show this, we need to show that the inverse of xe(x) exists and is equal to x~e(x). Indeed, x = x°(x) = xe e(x) — xe(x e(x)) = x e(xe(x)). 1 For our purposes, assume M is an open subset of 3?" 1.1.2 Infinitesimals A flow is completely characterised by its infinitesimal. D e f i n i t i o n 1.1.3 The infinitesimal of a flow x£{x) is given by v(x) = ^-x£(x) e=0 Given an inf in i tesimal, one can recover the corresponding t ransformat ion group th rough the fol lowing theorem: T h e o r e m 1.1.4 ( L i e ' s f u n d a m e n t a l T h e o r e m ) A flow is uniquely determined by its in-finitesimal and vice-versa. If x£(x) is a flow with the infinitesimal v(x) = ^x£(x) , then x satisfies ^x£(x) =v{x£{x)). (1.4) Conversely, if v(x) is analytic andx£(x) satisfies (1-4) with the initial condition x°(x) = x then x is a flow and v is its infinitesimal. P r o o f . F i rs t suppose that x is a flow. Then x£+5(x) = xs(x£(x)) and x°(x) = x. Di f ferent iat ing by S and evaluating at 5 = 0 we get (1.4). Conversely, f ix x and suppose x£(x) is a solut ion of (1.4) w i t h x°(x) = x. Let X = x£+s(x) and Y = xe(xs{x)). T h e n X\e=0 = Y | e = 0 = xs(x) and b o t h X and Y satisfy dX dY — = v { X ) , - = v ( Y ) . Since v is analyt ic, the above equations have an analyt ic and unique solut ion near e = 0. Thus X = Y is analyt ic for e in some neighbourhood of 0 by the existence and uniqueness theorem. Th is verifies properties (2-4) o f def ini t ion 1.1.2 • E x a m p l e 1.1.5 Let x£(x) be the rotat ional flow (1.3) as in Example 1.1.1. T h e n v(xi,x2) = — x£(xi,x2)=( ^ de\e=o V x\ J Conversely, fix x and denote xe{x) = y(e) = (^j) Then (1.4) w i t h x°(x) = x can be wr i t t en as a system: V = 0 -1 1 0 \ y, y(0) = x J whose solut ion is precisely the rotat ional flow. I t is convenient to introduce the generator X of xe(x) to be a first-order dif ferential operator XF = ^F(x) • v(x) act ing on any differentiable funct ion F : M —> 5R. Then d d X = Vi- h . --Vn-ox\ oxn where v; is the i - th coordinate of v. 1.1.3 Change of coordinates; canonical coordinates Given a flow xe(x), suppose that we make a change of coordinates, x = x(y) and let y = y(x) be the inverse of x(y). I n the y-coordinates, the flow xe(x) becomes y e ( y ) = y(xe(x{y))). Let v be the inf ini tesimal of x and let w be the inf ini tesimal of y: i w{y) = ^(ye(y)) e=0 « e e=0 Using the chain rule, we obta in : e=0 = v • S/yi{x) where wi and yi denotes the i-th coordinate of w and y respectively. Using the generator X of x, XF = v • \/F, this can be re-wr i t ten as w = Xv where Xv is the vector (Xvi)T. I f we choose a change of coordinates V = (n(x), ...,rn-1(x),s(x))T for which w = (0, . . . , 0 , 1 ) T , then by Theorem 1.1.4 the corresponding flow becomes ye(y) = (yi,2/2, • • • , y n - i , y n + e). The resul t ing coordinates are called canonical coordinates. Thus, canonical coordinates "straighten ou t " the flow. To find canonical coordinates, proceed as follows. F i rs t , find n — 1 funct ional ly independent solutions of a linear f irst-order P D E Xr{x) = 0. (1.5) Using the method of characteristics, this is equivalent to solving a system of n — 1 ODEs of first order . 2 One can then choose r i , . . . , r n _ i to be any n — 1 funct ional ly independent solutions of (1.5) Second, find a solut ion to the pde Xs = l. (1.6) Using the method of characteristics, 5 can be found by quadrature, once r\,..., rn-\ are found. 1.1.4 Invariance under continuous transformation Suppose that we are given a funct ion F : M —> A flow xe tha t leaves the curves F(x) — 0 invariant: F(x) = 0 ^  F{xe(x)) = 0, is called a symmetry of F. 2These solutions correspond to n — 1 constants of motion of the flow x. 6 I f we differentiate F(xe(x)) = 0 w i t h respect to e and evaluate at e = 0, we obta in : d de F{x%x)) e=0 = S/F{x) • v(x) = 0. (1.7) Thus i f xe is a symmetry of F then Conversely, \7F{x) • v(x) = 0. (1.8) Theorem 1.1.6 Suppose that (1.8) holds for some infinitesimal v, for all x € M. Let xe(x) be the flow that corresponds to the infinitesimal v, given by Lie's Fundamental Theorem 1.1.4-Then F(xe(x)) = 0 F(x) — 0 and thus xe(x) is a symmetry of F(x) = 0. Proof. Let y = xs(x). Then by assumption, 0=-F(x£(y)) e=0 Expand ing y and using the group property (2) of def in i t ion 1.1.2 we obta in : d = ^-F{xe+Sx) e=0 « e d = T,F(x°x). e=0 dd Thus F(xs{x)) is constant for al l <5. I n part icular F(x°(x)) = F(x) = 0 =^> F(xs(x)) =0. D I n terms of the symmetry generator, Theorem 1.1.6 states that a first order dif ferential operator X is a generator of a symmetry of F i f f XF = 0. Example 1.1.7 Cont inu ing w i t h Example 1.1.1, let (1.9) F(xi,x2) = xx + x2 - 1. T h e n v = ( ^ 2 ) is a solut ion to \/F • v = 0; the flow xe(x) corresponding to the inf in i tesimal v is jus t the ro ta t ional flow (1.3). I n part icular i f we choose xx = l,x2 = 0 then xe (J^J = (g°^). Hence by above theorem, F (c?*e£) =0, f rom which follows that cos 2 e + s i n 2 e = 1. This characterisation of infinitesimals is of fundamental importance: i t allows us to f ind the infinitesimals of a symmetry of F, and thus a symmetry itself. I n the next section we w i l l extend this result to ODEs. 1.1.5 Point symmetries of ODEs We want to study the invariance of an O D E G(x,y(x),y'(x),...,y^(x))=0 (1.10) under a point t ransformat ion t x = x€(x, y) V ' (1.11) ii = y€(x,y) which forms a flow and which maps solutions of (1.10) into solutions of (1.10). Thus dx dny where y = <p(x) denotes any solut ion of 1.10, and y, = , i > 1 (using convention yo = y ) . T h e expression y{ — c&n be wr i t t en using the total derivative operator, d d d Dx = — + y i — + y 2 - ^ + ... (1.12) ox ay oy\ as Di = D ^ Z ~ 1 , i > 1 w i t h yo = y. (1.13) Dxx The vector (x,y,yi, ...,yn) is called the n- th extension of the point t ransformat ion (1.11). I f (1.11) is a flow then so is its n- th extension, by the fol lowing theorem: Theorem 1.1.8 Suppose that (x,y) given by (1.11) forms a flow. Let yi be given by (LIS) (using convention yo = y). Then (x,y,yi, . . . , y n ) forms a flow on the n + 2 dimensional space spanned by (x, y, y i , y n ) . Proof. We w i l l prove here the case n = 1, the other cases being analogous. To do this we w i l l veri fy properties 2 and 3 of the def in i t ion 1.4. 8 Proper ty 3: We need to show that y\ = y\. Expand x,y i n Taylor series: x = x + e£ + 0 (e 2 ) , y = y + en + 0 (e 2 ) . Thus Vi Dxye Dxx6 yi + eDx( £ = 0 1 + eDxn e=0 Proper ty 2: Let xe = (xe,y£); note that x° = (x,y). We need to show tha t y?S(x.0,y°1)=yl(x6,yi) Since x forms a flow, i t follows that x e + 5 ( x ° ) = xe(x<5). Thus A^+*(x° ) D ^ x * ) y i ( x ° , y i ) . • We can now define: Definition 1.1.9 A point t ransformat ion (1.11) is a point symmetry of an n- th order O D E (1.10) i f i t is a flow and i f its n- th extension x = xe(x, y) y = ye(x,y) m = yY{x,y,yi) = % x Vn = yi€(x,y,yi,...,yn) - D x V n 1 Dxx (1.14) satisfies G(x, y , y n ) — 0 whenever G(x, y, ...,yn) = 0. I f we are only given infinitesimals 9x e=0 dy de e=0 dm de , i — l..n e=0 (1.15) 9 of some flow x = xe(x, y) y = y£(x,y) m = yY{x,y,yi) (1.16) Vn = yne(x,y,yi, ...,yn) then i t is possible to verify whether the contact conditions (1.13) hold, w i thou t comput ing the flow itself. To do that we expand (1.16) in Taylor series w i t h respect to e at zero. We obta in : & = x + e£ + 0 ( e 2 ) y = y + erj + 0 ( e 2 ) yi = yi + er?j + 0(e2),i = 1 , . . . , n Imposing contact condit ions (1.13) we get: _ Dxji _ yi+eDxy+Q(e2) y i Dxx ~ l+eDxZ+0(e2) = (yi + eDxn)(l - eDx£ + (eDx02 - ...) + 0 ( e 2 ) = yi+e{Dxrj-yiDx0 + O{e2). Hence m = DxV - y i -Dis-s im i la r l y by induct ion we obta in the extension formula Vi = DxVi-i ~ yiDx£,,i = 1,2, . . . , n , w i t h rj0 = rj. (1.17) (1.18) Thus i f the contact condit ions (1.13) hold then the infinitesimals of (1.16) satisfy (1.18). The converse is also t rue because of uniqueness of the flow corresponding to a given inf in i tesimal (see Theorem 1.1.4) and since (x,y, ...,yn) is a flow (see Theorem 1.1.8). Hence the inf initesimals of the symmetry, (£, 77,771,%), and thus the symmetry itself, is uniquely determined by £ and 77 through (1.18). Th is leads to a much more useful characterisation of a point symmetry of an O D E : 10 Theorem 1.1.10 (Lie's algorithm) To find point symmetries of an n-th order ODE 1.10 it suffices to find t;(x,y),rj(x,y) such that £GX + nGy + mGyi + ... + 7lnGyn = 0 {mod G=0) (1.19) where by (mod G=0) we mean that the equality holds whenever G(x,y,yi, ...yn) = 0 and rn is given by (1.18). Then (^,rj) is the infinitesimal of a point symmetry of G = 0. Using the symmetry generator X = i Y x + r , d - y + r ] l d y - l J r -equat ion (1.19) can be wr i t t en as XG = 0 (mod G=0) . (1.20) Of ten we w i l l omi t the extensions r]i,i > 0 and wr i te X = ^ 0 - x + ^ y To find a symmetry generator of an O D E in solved fo rm, G = y n - 9(x,y,yi,...,yn-i) = 0, (1.21) we wr i te down the condit ions (1.19) together w i t h (1.18), at the same t ime replacing any occurrence of yn by g. W h e n n > 1, the result ing linear system of PDEs is overdetermined and has only finitely many independent solutions. Consequently, i t is often possible to find them. Furthermore, computer programs (for instance " r i f " [17], [19] and "di f fa lg" [8]) are available for s impl i fy ing overdetermined systems of PDEs using compartabi l i ty condit ions. Even though not a l l second order ODEs have point symmetries, many physical ly relevant ODEs do. 11 1.1.6 L i e -Back lund symmetries of O D E s In the future, we will need transformations more general than point symmetries. To generalise the concept of point symmetries we allow the infinitesimals r\ and £ of a symmetry of ODE (1.21) to depend on x,y,yi,y2, ...yn-\. Then 771 = Dxr\ — y\Dx£ may depend on yn which one can replace by g and similarly for 772, ...,r)n. The extension law (1.18) remains the same, except that we replace all occurences of yn by g. Definition 1.1.11 A generator of a symmetry of an ODE (1.21) (or simply a symmetry) is first-order differential operator dx r>' dy ^l dy\ where £ , 7 7 may depend on y i , . . . , y n _ i : £ = £{x,y,yu• • • ,yn-i), v = •n{x,y,yi,---,yn-\) and rji is given by •m = Dxr]i^i - yiDxi (mod G=0),i > 1 with 770 = 77. (1.22) and which has the property that XG = 0 (mod G=0). If £ or 77 depend only on x, y then the symmetry is called a point symmetry. Otherwise it is called a Lie-Backlund symmetry. The generalisation of point symmetries to Lie-Backlund symmetries was introduced in [4]. 1.1.7 T r i v i a l symmetries Consider a symmetry with infinitesimals £ = 1,77 = yx. The corresponding infinitesimal gener-ator is the total derivative operator. d d d Dx = 7 r + y i 7 r + y21i- + ... (1-23) dx dy dy\ 12 Thus i f G{x,y,yu...,yn) = 0 is any O D E then DXG = ^ G = 0 (mod G=0) . ax So Dx is a t r i v i a l symmetry of any O D E . Geometrically, a t r i v i a l symmetry represents a trans-formation in the direct ion of the solut ion curves of the equation. More generally, Proposition 1.1.12 Let X be a symmetry generator ox oy Then X is a (trivial) symmetry of any ODE G = 0. Proof. One can show by induct ion that the extension formula (1.18) can be rewr i t ten as Vi = D ^ ( n - y i 0 + y i + i C . (1-24) Thus i f rj = y i£ then rn = £yi+\ and hence XG = (,DXG = 0 (mod G=0) . • 1.1.8 Evolutionary form for symmetry generators W h e n a symmetry does not t ransform the independent variable ux", i t is said to be in evolution-ary form. Geometrically, such a symmetry transforms the curves y(x) i n the vert ical d i rect ion only (see figure 1.1.8). The generator of a symmetry in evolut ionary fo rm is X = v — + V l — + ... (1.25) oy oyi where v may depend on yi,y2, ...,yn-i- The extension condi t ion (1.18) then becomes Vi = DxVi-i 13 Figure 1.1: A n evolut ionary symmetry transforms in the vert ical d i rect ion only. and hence XG = ^-G(y + ev) e=0 The expression on the r ight hand side corresponds to a l inearisation of G, i n direct ion v. D e f i n i t i o n 1.1.13 A Directional (or Lie) Derivative of G in the direction v, denoted by VVG, is defined by VVG = —G(y + ev) dG dG dG , = v— + vx—- + v27r- + ... (1.26) 6=0 dy dyi dy2 Thus a symmetry is in evolut ionary fo rm i f and only i f its symmetry generator X is a direct ional derivative Vv i n some direct ion v. Let X = ^YX+^dy-be a symmetry generator of an ODE G = 0 and let v = n- y i f . T h e n X - £ I > x = ( n - y i £ ) — + ... = X>„. (1.27) Since £D x is a ( t r iv ia l ) symmetry and a linear combinat ion of two symmetries is also a symmetry, T>v is also a symmetry generator of G = 0. Thus any symmetry can be " rewr i t ten" as a symmetry in evolut ionary fo rm. 14 I n general, two symmetry generators are equivalent i f f they differ by a t r i v i a l symmetry i f f their evolut ionary forms are equal. I n part icular a symmetry generator in evolut ionary fo rm (1.25) is equivalent to a point symmetry i f f v depends only on x, y, y\ and is at most l inear in y\. Since par t ia l derivatives commute, we have: L e m m a 1.1.14 A Directional derivative and the total derivative operator commute: DVDX — DXT)V. As a consequence, we have: L e m m a 1.1.15 Let V c 9 ^ 9 ox oy be a symmetry generator of G. Then XDX = DXX + {Dxi)Dx. Thus XDX = DXX (mod G=0). Proof. Wr i te X = Vv + (Dx where v = rj — Then DXX = DXVV + Dx{iDx) = VVDX + t;DxDx + (DX£)DX = XDX + (DX^)DX • 1.1.9 L ie A lgeb ra of symmetries I t is possible for an O D E G = 0 to admi t more than one symmetry. However one can show tha t an O D E of second or higher order admits only f in i te ly many point symmetries. I n par t icu lar , any second order O D E admits at most eight point symmetries, and an O D E of order n > 2 admits at most n+4 symmetries (see [7], [21]). Symmetry generators fo rm a vector space since they are solutions of a linear P D E (1.20). More interestingly, a commutator of two symmetry generators is again a symmetry generator: 15 T h e o r e m 1.1.16 If X,Y are two symmetry generators of G then their commutator, Z = [X, Y] = XY - YX, is also a symmetry generator of G. Proof. Let T h e n w i t h z — £z ^ + z ® + z ® + dx ^ dy ~*~ ^l dyi " i z = XiY - Y£x, vf = Xg - Ytf, i > 0 , m = v , is a f irst order differential operator. Furthermore, ZG = XYG - YXG = 0. Thus to show tha t Z is a symmetry generator, i t suffices to show that r j f = Dxri?_x - y%Dxiz\i > 1. Since X, Y are symmetry generators, we have rjf = Xtf - Yri = X{Dxrj{_x - yiDx6.Y) - Y(Dxr,f_x - yiDx£,X) Using L e m m a 1.1.15 and some algebra, one can show tha t this is equal to Dx(Xrj(_x - YVf_x) - yiDx{XiY - Yix) = Dxr,z_x - V i D x i z • Let x€,ye be two flows w i t h generators X, Y respectively, and let Z = [X,Y]. One can show tha t the generator of the flow Ze = X e o y e o xe o if 16 is Z. Note that i f X = Vv, Y = V w are in evolut ionary fo rm then their commutator Z — [DV,'DW\ is also i n evolut ionary fo rm, Z = Vu w i t h u given by u = Vvw — Vwv. Note also that a commutator of two point symmetries is a point symmetry. A Lie Algebra of symmetry generators is a vector space of symmetry generators tha t is closed under the commutat ion. The dimension of a Lie Algebra is its dimension as a vector space. B y Theorem 1.1.16, the set of a l l symmetry generators admi t ted by an O D E forms a L ie Algebra. 1.2 Reduction of order using symmetries One of the applications of symmetries is to help in f inding expl ic i t solutions of ODEs. I f we know a symmetry of an n- th order O D E then one can reduce its order by one. For a first order O D E this is is equivalent to finding a general solut ion of the O D E . We examine two wel l -known methods of reduct ion - canonical coordinates and dif ferential invariants. They are also described in [7] or [21]. 1.2.1 Canonical coordinates Suppose tha t an n- th order O D E wr i t t en in solved fo rm, G = yn -g(x,y,-,yn-i) = 0 (1.28) G{r,s,si,S2,...,S; "n ) = sn - g{r,s,su...,sn-i ) = 0 (1.29) admits a point symmetry r r, s = s + e (1.30) whose extensions are s$ = Sj. Then G{r,s,si,S2,...,S; "n 0 <^> G ( r , s + e , s i , . . . , a n ) = 0 (1.31) 17 and hence g — g(r, s \ , s n - i ) is independent of s. Thus i f we let z(r) = s'(r), then the O D E (1.29) becomes an O D E of order n — 1: Zn-i = g(r,z,...,zn-2)- (1-32) I f n = 1 then (1.32) becomes z(r) = g(r) and hence s(r) = JT g{t)dt + C is a general solut ion to (1.29) for n = 1. I n general, any point symmetry can be transformed into (1.30) by using canonical coordinates (cf. Section 1.1.3). Thus i f an O D E admits a point symmetry then its order can be reduced by one. We i l lustrate by example. E x a m p l e 1.2.1 Consider an O D E G = y2 + y31 + ^ ± y 2 l = 0 (1.33) where y = y(x),yi = y',y2 = y"• I t admits two point symmetries whose generators are: 1 ~ dy x = Pyr— + py— v?rpy— + w i t h [T, X] = X. Firs t consider what happens i f we use T to reduce the order of (1.33). T is already i n canonical fo rm; the reduct ion of order leads to a first-order Abe l O D E after a t ransformat ion z = y\ : Z l + z Z + ^±lz2 = r)_ ( L 3 4 ) X However the result ing first-order Abe l O D E does not have any apparent symmetries. The symmetry X is " lost" because Xz = z2xey contains y. However we can use X to reduce (1.33). To do this, we first find canonical coordinates. A solut ion of Xr(x, y) = eyxrx + eyry = 0, Xs(x, y) = eyxsx + eysy = 1 18 is found to be r = xe y , s = — e y y = — I n s , x = r/s. I n new variables s(r), we get s 4 G = - T 3 - ( r 5 2 + s?) = 0 (rsi — s)dr and the symmetry T becomes T = T 4 + T 4 + ... = - 4 * 0 a 9 + or os or os os\ 0S2 Th is t ime the reduced equation for z = s\, does inher i t a symmetry rzx + z2 = 0 (1.35) whose canonical coordinates (v,w(v)) w i t h = 0,Tzw = 1 are given by v = z, w = — l n r . Transforming (1.35) we get which has a general solut ion / -2 w = v w = — v 1 + K\. Untransforming we get z(r) 1 l n r + Ki sir) = I „ . , . + K2 /xe~y Ki + I n t -v 1 dt e V = l - T l n 7 + ^ which is a general solut ion to (1.33). 19 The above example i l lustrates an impor tan t point . I n general, i f an O D E admits two point symmetries X, Y w i t h [X, Y] = aX then one should start reducing its order using X? Then the resul t ing reduced O D E w i l l " inher i t " the symmetry Y. However i f Y is used f i rst , then the symmetry X w i l l be " l os t " 4 . See [7] or [21] for proof. See also [7] for an a lgor i thm to reduce the order of an O D E admi t t i ng an r-dimensional solvable Lie Algebra, by r. 1.2.2 Differential invariants I n th is section we shall only consider point symmetries. However the results generalise to L ie-Backlund symmetries as well. Given a point symmetry generator X = t:{x,y)^ + n{x,y)-^ + ..., any solut ion w of Xw — 0 is called an invariant of A'. A n invariant w is an invariant of order i, i > 0 i f i t depends expl ic i t ly (where yo = y) bu t does not depend on yj for j > i: w — w(x, y,... ,yi). A n invariant is a differential invariant i f i t is of order greater t han zero. A n invariant of order zero, u = u(x, y) must satisfy a P D E €{x,y)ux +rj(x,y)uy = 0. Hence there is exactly one funct ional ly independent invariant of order zero. Similar ly, there are exact ly i + 1 funct ional ly independent invariants of order at most i. I f two indepedendent invariants are known then one can generate an inf in i te sequence of inde-pendent invariants using the fol lowing theorem. Theorem 1.2.2 Let u,w be any two invariants of X. Then dw = DxW du Dxu 3 I f an O D E admits an n-dimensional Lie Algebra L, then it will be shown in Theorem 4.2.5 that for any Y 6 L, there exists X e L with X ^Y such that either [X, Y] = XX for some possibly complex number A or else [X, Y] = Y. Furthermore, A is real for n = 2. Thus a reduction of two orders is always possible for an O D E of order greater than one that admits a two-dimensional Lie Algebra of symmetries. 4 I n fact, it becomes a Lie-Backlund symmetry. See [7], Chapter 7.3 for details. 20 is also an invariant of X. Thus if w is a differential invariant of order i > I and u is an invariant of order 0, then ^ is a differential invariant of order i + 1. Proof. Since X is a differential operator of first order, the quotient rule holds so that fL\w\ = (Dxu)X(Dxw) - (Dxw)X(Dxu) \Dxu) (Dxuf and using Lemma 1.1.15 and the fact that Xv = Xw = 0 shows that the numerator of the resulting expression is zero. rj Now suppose X is a point symmetry of an n-th order ODE G — 0, and let u = u(x,y),w = w(x,y,y\) be invariants of order zero and one respectively5 and let DxWj-i dwj-i . ivi = —— = —-—,% = 1 , n , with WQ = w. ( l - J ' j Dxu du Then any other invariant of order at most n can be written as a function of u, w, w x , w n - \ . Since XG = 0 (mod G = 0) it follows that G = G(u,w,wi, ...,wn-i) (mod G = 0) for some function G. Thus G = 0 G = 0. But because of choice (1.37) of Wi, G{u,w, wi, ...,wn-i) = 0 (1.38) is an ODE of order n — 1 for w(u). Suppose we could solve (1.38) to obtain a general solution w = <f>(u,Ci,C2,...,Cn-i). Then w{x,y,yi) = <t>{u(x,y),Cx,...,Cn-1) is a first-order ODE for y(x). This first-order ODE still admits X and hence its solution can be found using the method of canonical coordinates. 5See [7] for an algorithm of obtaining w through quadrature, once u is known 21 Example 1.2.3 Consider the scaling symmetry, whose infinitesimals are £ = 0, rj = y. The corresponding symmetry generator is Vdy mdyi 2/2dy2 Note that u = rc is an invariant of X (of order 0). To find an invariant w(x, y, yi) of order 1 we need to find a solution to ywy + y\wyi = 0. Using the method of characteristics we find: dy dy\ y\ — = => ln(yi) — mly) = const. =>• — = const. y yi y Thus yi w — — y is a first-order invariant. An invariant of second order is then given by w =dw ^ Dxw = y2y - y\ = m _ ^ 2 du Dxu y2 y Note that w\ + w2 = y2/y, and in general yi/yj,i,j > 0 are also invariants of X. Example 1.2.4 Consider the harmonic oscillator . G = y2 + y = 0. G is invariant under X = y ^  so we can write G in terms of the invariants yi Dxw y2 u = x, w = — , wi = — — = x y Vxu y — u r of X that were computed in example 1.2.3. We obtain: y2 = (wi + w2)y and hence G = y2 + y = (wi + w2 + l)y = 0. 22 Thus we have to solve the equation dw + w2 + 1 = 0 du for w(u). Th is equation is translat ion-invariant in u and its solut ion is thus found to be u + / dt/(l + t2) = C /w dt/o or w = t a n ( C — u). Subst i tu t ing back u = x: w = yx/y we arrive at the equation yi = t a n ( C - x)y (1.39) which s t i l l admits the scaling symmetry Vdy V l dy{ Using canonical coordinates, r = x, s — Iny w i t h Xr = 0, Xs — 1 the O D E (1.39) becomes s' = t a n ( C - r ) and thus s = J t a n ( C - r)dr = ln(cos(C - r ) ) + K Untransforming we get the solut ion: y = Ci cos(C 2 - x). I n Section 2.1.3 we w i l l present a generalisation of the method of dif ferential invariants tha t also works for non-point symmetries and w i l l allow us to reduce the order of an O D E by two without changing coordinates. 23 1.3 Discussion I n Sections 1.1.1-1.1.4 we defined what is a symmetry of an O D E . A symmetry can be recovered f rom its symmetry generator using Lie's Fundamental Theorem. One can find al l symmetry generators of an O D E by solving a linear system of PDEs (1.20, 1.18). For po in t symmetries of order n > 1 the system (1.20, 1.18) is overdetermined and has only finitely many solutions. The problem of finding solutions to overdetermined systems is to a large extent algebraic, and algori thms are available (see [17], [19], [8]) to reduce such systems. The ou tpu t of such algori thms is an equivalent system, but much simpler, and whose solutions are often t r i v ia l l y found. I n Section 1.1.8 we showed how to represent any symmetry in an equivalent evolut ionary fo rm which leaves independent variables unchanged. The advantage of doing this w i l l become clear i n Chapter 3. A l l symmetries of an O D E form a Lie algebra under commutat ion. I ts st ructure is impor tan t i n applications tha t use many symmetries at once (for instance see [7] where an a lgor i thm for reducing an n- th order O D E admi t t ing a solvable r-dimensional Lie algebra of point symmetries is presented). I n Section 1.2 we have presented two methods of reducing the order of ODEs using symmetries. The first method, or iginal ly developed by Lie (Section 1.2.1), involves "straightening out " the flow using canonical coordinates. I n canonical coordinates the order of the O D E can be direct ly reduced using a quadrature. Whi le this technique works for point symmetries, i t does not generalise nicely to Lie-Backlund symmetries. However for first order ODEs this technique is general enough, since any symmetry of a f irst order O D E is a point symmetry. Another feature of th is method is that i t requires a change of coordinates to work. Al ternat ive ly , for ODEs of second order or higher, one can use the method of di f ferent ial invariants (Section 1.2.2), also or iginal ly developed by Lie. Wh i le this method generalises to 24 Lie-Backlund symmetries, i t requires one to f ind the general solut ion of an auxi l iary O D E (1.38) of order n — 1 before a successful reduct ion can be obtained! I n Section 2.1.3 a generalisation of the method of differential invariants w i l l be presented, by using the concept of conservation laws. 25 Chapter 2 Conservation laws and integrating factors Not a l l ODEs whose exact solut ion can be found admi t point symmetries. For example, i f we start w i t h a Bernoul l i equation, V- = j / " - 1 + f(x) y whose solut ion is known and differentiate i t , we get a second order O D E ^ - ^ = ( n - i ) y n " V + /'(x) which - except for very specific f(x) - does not admi t any point symmetries (see [12]). Nonethe-less, its solut ion can be found exactly since i t is equivalent to V- = yn-l + f(x) + C y which is a solvable (Bernoul l i ) O D E . The above reduct ion is an example of a reduct ion in the original variables. A l l such reductions can be obtained f rom conservation laws of an O D E . I n this chapter we study such reductions. I n Section 2.1.1 we define conservation laws and show how one can find them direct ly f rom their def in i t ion. The task of finding conservation laws can be simpli f ied when an O D E admits symmetries. I n par t icu lar , a symmetry generator applied to a conservation law again results i n a conservation law (see Section 2.1.2 or [21] or [20]). Based on this fact, two related ansatzes w i l l be developed i n Sections 2.1.3 and 2.1.4 that use a symmetry to look for a conservation law. 26 Instead of seeking the conservation laws, one can seek integrating factors which characterise conservation laws. Th is is done by seeking part icular solutions of the determining equations tha t any integrat ing factor must satisfy. I n Section 2.2 we review the developement of these determin ing equations (see also [16] and [1]). As a consequence of this developement, one can obta in a conservation law corresponding to a given integrat ing factor, using only integrat ion and ar i thmet ic operations. I n [3] i t was shown that one can also use an integrat ing factor and a symmetry to generate a conservation law. Th is is discussed in Section 2.3, where we also show that the conservation law thus generated is the same that one gets using the method of the previous paragraph. The relat ionship between symmetries and integrat ing factors w i l l be fur ther explored in Chapter 3, which w i l l discuss self-adjoint systems and Noether's Theorem. 2.1 Conservation laws and using symmetries to find them 2.1.1 Conservation laws A conservation law of an O D E G(x,y,yu...,yn) = 0 (2.1) is an expression P(x,y,yi, . . . , y n _ i ) such that DXP = 0 (mod G=0) . (2.2) Thus, P(x,y,yi,...,yn-i) = C for some constant C , for any solut ion y = ct>(x) of (2 .1) . 1 For example x — l n y = C is a conservation law of a differential equation y' — y = 0. Example 2.1.1 Let y = cb(x) be any solut ion of G = y2 + y = 0 (2.3) and let According to our definition of a conservation law, a constant is a (trivial) conservation law of any O D E . 27 We have d P t X = yi{y2 + y) = yiG. Thus P is constant for each solut ion y = 4>(x) of (2.3). Hence P is a conservation law of (2.3). Note tha t solving the or iginal equation (2.3) of second order is equivalent to solving P = C which is of first order. Thus by finding a conservation law we have reduced the order of the or ig inal O D E . Example 2.1.2 Let P = P(x,y,yi) = const, be a conservation law of a second order O D E in solved fo rm, G = y2 -g(x,y,yi) = 0. Then (2.2) becomes Px + yiPy + gPyi = 0. Any conservation law of an n- th order O D E G = 0 is a solut ion of (2.2) (and vice-versa) which is a linear first order P D E in n variables. Such a P D E has inf in i te ly many solutions (since a funct ion of any solut ion is again a solut ion), bu t only n of them are funct ional ly independent. 2 Hence any funct ion of a conservation law is once again a conservation law, and an n - th order O D E has inf in i te ly many conservation laws, bu t only n of them are funct ional ly independent. By using n funct ional ly independent conservation laws to el iminate derivatives of y one can obta in a general solut ion depending on n arb i t rary constants. Example 2.1.3 Consider G = y% + yi = 0. T h e n (2.2) becomes Px + yiPy - yiPyi = o. 2 Two expressions a(x) and b(x) are said to be functionally independent i f the only solution F to F(a, b) — 0 is the zero solution. For example a = xi — X2, b = x\ — 2x±X2 + x\ are functionally dependent whereas a — x\, b = X2 are not. 28 B y inspect ion, P = Px = y1 + y is one of the solutions to the above P D E . By looking for solutions P = P2 independent of y, we f ind that P2 = y\ex is another solut ion of the same P D E independent of P i . Hence P\ = y\+y = Ci,P2 = y\ex = C2 are two independent conservation laws of G. E l im ina t ing y\ we obta in the general solut ion, y = Ci- C2e~x. 2.1.2 Action of symmetry generators on conservation laws The fundamenta l relat ionship between symmetries and conservation laws is provided by the fol lowing lemma (see [21], [20]) Lemma 2.1.4 if X is a symmetry of G and P is a conservation law of G then X(P) (mod G=0) is a conservation law. P r o o f . Let X = + 7 7 ^ + . . . , and apply Lemma 1.1.15: DXXP = XDXP + (Dxf)DxP = 0 (mod G=0) . • Below we show that the converse is also true in the fol lowing sense: Lemma 2.1.5 Let Po be a (possibly trivial) conservation law and X be a symmetry ofG. Then there exists a conservation law P of G such that XP = Po (mod G~0). P r o o f . To simpl i fy notat ion, we shall assume that G is of second order. Any second order O D E has two funct ional ly independent conservation laws (cf. Section 2.1.1). Let Q and R be any two such independent conservation laws. 29 We first show that at least one of XQ (mod G=0) , XR (mod G=0) is non-zero. Note tha t the system of PDEs ( XP = 0 (mod G=0). (2.4) DxP = 0 has at most one funct ional ly independent solut ion for P when G is of second order. Bu t . R, Q are funct ional ly independent, and bo th satisfy Q',R' = 0 (mod G = 0 ) . Thus at most one of i?, Q can satisfy (2.4). Thus at least one of XQ (mod G=0) , XR (mod G—0) is non-zero. Now by Lemma (2.1.4), XQ (mod G—0),XR (mod G=0) are again conservation laws. Since there are at most two funct ional ly independent conservation laws of G, any conservation law of G must be a funct ion of Q, R. So let X(Q) = f{Q,R), X(R) = g(Q,R), P 0 = h(Q,R) for some functions / , g, h, w i t h at least one of / , g non-zero. Now let P(Q,R) be any non- t r iv ia l solut ion of a P D E f(Q,R)^P(Q,R)+g(Q,R)—P(Q,R) = h(Q,R). T h e n by chain rule we have X(P(Q,R))= X(Q)^P(Q,R)+X(R)^P(Q,R) = f(Q,R)^P(Q,R)+g(Q,R)^P(Q,R) = P0. Hence P(Q, R) is the desired conservation law. • I n the fol lowing section, this lemma w i l l be used to generalise the method of dif ferential invari-ants to L ie-Backlund symmetries. 2.1.3 Using symmetries to find conservation laws Lemma 2.1.5 provides an ansatz for looking for a conservation law P of G = 0: we seek solutions P o f a system 30 X(P) = 0 (mod G=0). (2.5) DXP = 0 This ansatz was f irst used by [12] and [11]. We i l lustrate w i t h examples. Example 2.1.6 For a second-order O D E G = y2 -g(x,y,yi) = 0 the system (2.5), when wr i t t en out , becomes Pxt + Pyr) + PyiVi = 0 ^ Px + VlPy + gPyi = 0 where P = P(x,y,yi) and rji = Dxn - yiDx£ (mod G=0). To solve system (2.5), f i rst f ind n — 1 differential invariants u, w,... of X (which can be obtained f rom any two independent invariants - see Theorem 1.2.2) and wr i te P = P(u,w,...). T h e n subst i tute into DXP = 0 (mod G=0) and solve the result ing system for P(u,w,...). Example 2.1.7 Let 's apply this method to the harmonic oscillator G = y2 + y = 0 which admits symmetries dx"1 dy' We first compute a conservation law P w i t h TP = 0. The invariants of T are u = y, w = y\; so P = P(u,w). Plugging this into P' = 0 (mod G = 0 ) we obta in Puw — Pwu = 0. Using method of characteristics: ^ + = 0 , udu + wdw — 0, u2/2 + w2/2 = const., we f ind P = y2 + v\ = C i is a conservation law invariant under T . I t is now possible to f ind the solut ion of P — C\ = 0 since i t is invariant under T. Al ternately, we w i l l compute a conservation law Q w i t h XQ = 0. 31 As before, u = x,w = y\jy are differential invariants of X so Q = Q(u,w). Tak ing to ta l derivative (mod G=0) we obta in u'Qu + w'Qw (mod G=0) = Qu + {-y2 - yj)/y2Qw = Qu - (w2 + 1)QW = 0. Using the method of characteristics we get du + = 0 o r ^ + i o 2 + l = 0 (which is exactly the same equation as obtained using differential invariants). Th is can be integrated and we obta in a solut ion for Q: Q = x + a r c t a n ( y i / y ) = Solving the system P = Ci, Q = C2 we obta in a fu l l solut ion of G: x + arctan( V 1 ~ V ) = C2. y Solving for y we obta in y = ±\/Ci cos(x — C2). Since this method works for any symmetry (point or Lie-Backlund ) , we can rewr i te the sym-met ry i n i ts evolut ionary fo rm, X — T>v. Then the first invariant is jus t u = x and another invariant is any solut ion to the P D E Vvw = 0 (mod G=0) . (2.6) A l l addi t ional invariants can be obtained through Theorem 1.2.2 which, for a symmetry i n evolut ionary fo rm, becomes W{ = DxW{^i (since u = x is an invar iant) . Example 2.1.8 For a general second order O D E G = y2-g(x,y,yi) = 0, a d m i t t i n g a symmetry d X = Vv = v{x,y,yi)— + oy 32 (2.6) becomes wyv + wyiv = 0 (mod G=0) (2.7) whose solut ion can be obtained using the method of characteristics: dy = dyi v v' where v = v{x, y, yi), v' = vx + yxvy + gvyi. Example 2.1.9 To i l lustrate that this method works for Lie-Backlund symmetries, consider an O D E G = y2- g = 0,g = — . (2.8) y , y 2x + 2y + 2yi-yxx 1 °> I t admits a Lie-Backlund symmetry X = y\-§^- Equat ion 2.7 becomes wyy\ + 2wyigyi = 0. Using the method of characteristics we f ind a solut ion: w = {yi - yix/2 + y + 2)e~^ and thus P = P{x,w) is a conservation law for some P. I n part icular, Dxw = ^ e-yil2{yi - y2(x + y + V l - ^yxx)) = 0 (mod G=0) and thus w i tself is a conservation law. So the or iginal O D E is equivalent to (yi - Vix/2 + y + 2)e^ = const. (2.9) This conservation law is s t i l l invariant under X. Wh i le in theory the method of canonical coordinates can be used to integrate (2.9) fur ther, i n practice i t is necessary to isolate y\ before f ind ing canonical coordinates. We w i l l overcome this di f f icul ty by f ind ing another conservation law of (2.8) directly, using X and the method described in the next section. 33 2.1.4 Using one symmetry to find two conservation laws I n the previous section we considered an O D E G = 0 admi t t ing a symmetry X and showed how i t may be possible to f ind a conservation law P of G by solving the system XP = 0 (mod G=0) . (2.10) DxP = 0 I n this section we w i l l show that i f i t is possible to f ind such a P then i t may be possible to find another non- t r iv ia l conservation law Q for which XQ = 1 (mod G=0) . (2.11) DxQ = 0 B y L e m m a 2.1.5, such a Q always exists, since 1 is a ( t r iv ia l ) conservation law. Furthermore, Q must be funct ional ly independent of P. I f G is of second order and P can be found, then Q can always be found using only two quadratures, w i thou t having to solve any addi t ional ODEs, as we now i l lustrate using examples. Example 2.1.10 Harmonic oscillator, G = y2 + y = 0, admits a symmetry X' = whose (differential) invariants are u = y,w = y\. I n (2.3) we found that P(u, w) — u2 + w2 satisfies (2.10). We now seek Q of the form Q = F{u,w) + s(x,y,yi) which satisfies (2.11). Imposing XQ = 1 XF + Xs = Xs = 1 we obta in sx — 1, whose par t icu lar solut ion is s = x. I n general, once u, w are known, 5 can always be obtained through quadrature using the method of characteristics. 34 Impos ing Q' = 0 (mod G=0) leads to Fuw-Fwu +1 = 0. (2.12) We already found a solut ion P = u2+w2 to the associated homogeneous problem Puw—Pwu — 0; thus the solut ion to (2.12) is obtained by solving Fuw = — 1 w i t h w = ±VP — u2 f r om which we find a solut ion to (2.12): F = a r c c o s ( u P " 1 / 2 ) . Thus Q = a r c c o s ( u P - 1 / 2 ) + x satisfies (2.11). Solving for u we obta in u = ±VP cos(Q — x) which leads to a general solut ion: y = C i cos(C 2 - a;). Example 2.1.11 Consider an O D E f rom example 1.2.1: G = y2 + yl + (l-x-1)yl=0. (2.13) I t has two symmetries: X = xey^- + ey^-,T=^-. ox oy oy We w i l l use the symmetry X to find a general solut ion of (2.13). The invariants of X of zero and first orders are found to be xyi - 1 u = xe y, w = . yi Thus a conservation law P w i t h XP = 0 must be a funct ion of u,w: P = P(u, w). I n add i t ion i t must satisfy DXP = (Dxu)Pu + {Dxw)Pw = 0 (mod G=0) (2.14) 35 where Dxu = e~y(l - xyi), Dxw= 1 X y i (mod G=0) and thus ^ = 1 (mod G=0) . Hence (2.14) is equivalent to: u P u -i-Pu, = 0. Thus P = ue~w (2.15) is a conservation law of (2.13). We now seek a conservation law Q w i t h XQ = 1. Once again, assume Q has fo rm Q = F(u,w) + s where F, s are to be found. Then XQ = Xs = 1. By inspection, satisfies X s = 1. I t remains to satisfy 0 = DXQ = (Dxu)Fu(u,w) + (Dxw)Fw(u,w) + Dxs (mod G = 0 ) . where and hence DrS 1 I t follows that F must satisfy uFu + P^, = . w The solut ion to the homogeneous part of this equation is given by P found in (2.15). Thus to find F one must solve u Fw = - w i t h ue~w = P w 36 or ewPw I ts solut ion is given by F = P which leads to a conservation law Q = P I t dt - e~y I f we now replace u = xe y,w = l n ( u / P ) = lna; — y — I n P we get a general solut ion In x—y—ln P „t -dt - e~y. t 2.2 Integrating factors and Euler Operator A conservation law can be characterised by an associated integrating factor. There is a close relat ionship between integrat ing factors and Euler-Lagrange equations f rom the var iat ional calculus. This relat ionship leads to determining equations for integrat ing factors. I n this section we define integrat ing factors and derive the determining equations for them. Th is mater ia l i n s tandard, see for instance Olver [16]. Following [1], we w i l l also show how to compute the conservation law corresponding to a given integrat ing factor using only integrat ion and ar i thmet ic operations. 2.2.1 Integrating factors Definition 2.2.1 The expression G(x, y , y n ) is exact or a divergence i f i t is a to ta l derivative of some expresson P(x, y, y i , y n - i ) ' -For example y\ and yyi are exact since they are to ta l derivatives of y and y 2 / 2 respectively. O n the other hand y or yy2 are not exact. G = DX(P) 37 Definition 2.2.2 w is an integrating factor of G i f wG is exact. B y def in i t ion of exactness, one can then f ind an expression P such that wG = DX(P). (2.16) Note tha t i f G is exact then i t has an integrat ing factor 1. The above condi t ion is equivalent to (2.2): DXP = 0 (mod G=0), when the order of P is less than the order of G. Thus P is a conservation law of G i f f there exists an integrat ing factor w w i t h DXP = wG. Example 2.2.3 Consider an O D E G — y\ — y = 0, let P = ln(y) - x and let w = 1/y. T h e n DXP = y\/y — 1 = wG. Hence P = C is a conservation law of G = 0 corresponding to the integrat ing factor w. Solving P = C for y we obta in the general solut ion to the O D E , y = Cex. 2.2.2 Adjoint Directional Derivative and Euler Operator Recall f rom Section 1.1.8 tha t the direct ional derivative VVG is defined by VVG = —G(y + ev) dG , dG „ dG = v—+v'— + „ " _ _ + ... e = 0 dy dyi dy2 Definition 2.2.4 A n adjoint of a directional derivative is an operator V*w such that wVvG — vT>*wG is a to ta l derivative, for any w,v, G. A n expl ic i t formula for V*w is obtained using integrat ion by parts: Theorem 2.2.5 Let V*WG = wGy - Dx(wGyi) + D2x(wGy2) - ... (2.17) Then V*w is an adjoint of a directional derivative and satisfies the identity: wVvG = vV*wG + DxS(w,v,G) (2.18) 38 where S is given by S(w,v,G) = wGyiv +wGV2v' — (wGy2)'v + WGy3V" - (wGy3 ) V + (wGyJ'v + WGy4V'" ~ (wGyJv" + (wGyJ'v' - (wGyJ"v + ... = El>lEt-J0(-l)W^\wGyi)DtJ'1)(v) where (*)' = Dx* and Gyi = Proof. The theorem follows f rom the recursive appl icat ion of the Leibni tz rule ba! = (ab)' — ab': wVvG = wGyV + wGyiv' + wGy2v" + wGy3v'" + ... = WGyV +(wGyiv)' - (wGyi)'v + (wGy2V' - (wGy2)'v)' + [wGy2)"v + (wGy3V" - {WGyJv' + {wGy3)"v)' - (wGy3)'"v +... The r ight border of the above tr iangle gives vV^G, the rest is S'(w,v, G). • O f special interest w i l l be the case w = 1: Definition 2.2.6 The Euler operator E is B =* = £ - B - £ + ( 2 - 2 0 ) The Euler operator satisfies Euler identity: VVG = vEG + DxS(l,v,G). (2.21) Example 2.2.7 For a general t h i r d order O D E we have V*WG = wGy — (wGyi)' + (wGy2)" — (wGy3)'" and S(w,v, G) = wGyiv + wGy2v' - (wGy2)'v + wGy3v" — (wGy3)'v' + (wGy3)"v. 39 Figure 2.1: A maximiser and curves nearby. Let P = 2/2 + V, G = (y3 + y i ) / y 2 , w = y 2 , v = y. Then v is a symmetry of G = 0, P' = wG and so w is an integrat ing factor w i t h P being the corresponding conservation law. To compute V*WG and S(w, v, G) we first compute wGy = 0, wGyi = 1, w G K = — G, wGy3 = 1, so £>;G = 0 - 0 + ( - G ) " - 0 = 0 (mod G=0) and S(w, v, G) = y-Gyi+G'y+y2 = P (mod G=0) . 2.2.3 Euler-Lagrange Equations We have introduced the Euler operator above to treat the fol lowing basic problem of the calculus of variations: F i n d a curve y = (f>(x) that minimizes a funct ional A [y ] : rb A[y]= L(x,y,yu...,yn)dx (2.22) Ja w i t h y i = y',...,yn = y^• Here L is called a Lagrangian, and we minimize over al l possible funct ions y = (/>(a;) 6 5 where S is the set of al l smoo th 3 functions w i t h prescribed values at fixed endpoints: S = [tb{x) € G°° | <f>(a) = A,<f>{b) = 5,</> ( i )(a) = Ai,</>®(b) = Bui = l . .n - l } . (2.23) Example 2.2.8 I f we choose L(x,y,yi) — \Jl + y\ then (2.22) is the length of any curve y = <j>(x) f rom a to b and hence the m i n i m u m of A is the shortest smooth pa th between two points (a, A) and (b,B). 3 To simplify presentation, we define smooth to mean C°°, although only Cn is required. 40 The idea to find a minimiser y = cb(x) is as follows. I f y = <p(x) is a minimiser, A[y] must be less then Ajany curve nearby] (see Figure 2.2.3). I n part icular , i f we take any variation, i.e. any smooth funct ion h w i t h y + h €E S then we expect tha t A[y] < A[y + eh] for any sufficiently smal l e. So i f we let / ( e ) = k[y + eh] then / has a m i n i m u m at e = 0 and hence / ' ( 0 ) = 0. Thus i f y = cb(x) is a minimizer o f (2.22) then f'(0) = d A [ y + e h \ e = Q = VhjbLdx= fbVhLdx = Q where L = L(x,y, . . . , y n ) . Th is must hold for a l l h. Using (2.21) we get: fbVhLdx= jbhE{L)dx+ f S'{1, h, L)dx J a J a J a Note tha t <S(1, h, L) is l inear i n h and its to ta l derivatives. Also note tha t hi(a) — 0 = hi(b), 0 < i<n-l. Thus / S'(l, h, L)dx = S(l, h, L)x=b - S(l, h, L)x=a = 0 J a So a necessary condi t ion for y = <f>(x) to be a minimizer of (2.22) is tha t j j hE{L)dx = 0 for an arb i t ra ry var iat ion h. Th is forces the integrand to be zero, and hence we obta in Theorem 2.2.9 If y = cb(x) is a minimizer of (2.22) over a set S given by (2.23) then E(L) = 0. (2.24) Note tha t the converse does not hold. Definition 2.2.10 The O D E (2.24) is called Euler-Lagrange equation. Example 2.2.11 For a general L = L(x,y,y\), EL = Ly — DxLyi. Tak ing L = A / 1 + y i 2 as i n example (2.2.8), we f ind EL = (1 + y 2 ) ~ 2 y 2 = 0 => yi — 0. Thus the shortest (smooth) pa th between two points, i f i t exists, must be a straight l ine. 2.2.4 Kernel of Euler Operator We now show that the Euler operator annihilates to ta l derivatives. Th is w i l l lead to the determin ing equations for integrat ing factors. The proof presented here is similar to tha t in [16] and [1]. 41 Suppose tha t L is exact, i.e. L = P' for some P. Then the funct ional (2.22), A[y] = fb Ldx = P{y)\x=b - P(y)\x=a Ja is independent of the pa th , depending only on the value of y and its derivatives at the endpoints. So A[y] is constant for al l y G S (where S is given by (2.23)). Thus any y G S is a minimiser. Thus by Theorem 2.2.9, EL = 0 for y G S. B u t since the restr ict ion on the endpoints was arb i t rary , EL = 0 for all y. Conversely, suppose that EL = 0. Then by (2.21) we have VhL = ±S{l,h,L) and so Thus T>hL\y=y+\h — —S(l,h,L)\y=y+Xh ^L{(y + Xh) + €h)\e=0 = ~L{y + Xh)= ^-S{l,h,L)\y=y+Xh. Using the Fundamental Theorem of Calculus we obta in: f1 d d f1 L(y + h) - L(y) = —S{l,h,L)\y=y+XhdX = — J S{l,h,L)\y=y+xhd\. We now choose h = —y + h(x) for some h(x) such that L(h) is f ini te then we obta in L(y) = -ihl! s{1> ~y+k L)\v=vn-^dX+tSL{h{x))dx Thus we obta in the fol lowing three theorems. Theorem 2.2.12 (Kernel of Euler Operator) Let H{x, y, yi,...) be any differential expres-sion. Then H is exact if and only if EH = 0 for all y G C°°. Theorem 2.2.13 (Determining equations for Integrating Factors) w is an integrating factor of G if and only if E(wG) = 0 for all y G C°°. (2.25) 42 Theorem 2.2.14 If H is exact, H = DXP, then P is given by P = - £ S(l,-y + h,H)\y=y{1_x)+hdX + J H(h(x))dx (2.26) where h(x) is any function such that the above expression is finite. Theorem 2.2.14 provides a way of f inding conservation laws of an O D E f rom integrat ing factors: i f w is an integrat ing factor of G then the corresponding conservation law P w i t h P' = wG can be found by apply ing formula (2.26) to H = wG. Theorems 2.2.12, 2.2.13 appear in [16] and [3]. Theorem 2.2.14 appears in [3]. The di f f icul t step is to f ind the integrat ing factor itself: one needs to seek par t icu lar solutions of the P D E E(wG) — 0. I f one assumes that G = yn — g{x,y,yi, . . . , y n _ i ) then the integrat ing factor must be of the form w = w(x, y, yi,yn-i). Consequently, since E(wG) — 0 must hold for a l l y, yi,y2, i t splits into a system of PDEs by equating the coefficients of y „ , y n + i> •••) yin to zero. A more natura l sp l i t t ing suggested in [16] and [1] is to f irst solve E(wG) = 0 (mod G=0). Using the product rule for derivatives one obtains: E(wG) = V*WG + V*Gw. (2.27) B u t VQW = Gwy — (Gwyi)' + (Gwy2)" — ... = 0 (mod G=0). Thus i f w is an integrat ing factor of G then one must have V*WG = Q ( m o d G = 0 ) . (2.28) I f w satisfies above then i t is called an adjoint symmetry o f G = 0. Th is leads to the fo l lowing: Proposition 2.2.15 An integrating factor is necessarily an adjoint symmetry. Conversly, an adjoint symmetry w is an integrating factor of an ODE G = 0 if it satisfies E(wG) = 0 for all functions y. 43 Since an O D E admits inf in i te ly many integrat ing factors, the system E(wG) = 0 is not overde-termined. So in general, i n order to find a part icular integrat ing factor, one needs to assume some ext ra condi t ion on w, for example, its polynomial dependence on one of x, y, y \ , y n - i -The s i tuat ion is similar to that of symmetry methods: to find a Lie-Backlund symmetry, one needs to assume some extra condi t ion on its form. For symmetries, a "na tu ra l " (geometric) condi t ion is to seek point symmetries. This leads to an overdetermined system for ODEs of order two or higher. I n general, no geometric interpretat ion of integrat ing factors is known, and hence there is no "natura l " extra condi t ion that can be imposed. However an impor tan t special case occurs when an O D E is self-adjoint as w i l l be discussed in Chapter 3. 2.3 Relationship between integrating factors, conservation laws, and symmetries Given an integrat ing factor, a conservation law can be found using a quadrature. A different way of finding conservation laws without quadrature is possible when a symmetry is also known. The fol lowing theorem was first proved in [3]: Theorem 2.3.1 Given an ODE G = 0, let w be an adjoint symmetry of G and let Vv be a symmetry of G. Then S(w,v,G) (mod G=0) is a conservation law. Proof. Since v is a symmetry and w is an adjoint symmetry of G, one has VVG = 0 (mod G= 0) and V*WG = 0 ( m o d G = 0). Hence by (2.18), DxS(w,v,G) = 0 ( m o d G = 0) and. thus S(w,v, G) (mod G=0) is a conservation law. r j As was shown in Theorem 2.1.4, given a conservation law P and a symmetry X of G = 0, Q = XP is also a conservation law of G = 0. We now show how this is related to the preceeding theorem: Theorem 2.3.2 Given an ODE G = 0, let w be an integrating factor of G and let P be the conservation law corresponding to w. Then VVP = S(w,v,G) (mod G=0). (2.29) 44 where S(w,v, G) is given by (2.19) for any v. Furthermore, ifVv is a symmetry of G = 0, then the above expression is a CL. To prove this theorem we w i l l need the fol lowing lemma. L e m m a 2.3.3 If G is an expression that depends at most on x, y,yi,yn and if DXG = 0 then G is constant. Proof. Note tha t DXG = GX + yiGy + yiGyt + ... + yn+iGyn = 0 and since G does not depend on y n + i > one must have GVn = 0. Proceeding by induct ion we have GVn = 0 =• Gyn_, = 0 => ... Gyi = 0 => Gy = 0 Gx = 0. n The proof of theorem 2.3.2 now consists of three steps. Step 1 We first show tha t DXVVP = DxS{w,v, G) (mod G=0). (2.30) Since Dx and Vv commute (see Lemma 1.1.14) and since DXP = wG we have DXVVP = VVDXP = Vv{wG). We now apply the product rule to the differential operator Vv; we get Vv(wG) = wT>vG + GVvw. A p p l y i n g equation (2.18) to b o t h terms on the r ight hand side, we get: wVvG + GVvw = DxS(w, v, G) + DXS(G, v, w) + vV*wG + vV*Gw. Combin ing the last two terms by using (2.27) we obtain DxS(w, v, G) + DXS{G, v, w) + vV*wG + vV*Gw = DxS(w, v, G) + DXS{G, v, w) + vE(wG). 45 Since w is an integrat ing factor of G, the te rm vE(wG) vanishes (see Theorem 2.2.12). P u t t i n g i t al l together, we get DXVVP = DxS{w, TJ, G) + DXS{G, TJ, w), (2.31) th is being t rue for al l x, y, y\,... Now S(G,v,w) is linear in G and its to ta l derivatives; i t can be w r i t t e n as S(G,v,w) = Gc0 + (G) ' c i + ( G ) " c 2 + ... where Cj are some expressions involving v,w bu t independent of G. Thus S(G,v,w) = 0 (mod G=0). Th is proves (2.30). 2 Let L{v) = Z ^ P - S(>, 7j , G) - 5 ( G , TJ, w). We w i l l show that L{v) = 0 for al l TJ. Equat ion (2.29) then follows since, as was shown in Step 1, S(G,v,w) = 0 (mod G=0). Since L is linear in TJ and its par t ia l derivatives, one can wr i te L(v) = va0 + vxa\ + T j y a 2 + vyia3 + ... + vxxau + vxyai2 + vxyiai3 + ... (2.32) where ai — ai(x,y,yi, . . . , j / 2 n ) a n d the equality holds for arb i t rary TJ = v(x, y, yi, yn) and arb i t ra ry x,y,yx,.... Because of (2.31), Dx(L(v)) = 0, for any TJ. Thus, by Lemma 2.3.3, L(v) is independent of x, y, y i , f o r any v = v(x, y, yi,ym). Th is forces al l of the coefficients to be zero, as the fol lowing argument demonstrates. Note that 1,(1) = ao and thus ao is independent of x. Similar ly, L(x) = CLQX + a\ is independent of x and hence a\ = —a^x + k for some constant k, independent of x. T h e n L(x2) = aox2 + 2a\x + 2a\\ is independent of x and thus a n is at most quadrat ic in x. Similar ly, one can show that al l the coefficients aj are at most po lynomia l i n x,y,yi,.... I f one now takes v = Inx then L ( l n x ) = a o l n x + rat ional expression in x. B u t the result ing expression must also be independent of x. Thus ao = 0. Using similar argument, one can show tha t al l coefficients in (2.32) are zero. Thus -L(TJ) = 0. 46 Step 3 The fact tha t VVP is a conservation law is a direct consequence of Lemma 2.1.4. A l ter -nately, S(w, v, G) is a conservation law by Theorem 2.3.1. • A n integrat ing factor by itself always leads to a conservation law by using a quadrature (see Theorem 2.2.14). The advantage of the above theorem is that given an integrat ing factor and a symmetry, a conservation law can be obtained without quadrature. The disadvantage is tha t of ten the resul t ing conservation law may be t r i v ia l . For example, i t can be shown (see [1]) tha t a first order O D E G = y\ — g(x, y) = 0 that admits a symmetry Vv also admits an integrat ing factor w = ^ . I n this case S(w, v, G) = 1 is a t r i v ia l conservation law. E x a m p l e 2.3.4 Consider an O D E G = y2 - yj - b{x) = 0. (2.33) A search for integrat ing factors of the form w = w(x, y) reveals tha t w = e~yc(x) is an integrat ing factor of (2.33) i f f c(x) satisfies c"(x) + b{x)c(x) = 0. (2.34) As well , the O D E (2.33) is independent of the dependent variable and thus admits a symmetry Vv w i t h v = l. Thus S(w,v,G) = (-yic(x) - c'{x))e-y = C is a conservation law of G = 0. Let c\(x),C2{x) be two independent solutions of (2.34). T h e n for any solut ion y = (f>(x) of (2.33), there exist constants C\, C2 such tha t (-y'Cl(x) - c[(x))e-y = Cu (-y'c2(x) - c ' ^ ) ) ^ = C2 El im ina t ing y' f r om these equations and using the fact that cic'2 — c2c[ is constant, one obtains y = - l n ( c i i \ i + c2K2) 47 is a solut ion of (2.33) for arb i t rary K\^K2- Thus y = - l n (c (a ; ) ) is a solut ion of (2.33) iff c satisfies (2.34). Theorem 2.3.2 can also be used to generate an ansatz: Given an integrat ing factor w (or a sym-metry v), one can seek a symmetry v (or an integrat ing factor w) for which either S(w, v,G) = 0 or S(w, v, G) = 1. We w i l l i l lustrate this in Chapter 4, in connection w i t h classification of ODEs. Theorem 2.3.1 was first discovered by Anco & B luman in [3]. The connection made in Theorem 2.3.2 is new. 2.4 Discussion I n this chapter we have discussed how to f ind conservation laws of ODEs. We have discussed a direct way of looking for conservation laws by seeking part icular solutions of the P D E (2.2), as well as an indirect approach, by looking for integrat ing factors. These can be found by seeking par t icu lar solutions of the P D E (2.25). Once an integrat ing factor is found, a corresponding conservation law can be found through a quadrature, using the formula (2.26), due to Anco & B l u m a n [1]. We have also discussed how to generate a conservation law f rom a known conservation law and a known symmetry (Lemma 2.1.4). Based on i t , we developed a method of using symmetries to generate an ansatz for seeking conservation laws (see Section 2.1.3). Such a method is equivalent to the method of differential invariants for point symmetries, but works for general L ie-Backlund symmetries as well . Furthermore, a related method discussed in Section 2.1.4 can be used to f ind a second conservation law i f the method of Section 2.1.3 succeeds. The method of Section 2.1.3 was first presented by Gonzales [12] and by Cheb-Terrab et al . [11]. Simi lar methods can be developed to use a symmetry to generate an ansatz to look for inte-grat ing factors, based on Theorem 2.3.2. This w i l l be discussed Section 4.2 for the purposes of 48 classification of solvable ODEs. Theorem 2.3.2 is also of interest in itself, and will be used in the next chapter, in connection with Noether's Theorem. 49 Chapter 3 Self-adjoint systems 3.1 Introduction I n this chapter we w i l l s tudy a special class of ODEs that are called self-adjoint ODEs, for which an integrat ing factor is also a symmetry. Such ODEs have a Lagrangian formula t ion and a celebrated result of Noether characterises precisely those symmetries that lead to conservation laws. I n Section 3.2 we show that Euler-Lagrange equations are self-adjoint. We then state and prove Noether's theorem. Noether's theorem is used to f ind conservation laws for variational symmetries of Euler-Lagrange equations. Furthermore, we show tha t the result ing conservation law admits the var iat ional symmetry that generated i t , and hence every var iat ional symmetry provides a two-fold reduct ion of order [16], [21]. There are two versions of Noether's theorem: the or ig inal version, due to Noether [15] and a more modern presentation due to Bessel-Hagen [6]. I n this chapter we w i l l only cover the Bessel-Hagen version. I n i ts fu l l generality, Noether's theorem relies on a quadrature to find a conservation law ( though often the quadrature is t r i v ia l to per form). However when a self-adjoint O D E admits two or more var iat ional symmetries, we show in Section 3.2.2 tha t the conservation law corresponding to their commutator can be obtained w i thou t any quadrature [3]. I n Section 3.3 we also discuss how a scaling symmetry can be used to obta in conservation laws corresponding to known variat ional symmetries, w i thout using quadrature. 50 3.2 Self-adjoint systems and Noether's theorem 3.2.1 Characterisation of a self-adjoint system I n this section we w i l l use the notat ion f rom previous chapters. Namely, we w i l l make use of the Euler operator E defined by (2.20), the direct ional derivative Vv defined by (1.26) and the adjoint of a direct ional derivative, V*, defined by (2.17). Definition 3.2.1 A n O D E G = 0 is self-adjoint i f f V%G = VVG for any v. The mot iva t ion for this def in i t ion is as follows: Proposition 3.2.2 Let G = 0 be a self-adjoint ODE and let v be its integrating factor. Then T>v is a symmetry generator of G. Proof. By proposi t ion 2.2.15, I f v is an integrat ing factor of an O D E G -0 (mod G = 0 ) . Since G is self-adjoint, V*VG = VVG. Hence VVG = 0 (mod G is a symmetry of G. There indeed exist non- t r iv ia l self-adjoint ODEs as the fol lowing theorem shows: Theorem 3.2.3 Let L — L(x,y,y\, ...,yn),v = v(x,y,y\, ...ym) be any expression. Then V*VEL = VVEL for any v. Olver [16] proves this theorem using Var iat ional Complex. Here we provide a more elementary proof. I t consists of a sequence of lemmas. Lemma 3.2.4 Let f — f(x) be a function independent of y and its derivatives. Then E and T>j commute: EVfL = VfEL = 0 then V*G = = 0 ) and thus Vv • 51 Proof. Because / is independent of y and its derivatives, T>f and -^p commute. B u t to ta l and direct ional derivatives also commute (see Lemma (1.1.14)). Thus for any expression L. • Lemma 3.2.5 EVVL = V*EL + V*ELv. Proof. B y (2.21) and Theorem 2.2.12 one has EVVL = E(vEL + divergence) = E(vEL). A p p l y i n g (2.27) to the expression on the r ight proves the lemma. • Lemma 3.2.6 Let f = f(x) be independent of y and its derivatives. Then VJEL = VjEL. Proof. Using Lemma 3.2.5 we get V)EL = EVfL-V%Lf. Since f(x) is independent of y,y%,... i t follows f rom (2.17) that V*ELf = 0 and thus the second te rm on r ight hand side vanishes. Using Lemma 3.2.4 on the first t e rm completes the proof. • We now re tu rn to the proof of Theorem 3.2.3. F i rs t take v = f(x); then by Lemma 3.2.6 we have: vVfEL = vV)EL. Now apply (2.18) to b o t h sides to obta in fV*EL = fVvEL + R where R = —S'(f,v,EL) — S'(v, f,EL). Since S is linear in the first two arguments and their to ta l derivatives, one can wr i te R = vao + v'ai + v" a2 + ... + v^ar 52 where aj are independent of v. Now i f we take any v = v(x) independent of y and its derivatives, then R = 0 by L e m m a 3.2.6. Th is , and the fact tha t the are independent of v implies tha t a\ — 0 for a l l % (for example choose v = l=>R = ao = 0, then choose v = x and so on). Hence R = 0 for al l v. • One can also show that a self-adjoint equation is necessarily an Euler- Lagrange equat ion; see Olver [16] Theorem 5.68. A n explici t formula for the corresponding lagrangian L is also given there. 3.2.2 Variational symmetries and Noether's theorem Since the Euler-Lagrange equation is self-adjoint, an integrat ing factor of an Euler-Lagrange equation is also a symmetry of i t (see Theorem 3.2.2). However not every symmetry is an integrat ing factor. Noether's Theorem provides a characterisation of those symmetries of an Euler-Lagrange equation that are integrat ing factors. Definition 3.2.7 A symmetry Vv of an Euler-Lagrange equation EL = 0 is variational i f there exists an expression A such that VVL = DXA. Noether's Theorem provides just i f icat ion for call ing v a symmetry: Theorem 3.2.8 (Noether's Theorem, Part 1) Let G — EL = 0 be an Euler-Lagrange equation. Then the following are equivalent: (a) Vv is a variational symmetry of G (b) E(vG) = 0 (c) v is an integrating factor of G. 53 Proof. We first show tha t (a) => (6). By def ini t ion of a var iat ional symmetry, there exists some A for which T>VL = DXA. App ly ing Theorem 2.2.12 one obtains: EVVL = 0. Using (2.21) th is becomes: E{vEL + DxS{l,v,L)) = 0 Using l inear i ty of E and invoking Theorem 2.2.12 for a second t ime we get: E{vEL) = 0. To show that (b) (a) s imply r u n the preceeding impl icat ions backwards. The equivalence of (6) and (c) follows by Theorem 2.2.13. • Once an integrat ing factor v of a self-adjoint O D E G = EL = 0 is known, one can find a corresponding conservation law P using Theorem 2.2.14 w i t h H = vG. A l ternate ly one can use the Lagrangian L to signif icantly reduce the computat ion of P as follows. Theorem 3.2.9 (Noether's Theorem, Part 2) Suppose that v is an integrating factor of a self-adjoint ODE G = EL = 0 and, in view of Theorem 3.2.8, let A be such that VVL = DXA. (3.1) Then P = A-S{l,v,L) (3.2) is the conservation law corresponding to v so that vEL = DXP. Proof. The formula for P follows immediately f rom (2.21): DXP = vEL = VVL - DxS{l,v,L) = DX{A - 5(1,v,L)). • 54 Note tha t i f v is a var iat ional symmetry of G then the expression A f rom (3.1) i n the preceeding theorem can be obtained th rough a quadrature by apply ing Theorem 2.2.14 to H = VVL. Since the order of EL is i n general twice that of L, Theorem 3.2.9 is more effective for comput ing a conservation law corresponding to a given integrat ing factor than a direct appl icat ion of Theorem 2.2.14 to H = vG. I n add i t ion to g iv ing a conservation law, an integrat ing factor of of a self-adjoint O D E is also its symmetry. The fol lowing theorem shows how to take advantage of this and get a two-fo ld reduct ion of order. Th is w i l l be i l lustrated by an example. Theorem 3.2.10 Let G = 0 be a self-adjoint ODE and let v be its variational symmetry. Let P be a corresponding conservation law: P' = vG. Then VVP = 0 (mod G = 0 ) where C is any constant. Thus v is also a symmetry of P — C where C is any constant. Proof. B y Theorem 2.29, VVP = S(v,v,G) (mod G = 0). Hence the proof follows f rom the fol lowing lemma. • Lemma 3.2.11 G is self-adjoint iff S(v,v, G) = 0 for all v Proof. By (2.18), S'(v, v, G) = vVvG-vV*vG = 0 for al l v i f f G is self-adjoint. B y an argument simi lar to the argument given in Step I I in the proof of Theorem 2.3.2, RemD(v, v, G) = const, for a l l v i f f S(v, v, G) = 0 for a l l v. • A direct proof of this theorem for ODEs of second order is given in Sheftel [20]. Olver [16] gives another proof of this theorem. Example 3.2.12 Consider a self-adjoint O D E G = y2 + x V = 0. (3.3) I ts Lagrangian is given by ' _ y± B + -i T - J 0+1 2 ' P T 1 2 xa lay- \,P = - 1 5 5 I t admits a scaling symmetry W r i t t e n in evolut ionary fo rm, X becomes Vv w i t h v = (a + 2)y-x(l - f3)yx. To check i f v is a var iat ional symmetry, one can compute E(vG) = - ( 2 a + /3 + 3)G and thus v is a var iat ional symmetry i f f 2a + p + 3 = 0. Al ternately, one can check under what conditions T>VL is exact. B y (1.27), VVL = XL - ZDXL = XL — DX(£L) + where £ = (1 — (3)x. By direct computat ion, XL = 2(ct + / ? + 1)L and thus 2?„L = (2a + /? + 3)L - £ > x ( f L ) . Since EL ^ 0, L is not exact and thus VVL is exact i f f (3.5) holds. E x a m p l e 3.2.13 Assuming (3.5) holds, G becomes G = y2 + xay-3-2a and admits a var iat ional symmetry d d X = 2x— + y — • ox oy The conservation law P w i t h P' = (y — 2xy\)G can be computed using Theorem 3.2.9 P = -2xL - 5 ( 1 , v, L) = -y\x + Vly +  J— , a + - 1 . a + 1 56 Thus G = 0 is reduced to a first order O D E y\x - yiy y— = C,a^-l (3.9) a + 1 tha t , by Theorem 3.2.10, admits X. The canonical coordinates of X are x = e , y = re under which (3.9) becomes - 2 - 2 a \ - V 2 s'(r)= ^(r2 + 4C) + ^ T r J « ^ - 1 and hence a general solut ion to G = 0 is given impl ic i t l y by rv*-1'2 ( , 4 r - 2 - 2 a \ - V 2 \nx = 2j f(r 2 + d ) + ^ j dr + C 2 , a ^ - l where Ci ,C2 are arb i t ra ry constants. Thus using Noether's Theorem, a single var iat ional symmetry X led to a reduct ion of order of two. One can show that G = 0 has no other point symmetry, except for the t r i v i a l cases a = 0,a — -2, a = — | (See Section 4.1.1). Noether's Theorem was first proved in a sl ightly different version by Noether [15]. She consid-ered general var iat ional symmetries of the Lagrangian, involving bo th dependent and indepen-dent variables and d id not note invariance of L to w i t h i n a divergence. Bessel-Hagen [6] was the first to notice this impor tan t generalisation. His version is presented here. 3.3 Obtaining conservation laws without integration Once a var iat ional symmetry is known, a conservation law can be found th rough a quadrature using Theorem 3.2. However, when more then one symmetry is known, i t is often possible to obta in the corresponding conservation laws w i thou t any quadrature. Accord ing to Theorem 2.3.1, i f w is an integrat ing factor of G = 0 and v is i ts symmetry, then S(w, v, G) given by (2.19) is a conservation law of G = 0. However the result ing conservation law 57 may be t r i v ia l . I n the case when G is self-adjoint, more can be said about such a conservation law. i f b o t h v, w are var iat ional , then such a conservation law corresponds to their commutator . Th is was first observed in [3] where a direct proof was given. I n Section 3.3.1 we give an alternat ive proof which is based on Theorem 5.48 of Olver [16] and on Theorem 2.3.1. We then consider non-var iat ional symmetries. I n Theorem 3.3.5 we give necessary and sufficient condit ions for a point symmetry to be variat ional. As far as we know, this theorem is new. I n Section 3.3.3, given any point symmetry v and any var iat ional symmetry w, we define an expression u(v,w) which results in another (possibly new) var iat ional symmetry. W h e n v is also var iat ional , we show that u(v,w) = [v,w\. The conservation law corresponding to u(v,w) is given by P = S(w, v, G). 3.3.1 Commutator of variational symmetries For convenience, f rom now on we shall refer to v as a symmetry of G = 0 when T>VG = 0. T h e n v is a point symmetry i f f Vv is the evolut ionary fo rm of a po in t symmetry i f f v = n(x, y) — y i £ ( x , y) for some £(x,y),r](x,y). Also v is a var iat ional symmetry i f f i t is an integrat ing factor. Theorem 3.3.1 Let G = 0 be a self-adjoint ODE. Suppose that v, w are variational symmetries of G = 0. Then their commutator, u = [v, w] = Vvw — Vwv is also a variational symmetry of G = 0. Furthermore, suppose that Q is a conservation law corresponding to w, so that Q' = wG. Then the expression R = S{v,w,G) =VVQ [mod G=0) , where S is defined by equation (2.19), is the conservation law corresponding to u: R' = uG. Proof. The proof of this theorem is essentially the same as tha t given in Theorem 5.48 of Olver [16]. 58 Let Q be the conservation law corresponding to w so that Q' = wG. As w i l l be shown in the fol lowing lemma, for any v one has: DXVVQ = [v,w]G + DxS(G,w,v) +wE[vG). Since a var iat ional symmetry v is an integrat ing factor, E(vG) = 0 (see Theorem 2.2.13) and hence the last t e rm on the r ight hand side vanishes. Le t t i ng R = VVQ — S(G,w,v) we thus get DXR = uG. B u t S(G,w,v) = 0 (mod G = 0 ) since S is linear in its first argument and its to ta l derivatives. Thus R = VVQ (mod G = 0). A n appl icat ion of Theorem 2.3.2 completes the proof. • Lemma 3.3.2 Let G be self-adjoint, and let w, Q be such that Q' = wG. Then for any v, DXVVQ = [v, w]G + DXS{G, w, v) + wE{vG). where [v, w] = Vvw — Vwv. Proof. Using Lemma 1.1.14, def ini t ion of w, Q, product rule for the operator Vv, and self-adjointness of G we obta in : DXVVQ = VVDXQ = Vv{wG) = GVvw + wVvG = GVvw + wV*vG. Using (2.27) and (2.18) we get WV*VG = -wV*Gv + wE{vG) = -GVwv + DXS{G, w, v) + wE(vG). Thus DXVVQ = GVvw - GVwv + DXS(G, w, v) + wE(vG) = [v, w]G + DXS{G, w, v) + wE(vG). • Theorem 3.3.1 can simpl i fy the appl icat ion of Noether's Theorem: the conservation law corre-sponding to the commutator of two variat ional symmetries can be obtained w i thou t any inte-59 grat ion. Thus i f a Lie Algebra of var iat ional symmetries is s imple 1 then al l its corresponding conservation laws can be obtained w i thout integrat ion. E x a m p l e 3.3.3 ( S h e f t e l ) Consider an O D E that was analysed in Sheftel [20], p.116: G = y4-y-3=0. (3.10) Th is equation is self-adjoint since i t is in solved form and is independent of odd derivatives of y. A symmetry analysis of this equation (see [20]) reveals that i t admits three po in t symmetries: X\ = —, X2 = 2x— + 3 y — , Xz = x 2 — + 3 x y — ox ox oy ox oy w i t h commutators given by [Xi,Xj] Xi x2 x3 Xi 0 2Xi x2 x2 -2Xi 0 2X3 x3 -x2 - 2 X 3 0 where the entry in the i-th row and the j-th column corresponds to [Xi,Xj]. Thus point symmetries of the O D E (3.10) fo rm a simple Lie Algebra. The evolut ionary forms corresponding to X\,X2,X3 are: v\ = -yi, v2 = Sy - 2xyi, v3 = 3xy - yxx2. I t w i l l be shown by Theorem 3.3.5 tha t the above symmetries are var iat ional ( this is easily checked direct ly by ver i fy ing that E(v{G) = 0). Let Pj be the corresponding conservation laws, so tha t P[ = ViG. Then by Theorem (3.3.1) one has: 2Pl=S(v1,v2,G), P2 = S(vuv3,G), 2P3=S(v2,v^G) : A Lie Algebra is simple if it is equal to its derived algebra. A derived Lie Algebra is the Lie Algebra obtained by taking all possible commutators of the original Lie Algebra 60 f r om where one can compute: Pi = -vm + \vl - \v~vz P2 = +2xy3yx + y2yi + 3y~ 2 / 3x - y\x - 3yy3 3 1 P 3 = -3y£y 3 + - x 2 y ~ 2 / 3 + yi2 2y 3 + 3yy2 + .xy iy 2 - - a ; 2 y | - 2y\. The fourth-order O D E G = 0 is thus equivalent to the first order O D E H(x,y,y1,C1,C2,C3) = 0, (3.11) tha t can obtained by e l iminat ing y 2 , y 3 f rom the system Pi = C1,P2 = C2,P3 = C3. (3.12) The resul t ing O D E H does not admi t XUX2 or X3. Nonetheless, Sheftel showed how to obta in a symmetry of H, and thus was able to obta in a general solut ion of G. The idea is to seek a symmetry X — \\Xi + X2X2 + A 3 X 3 such tha t XPi = 0 (mod G = 0 ) , i = 1,2,3. Since H = 0 is equivalent to system (3.12), X is then also a symmetry of H. B y Theorem 3.3.1 and since VVi = Xi (mod G=0) one has X i P i = 0, X2PX = - 2 P i , XZPX = -P2 (mod G=0) since = 0, [X2,XX] = -2XU [ X 3 , X i ] = - X 2 . Thus X P i = — 2 A 2 P i — A 3 P 2 = 0 Similarly, one obtains a linear system for A^: XP2 ( 0 - 2 P i - P 2 2Pi 0 - 2 P 3 P 2 2P 3 0 V A 3 1 = o. 61 This system admits a non-zero solut ion Ai = - P 3 , A 2 = P 2 , A 3 = - 2 P 1 and hence X = —C3X\ + C2X2 — 2 C i X 3 is a symmetry of (3.11). Using X, one can ob ta in a general solut ion to H = 0 and thus to G = 0. A generalisation of the preceeding example leads to the fol lowing conjecture: Conjecture 3.3.4 Let G = 0 be a self-adjoint ODE admitting a r-dimensional Lie Algebra of variational symmetries Xi, ...,Xr. Let P\,...,Pn be the corresponding conservation laws. Let H be the system P\ — C i , P n = Cn, equivalent to the ODE G. Let A be the matrix with entries Aij — \Xi,Xj\ = 2~2,kCijkXk Let R be the rank of A. Then using linear algebra only, one can find r — R symmetries of H. I n part icular , i f G admits an r-dimensional abelian Lie Algebra of var iat ional symmetries that correspond to r funct ional ly independent conservation laws then 2r reductions of order are possible. The preceeding conjecture is t rue for simple var iat ional Lie algebras. 3.3.2 Characterisation of variational point symmetries Given a self-adjoint O D E G = 0 and its symmetry v, one can check i f v is a var iat ional symmetry of G by checking i f i t verifies E(vG) = 0. However this check involves G itself. I n this section we develop a new, much simpler check which does not reference G. We w i l l prove the fol lowing theorem: Theorem 3.3.5 Let G = yn - g{x,y,yi,...,yn-i ) = 0 (3.13) be a self-adjoint ODE in solved form <? and let v = r)(x,y) -yi£(x,y). (3.14) 2 Note that n must be even for G to be self-adjoint 62 If v is a point symmetry of G, then E(vG) = (2Vy + (1 - n)tx - (n + l ) £ w j / i ) G for all x, y, yi, ... (3.15) Consequently, a point symmetry v = n — £yi of G is a variational symmetry iff 2r]y + (1 - n ) £ x = 0 and £y = 0. (3.16) The proof of this theorem w i l l be based on the fol lowing lemmas. Lemma 3.3.6 Let X = ^{x,y)-§^ + "n(x,y)-^ be a point symmetry generator. Then its n-th extension r\n given by (1.18) has the form Vn = yn(r]y ~ < x - (n + l ) £ j , y i ) + p{x, y, yu yn-X) where p is some polynomial in yi, . . . y n _ i . Proof. Follows by induct ion f rom (1.18). • Lemma 3.3.7 Let X = i{x,y)-§^. + t){x,y)-^ be a point symmetry of an ODE (3.13). Then . XG = {riy - n £ x - (n + l)£yyi)G. Proof. Since X is a symmetry generator of G, one must have XG = 0 (mod G = 0 ) . B u t b o t h XG and G are at most linear in yn. Thus XG = aG = a(yn - g) (3.17) for some expression a = a(x,y,y\,yn-i), f ° r ah x,y,y\, ...,yn. Also by L e m m a 3.3.6 XG = yn(rjy - n £ x - (n + l ) ^ y i ) + f[x,y,yu ...,yn-i) (3-18) where / is some expression of order at most n — 1. Equat ing (3.17) and (3.18) and then collecting the yn coefficient, one obtains the desired result. n 63 We now re tu rn to the proof of Theorem 3.3.5. Using (2.27) and self-adjointness of G one has: E{vG) = V*VG + V*Gv = VVG + V*Gv. Using (2.17): VGv = vyG - (vyiG)' = vyG + (£G) ' . Using (1.27) and Leibni tz rule: VVG = XG- £.DXG = XG- (£G) ' + £'G. Thus E{vG) = XG + VyG + i'G. A n appl icat ion of Lemma 3.3.7 proves (3.15). Equat ion (3.16) follows immediately f rom (3.15) and Theorem 2.2.13. n E x a m p l e 3.3.8 A point symmetry A" is a var iat ional symmetry of an n- th order self-adjoint O D E (3.13) i f f i t is of the fo rm x = a x ) T x + {^tWv + f{x)) Yy (3"19) for some functions £(x),f(x). E x a m p l e 3.3.9 Any translat ional symmetry of (3.13) is always var iat ional . A scaling symmetry of (3.13) is var iat ional i f f i t is a constant mul t ip le of Th is provides another method of checking that a scaling symmetry (3.4) of a self-adjoint O D E (3.3) is var iat ional i f f (3.5) holds. 3 .3 .3 Using non-variational symmetries in conjunction with variational sym-metries As we have seen in Theorem 3.3.1, a commutator of two var iat ional symmetries is a var iat ional symmetry. Wh i le a commutator of two non-variat ional symmetries need not be var iat ional , i t sometimes is. The fol lowing theorem identifies when this is the case. 64 Theorem 3.3.10 Let G = 0 be a self-adjoint ODE. Let v = w(x,y) — yi^{x,y) be a point symmetry of G and let w be a variational point symmetry of G. Let [v, w] = T>vw — T>wv be the commutator of v, w. Then u = [TJ, W] + w{2r]y + (1 - n)£x - (1 + n)^yi) (3.20) is a variational point symmetry of G. Its conservation law is given by P = S(v,w, G) [mod G= 0). Proof. Th is is a direct consequence of Lemma 3.3.2 and Lemma 3.3.5. • Th is theorem is par t icu lar ly interesting i f G admits a scaling symmetry X = a x ^ + which, expressed i n evolut ionary fo rm, is v = by — axy\. Using the notat ion f rom the preceeding theorem we obta in u = [v, w] + cw, c = 26 + (1 — n)a is a var iat ional symmetry corresponding to a conservation law S(v, w, G). I n par t icu lar , [v, w] = u — cw is also a var iat ional symmetry since a difference of two var iat ional symmetries is also a var iat ional symmetry. Furthermore, suppose, as is often the case, that v i tself is non-var iat ional and tha t [v,w] = av + (3w. T h e n automat ical ly a = 0 and u = (f3 + c)w and, provided tha t 0 + c 7^  0, one can obta in a conservation law for u w i thout integrat ion. Example 3.3.11 A classification of a self-adjoint O D E G = y" - xay2 = 0 (see Section 4.1.1 or [21]) reveals that there are exactly four values of a for which G has point symmetries other than the scaling symmetry v = xyi + (a - 2)y. These values are a = 0 , - 5 , — - , — - . 65 I n a l l four cases, the only other point symmetry admi t ted is var iat ional. For instance, consider the case a = —5. T h e n G admits a var iat ional symmetry w = xy — x2yi w i t h [v, w] = —w. Thus, c = 7, (3 = — 1 , and u = ((3 + c)w = 6w. Thus the conservation law for w is P = ls(v,w,G) (mod G=0) = \(vw' - wv1) (mod G=0) 6 6 f rom where JP = y i y ^ - ^ ( y 2 + x2yl) + \y*x~z-Note tha t no integrat ion was required to obta in P. By comparison, Noether's Theorem relies on Lagrangian formulat ion as well as being able to find the divergence A f rom Theorem 3.2. I n [3] the authors showed that a scaling symmetry of a linear self-adjoint P D E can be used to obta in w i thou t integrat ion the conservation law corresponding to a given var iat ional symmetry. The preceeding theorem is a generalisation of this. 3 . 4 Conclusions I n this chapter we have studied symmetries and integrat ing factors of self-adjoint ODEs. We started by showing that the Euler-Lagrange O D E is self-adjoint and that i ts integrat ing factors are var iat ional symmetries and vice-versa. We presented Noether's Theorem 3.2.9 tha t can be used to find a conservation law using a var iat ional symmetry and a Lagrangian. The result ing conservation law admits the var iat ional symmetry that was used to find i t . Thus two reductions of order are possible using a single var i -at ional symmetry : one reduct ion in or iginal variables and one symmetry reduct ion. Theorem Theorem 3.3.4 generalises this result to self-adjoint ODEs admi t t i ng r var iat ional symmetries. I n Example 3.3.3 a simple three-dimensional L ie Algebra of var iat ional symmetries is used to obta in four reductions of order. 66 For a self-adjoint O D E of the fo rm G = yn ~ g(x,y,yi, . . . , y n - i ) , (3.21) a commutator of a scaling symmetry and a var iat ional symmetry is always a var iat ional symme-try . I n most cases the conservation law corresponding to such a commutator can be obtained w i thou t any integrat ion (see discussion after Theorem 3.3.10). More generally, a commutator of a var iat ional symmetry w and a non-variat ional point symmetry v need not be a var iat ional symmetry. However, there is an expression u(v,w) given by (3.20) which results i n a var iat ional symmetry. W h e n v is also var iat ional, u(v,w) = [v,w]. For a self-adjoint O D E (3.21) i t is possible to te l l when a given point symmetry is var iat ional , w i thou t using G. Th is provides an ansatz for looking for var iat ional symmetries, given by (3.19). 67 Chapter 4 Classification of solvable O D E s A n O D E is said to be solvable i f its general solut ion can be expressed using quadratures only. Given a fami ly of ODEs, the classification problem is to f ind as many solvable ODEs i n tha t fami ly as possible. I n this chapter we w i l l consider the classification problem for two families of second order ODEs: The f irst fami ly is known as the Emden-Fowler equation and is chosen because 1. Solvable cases are known which do not admi t two point symmetr ies. 1 (see [22]) 2. W h e n I = 0 the O D E is self-adjoint. 3. A n y O D E i n this fami ly admits a scaling symmetry. The second fami ly is chosen because 1. Any O D E in the fami ly admits a translat ional symmetry. 2. A n y solvable case leads to a solvable Abel equation y" = Axnymy'1 and y" = f(y)y' + g(y). u'{t) = -g{t)uZ -f(t)u2 (4.1) th rough a change of variables x = s(t),y(x) = t,u(t) = s'(t). (4.2) 1Note that any ODE admits Lie-Backlund symmetries. 68 To classify these ODEs we w i l l apply the theory of symmetries and integrat ing factors developed so far. We shall make use of the the symmetries admi t ted by the above ODEs when seeking other symmetries or integrat ing factors. To this end, in Section 4.2 we develop ansatzes that use known symmetries or integrat ing factors. I n Section 4.1 we classify the f irst fami ly for symmetries and integrat ing factors. The second fami ly w i l l be classified in Section 4.3. 4.1 Classification of the Emden-Fowler Equation, y" = Axnyrny'1 The goal of this section is to find solvable cases of the Emden-Fowler Equat ion, G = y" — Axnymy'1 = 0. (4.3) We shall denote such equation by a t r ip le (l,m,n). Before proceeding, we make several useful remarks tha t hold for any (l,m,n). F i rs t , note that a change of variables y(x) = t,x = u(t) (4.4) maps G = 0 into another Emden-Fowler Equat ion, u" + Atmunu'3'1 = 0. (4.5) Thus a solvable case (l,m,n) leads to a solvable case (3 — l,n,m). Second, note tha t G always admits a scaling symmetry (l-rn-l)x— + (2 + n-l)y—. (4.6) I n Section 4.1.1 we w i l l classify al l possible (I, m , n) which admit more then one point symmetry. I n Section 4.1.2 we w i l l f ind al l cases for which G admits an adjoint symmetry of the form w = a(x,y)+ b(x,y)yi. (4.7) Most adjoint symmetries of the form (4.7) w i l l t u r n out to be integrat ing factors. 2 Th is w i l l 2 Note that by Proposition 2.2.15 an integrating factor is an adjoint symmetry. An adjoint symmetry is not necessarily an integrating factor, but often it is. 69 lead to more solvable cases, some of which w i l l be different f rom those found using a symmetry classification. 4.1.1 Point symmetry classification of (4.3) We begin w i t h the point symmetry classification of (4.3). Using Lie's a lgor i thm, this amounts to solving the overdermined P D E XG = 0 (mod G = 0) for X = (,(x,y)-^ + rj(x,y)-^ + ... W r i t t e n i n fu l l , the result ing P D E is Vxx + {2l]xy — (,xx)yi + (Vyy ~ 2£XT/)?/I — ZyyVl -Alr)xxnymyl^x + A(l - 3)^yxnymy[+1 -A (mrixnym-1 + n£xn-lym + {2£x - rjy + lr)y - l£x) xnym) y[ = 0 For a fixed I, the result ing system splits by equating the coefficients of the like powers of y\ to zero. The sp l i t t ing depends on whether I is arb i t rary or one of / = — 1 , 0 , 1 , 2 , 3 , 4 . Since the change of variables (4.4) maps (4.3) into (4.5), the classification of the cases I = 2, 3,4 can be obtained f rom the classification of the cases / = 1 , 0 , - 1 respectively. A f te r considering al l possible subcases, one eventually obtains the fu l l symmetry classification of (4.3) l isted i n Table 4 .1 . Note tha t the cases I = — 1 , 4 do not y ie ld any addi t ional symmetr ies. The cases (I, m, n) = (0,2, — y ) , (0,2, — y ) were previously classified in Stephani [21]. Chapter two of the standard reference of solvable ODEs by Kamke [14] lists 246 non-l inear ODEs. O f those, seven are are Emden-Fowler ODEs w i t h n , m ^  0. They are O D E number 96 (0, n, — 4 ) , 100 (0,3/2, -1 /2 ) , 102 (0,1 -n,n), 105 (0,-1,1), 106 (0,-1,2), 205 (0,-2,-1), and 229 (-1,2,-3). Thus the non- t r iv ia l cases (0, m, 3 — m), (1,1,-1) as well as the cases obtained f rom them using the t ransformat ion (4.4) are not found i n [14] or in any other l i terature cited i n the bibl iography. 4.1.2 Adjoint symmetry classification of (4.3) I n this section we list a l l of the cases for which (4.3) admits an adjoint symmetry of the fo rm w(x,y,yx) = a{x,y)+yib{x,y). (4.8) 70 Condition Point symmetries of y" = Axnymy'1, other than (4.6) m = 0 I = 0, m + n + 3 = I = 0, m = 2, n = Z = 0, m = 2, n = / = o,m = 1 or Z = 1, m = 0 Z = 1, m = 1, n = 15 7 20 7 _3_ ^ 6 / 7 ^ + ( l + ^ - 1 / 7 ) * G is linear and thus admits eight symmetries. -Ax\nx£- + {l + Ay)^ n = 0 Z = 3,m + n + 3 = Z = 3, n = 2, m = I = 3, n = 2, m = Z = 3, n = 1 or Z = 2, n = 0 Z = 2, n = 1, m = dx xy-s-x + y2ik 15 7 20 7 i/ + ^ y 1 / 7 ) j l + ^Sy8/7^ G is linearisable using (4.4) and admits eight symmetries. (l + Ax)fx-Aylnyl Table 4.1: Symmetry classification of (4.3) 71 The process of finding adjoint symmetries is similar to the process of finding symmetries. F i rs t wr i te out the determining equation V*WG = 0 (mod G=0) . (4.9) The restr ic t ion (4.8) makes (4.9) an overdetermined system. Solving i t leads to the classification l isted in Table 4.2. There are two ways in which an adjoint symmetry may lead to a conservation law. 1. Note tha t (4.3) always admits a scaling symmetry which, w r i t t en in evolut ionary fo rm, is v = {l-n-2)y-(l + m-\)xyx (4.10) B y Theorem 2.3.1, i f w is an adjoint symmetry then the expression S(w, v, G) (mod G=0) defined by (2.19) is always a (possibly t r iv ia l ) conservation law. The last co lumn in the Table 4.2 lists such an expression. 2. A n integrat ing factor is always an adjoint symmetry. Conversely, some (but not all) adjoint symmetries are integrat ing factors. I f w is also an integrat ing factor then the t h i r d co lumn in Table 4.2 lists the corresponding conservation law. Th is classification identifies two cases which admit an integrat ing factor w i thou t admi t t i ng two symmetries. Case 1: The case I = l , n = — 1, corresponding to the O D E Y X admits an integrat ing factor w3 = x leading to a reduct ion of order P3 = C. From Table 4.2 we see that S(w3, v, G) = 0 (mod G=0). Thus by Theorem 2.3.2 i t follows tha t VVP3 = 0 (mod G= 0) and hence P% = C inherits the symmetry v given by (4.10)! Thus in this case, G — 0 is completely solvable. Using standard symmetry methods, the solut ion is found to be X = C 2 6 X P (ft + Af^dt + d) 72 Condition Adjoint symmetry Conservation law S(w,v,G) I - -l,m = 2 wi = 2/2/1 N/A 0 I = l , n = 0 7X»2 = 1 P2=yi-Afyndy (m + 1)P2 / = l , n = -1 w 3 = X P3 = x y i - y - AJ ymdy 0 Z = l ,n = — i , m = — 2 I w4 = 2xyx + 2A?f- - y Pi = y21x + y1(2A^-y)+A^ -Pi Z = 0 v = (n + 2)y - (1 - m)xyi N/A 0 Z = 0 , 7n = -3 - 2n v = (n + 2)y - 2{n + 2)xyx P 5 = -y\x + y y i + A f xndxy-2-2n 0 I = 0, m = — 3 — n w6 =xy- x2yi Pe = - \{y2 +x2yl) +xyyx -A J xn+1dxy-2-n (4 + 2n)P6 1 = 0,77i = 2,n = w7 = l + fAx-Wy -7r2Ax^yi P7=y1 + fAx-Wyy! - ^Ax^y2 + lAy2x~8/7 + ^A2x-9/7y3 6 p Z = 0,777, = 2,77 = W s = - X + fAx-1/^ Ps = -xVl + fAx'/7yyi - ^Ax*fyl -\Ay2x-%<1 + ^A2x-12/7y3 6 p — 7 " 8 Table 4.2: Adjoint symmetries of (4.3) of the form (4.8). If the adjoint symmetry listed is also an integrating factor, then the third column lists the corresponding conservation law. If an adjoint symmetry is not an integrating factor, N / A is present in the third column. This table does not include cases that are linear or obtained from the linear case by the transformation (4.4). See text for description of the fourth column. 73 Case 2: The case I = l , n = - 1 / 2 , m = - 2 , corresponding to the O D E G = y" - x - V \ S = 0 admits an integrat ing factor leading to a reduct ion of order P 4 — C = 0. However this equat ion does not inher i t v since VvPt± = <S(t/j4, u, G) = — P 4 (mod G=0) . Nevertheless, one has X \ ,P i = 0 (mod P i = 0 ) and hence the equation P 4 = 0 does inheri t the symmetry v. Thus one can find a particular solut ion of G = 0 by solving P 4 = 0 which admits the symmetry v. The resul t ing par t icu lar solut ion is Note that the two above cases are not obtainable through a point symmetry classification of (4.3). Neither are they found in Kamke [14] or any other l i terature cited in the bibl iography. 4.2 Using known symmetries or integrating factors as ansatzes 4.2.1 Using a symmetry to generate an ansatz for an integrating factor I n Section 2.1.3 we have discussed how to use a symmetry as an ansatz when looking for conservation laws. Since to every conservation law there corresponds an integrat ing factor, one can also use a symmetry as an ansatz for an integrat ing factor directly, w i thou t finding a conservation law. L e m m a 4 .2 .1 Let X = ^ (x,y)-^ + rj(x,y)-J^ be a point symmetry of an ODE Let P be a conservation law of G and w the corresponding integrating factor. Let Q = XP (mod G=0). Then Q is a conservation law and G = yn - g(x,y,yi,...,yn-i). (4.11) v = w (Vy - niyVl - (n - 1)6;) + Xw (4.12) is the integrating factor corresponding to Q. 74 Proof. Note tha t Q = X P (mod G=0) is a conservation law by Theorem 2.1.4. Since X is a point symmetry and P = P(x, y , y n - i ) , i t follows that XP is independent of yn. Thus X P (mod G = 0) = X P . Dif ferent iat ing, using Lemma 1.1.15 and then using P' = wG, we obta in : Q' = (xpy = x(p') + e'p' = x(u>o + e'wG. Using Q' = vG and the product rule we get: vG = wXG + GXw + i'wG. App l ica t ion of Lemma 3.3.7 results in (4.12). • Theorem 4.2.2 Let X , G' be as in Lemma 4.2.1. Then for any given constants a, (3, there exists a conservation law P of G which satisfies XP = aP + p (modG=0). (4.13) Furthermore, let w = w(x,y,y\, ...,yn-\) be an integrating factor of G with P' = wG. Then w satisfies the following PDE for all x, y, y\,yn-\: w (•% - n£yVl - (n - 1)6 - a ) + Xw = 0. (4.14) Conversely, If w satisfies (4-14) then the conservation law P with P' = wG satisfies (4-13) for some constant p. In addition, if a ^ 0 then the conservation law P can be obtained without any integration: P = SJ^l91 { m o d G=o) (4.15) a where v = n — y i £ is the evolutionary form of X. Proof. Step 1: We first show tha t there exists a conservation law P satisfying (4.13). By Lemma 2.1.5 w i t h PQ = 1, there exists a conservation law Q w i t h XQ = 1. So P = f(Q) is also a conservation law for any funct ion / . I n part icular , choose / to be a solut ion to f'{Q) = <xf(Q) + 0. Then X ( P ) = X(f(Q)) = f'(Q)XQ = f'(Q) = af(Q) + (3 = aP + p. 75 Step 2: Equation (4.14) is obtained by differentiating both sides of (4.13) in exactly the same fashion as in the proof of the preceeding lemma. Step 3: Equation (4.15) follows from application of Theorem 2.3.2. • Remark: Suppose that we found a conservation law P of G = 0 with XP = aP + /3 and with a 0. Letting Q = P + we see that XQ = aQ and hence XQ = 0 (mod Q=0). Thus X is a symmetry of the ODE P + - = 0. (4.16) a This fact can be used to find a reduction of order of (4.16) which leads to a particular solution depending on an arbitrary constant, if G is of order two. See Section 4.1.2, Case 2 for example. The following table lists commonly encountered symmetries and the corresponding solution of (4.14), for the case n = 2. X = -§i: w = eaxF(y,yi) X = ^ : w = eayF(x,yi) X = ai; + b c k : w = eXxF(ay-bx,yi),X = ^  X = V£-: w = yxF(x,^),\ = a-l y ' dx X =  ax£  + hy-L-  w = xxF(x~ by a, x a- byf), A = s±&zzb X = x-£-: w = xxF(y,xyi),\ = a+l For example, an ODE G = 0 admitting X = ^ will have an integrating of the form w = eaxF(y,yi) for any a, for some function F. Note however, that not every integrating factor of G is of that form. Nevertheless, one has the following theorem. Theorem 4.2.3 Let G = yn — g(x,yi,yn_i) = 0 be an ODE of order two or higher, and suppose that G admits a point symmetry of the form 76 Suppose also that G admits an integrating factor of the form w = a0{x,y) +yiai(x,y) + y(a2(x,y) + ... + y f aN(x, y) (4.17) where N is an arbitrary fixed positive integer if ^y = 0 and N = n if £y ^ 0. Then G admits an integrating factor of the form 4-L7 which in addition satisfies (4-14)-Before prov ing this theorem, we w i l l need the fol lowing lemma: L e m m a 4.2 .4 Let A : 5Rn —> 5Rn be a linear transformation and let x G 5R",x ^ 0 be such that Ax = 0. Then there exists y £ ffl1 independent of x such that either Ay = Ay for some constant X or else Ay = x. Proof. A n y linear t ransformat ion must admi t at least one eigenvector. I f A admits an eigen-vector independent of x then choosing y to be such an eigenvector proves the theorem. So assume w i thou t loss of generality that A does not admi t an eigenvector independent of x. T h e n x is the only eigenvector of A and zero is its only eigenvalue. B y a change of basis, we may assume w i thou t loss of generality that A is in its Jordan-Canonical fo rm, 0 a i A = 0 where aj is either zero or one, i = l . .n — 1. By assumption of uniqueness of the eigenvector, i t follows tha t ai = 1 for i = l . .n — 1. Thus r 0 1 A = (4.18) Also by uniqueness of the eigenvector, x = (a, 0 , 0 ) T for some a ^ 0 is the unique eigenvector of (4.18). Choosing y = ( 0 , a , 0, . . . , 0 ) T we obta in Ay = x as desired. • 77 Proof of Theorem 4.2.3. Step 1: Let w be an integrat ing factor of the fo rm (4.17) w i t h corresponding conservation law P. By Lemma 4.2.1, the integrat ing factor corresponding to XP is given by w (riy - n£yyi - (n - l ) £ x ) + Xw. Expand ing Xw using Lemma 3.3.6 one can show that such an integrat ing factor is also of the fo rm (4.17). Step 2: We now show that there exists an integrat ing factor w satisfying b o t h (4.17) and (4.14). One can show that any O D E of second order or higher admits f in i te ly many l inearly independent integrat ing factors of fo rm (4.17). Let wi,...,wr be such integrat ing factors and let Px,...,Pr be their corresponding conservation laws. Let PQ = 1 be a t r i v i a l conservation law. B y Step 1, XPi is a linear combinat ion of PQ, Pr. Thus X defines a linear t ransformat ion on the vector space spanned by P Q , . . . , P T . By Lemma 4.2.4 there exists P independent of PQ such tha t either XP = aP for some a. or XP = PQ = 1. Since P is independent of Po, i t is non- t r iv ia l . Hence there exists a non- t r iv ia l conservation P w i t h XP = aP + (3 (where /? = 0 when a ^ O and (3=1 when a = 0). App ly ing theorem 4.2.2 concludes the proof. • See section 4.3.2 for example. 4.2.2 Using a symmetry to generate an ansatz for a symmetry To find a symmetry X of an O D E G = 0, one needs to solve a linear P D E XG = 0 (mod G=0) for X. W h e n another symmetry is known, an ansatz can be made tha t decreases the number of independent variables in the above P D E by one. Theorem 4.2.5 Suppose that an ODE G = 0 admits a finite Lie algebra C of symmetries of dimension at least two. Then given any symmetry X G C, there exists another symmetry Y G C independent of X such that either [X,Y] = XY for some (possibly complex) constant X or else [X,Y]=X. 78 Proof. Th is is a direct consequence of Lemma 4.2.4 applied to the linear t ransformat ion A : Y -> [X,Y] and x = X. n E x a m p l e 4 .2 .6 Suppose an O D E admits a translat ional symmetry X = J j . I f i t admits any other point symmetries, then by Theorem 4.2.5 i t must also admi t either a non- t r iv ia l symmetry of the f o r m Y = (a(y)+x)l + o(y)ly. or else a non- t r iv ia l symmetry of the form Y = ex* (a(y)f- + b{y) d_ dy for some functions a(y),b(y) and some (possibly complex) constant A. See Section 4.3.1 for fur ther appl icat ion of this example. I n his thesis, Bou l ton [9] shows how to uti l ise structure constants of a Lie Algebra to generate ansatzes for symmetries. His a lgor i thm requires an explici t computat ion of these structure constants. 3 By contrast, no a-priori knowledge of the structure constants is required to use Theorem 4.2.5. 4.3 Classification of y" + f(y)y' + g(y) = 0 Consider the fami ly of ODEs G = y" + f(y)y' + g(y) = 0. (4.19) Note tha t (4.19) admits a point symmetry dx for any f(y),g{y)- We wish to find such f(y),g{y) for which (4.19) admits another symmetry or an integrat ing factor. We w i l l use ansatzes developed in Section 4.2 to simpl i fy th is task. 3 See Reid [18] for an algorithm that finds structure constants by reducing the overdetermined system of determining equations to standard form and without explicitly solving the determining equations 79 4.3.1 Symmetry classification of (4.19) I n th is section we w i l l classify al l possible cases for which (4.19) admits a symmetry other than Jj. B y Example 4.2.6, i f (4.19) admits another point symmetry then i t must also admi t a point symmet ry X of the fo rm either X = (a(y)+x)^ + b(y)^y (4.20) or We analyse the two cases separately. Case 1: Assuming (4.20), the determining equations XG = 0 (mod G=0), s impl i fy to a system of ODEs for a(y),b(y)J(y),g(y): a" = 0, b" + 2a'f = 0, g'b - b'g + 2g = 0, f'b + 3a'g + f = 0 (4.22) Solving (4.22) leads to the fol lowing. T h e o r e m 4 .3 .1 The ODE G = y" + f{y)y' + g(y) = 0 admits a symmetry T = ^ and a symmetry X with [T, X] =T iff X = ( a ( y ) + X ) l + b ( y ) l y and a(y),b(y),f(y),g(y) satisfy one of: 1. a ( y ) = a , b(y) = 0, f(y) = Q, g(y)=0 2. a(y)=a, b(y) = A, f(y) = B e ^ , g(y) = Ce'^, A + 0 3. a(y)=a, b(y) = A + By, f(y) = C(A + By)-1/B, g(y) — E(A + By)l~2lB, B^O 4. a(y) = Ay + a, f{y)-±b"(y), g(y) = ^  (b"(y)b(y) + b"(y)), A ± 0 where b(y) satisfies b""b2 + m'" - b'b" + 2b" = 0 (4.23) where capital letters represent arbitrary constants. 80 Remark 1: One can show that case 2 can be obtained f rom case 3 by tak ing the l i m i t as B —> 0. Also, case 1 is jus t a special case of case 2. Thus there are actual ly two dist ict cases: either a' = 0 (cases 1,2,3) or a' ^ 0 (case 4). Remark 2: Cases 1,2 and 3 lead to solvable Abe l ODEs (4.2) . through a t ransformat ion (4.1). However the resul t ing O D E is also solvable by the method of " A b e l invar iant" described i n Kamke [14], page 26. 4 Thus Theorem 4.3.1 does not lead to any solvable Abe l ODEs not found in [14]. We now derive a sequence of transformations that transforms the fourth-order O D E (4.23) into an A b e l O D E u'(t) + t(t + 2)(2t + 3)u(t)3 - (7 + 3t)u{t)2 + 3 ^ - = 0. (4.24) A point symmetry analysis reveals that the O D E (4.23) admits two symmetries: X l = y Y y + h W X 2 = dy w i t h [X2,X1] = X2. Thus a change of variables r = b{y),s(r) =y leads to a 3rd order O D E for z(r) = s'(r) that inheri ts the symmetry X\ —f-^. The canonical coordinates v(q) = l n r , g = z,w(q) = v'(q) then lead to a second order O D E w"wq2 - 3 g V 2 + 3o 2 (q - 1) w2w - l O o W + q{9q - 10)w3 - 2q2 (q - l ) 2 wi. (4.25) 4 The method of "Abel invariant" is the only general algorithm in Kamke to solve Abel ODEs. I t shows how to find, i f exists, a transformation y = F(t)u(t) + G(t),x — H(t) which maps an Abel ODE into a separable ODE. 81 A search for point symmetries of this O D E reveals a symmetry d d X3=q(q-l)-+w(l-3q) — (4.26) whose canonical coordinates are t = q{q-l)2w,p = \n(^-^j lead to an Abe l equation (4.24) w i t h u(t) = p'(t). No solut ion of this Abe l O D E is known. Using a method described in [7], a symmetry X3 leads to three part icular solutions of (4.25): Q w{q) = o w i t h c = — 2 or — 3/2 or 0. « ( « - ! ) Corresponding part icular solutions for b(y) can then be obtained when c ^ 0. Case 2: Assuming (4.21), the determining equations XG = 0 (mod G=0), s impl i fy to a system of ODEs for a(y),b(y),f(y),g(y): a" = 0, b" - 2Xa' + 2a'f = 0, X2b + bg' - b'g + A / 6 + 2Xag = 0, bf + 3a'g + Xaf - A 2 a + 2A6' = 0. Several subcases result, summarised below. Theorem 4.3.2 The ODE G = y" + f(y)y' + g(y) = 0 admits a symmetry T = ^ and a symmetry X with [T, X] = A X iff X is a constant multiple of 82 and a(y),b(y),f(y),g(y) satisfy one of: 1. a ( y ) = 0 , b(y) = l, f(y) = A, g(y) = - A (A + A)y + B 2. a(y) = 0, b(y)=y + A, f(y) = - 2 A In (y + A) + B, 9{y) = - A (y + A) ((A + 5 ) In (y + A) - A I n 2 (y + A) + 6) 3. a(y) = l , 6(2/) = 0, / ( y ) = A, y(y) = 0 4. a (y ) = l , b(y) = A, f(y) = A ( i J e " ^ + l ) , g ( y ) = - A (A + BAe~h + C e " ^ ) , A ^ 5. a(y) = l , b(y)=A + By, f(y) = A - 2B + (A + By)~x/B C g(y) = {A + By)l-2X'B E-(A + B y ) 1 ^ 8 C - (A + By) (A - B) , B^O 6. a(y)=y + A, j'(y) = A - \b"\y) . 9(y) = \b{y)b"'{y) + xyb"(y) + xAb"{y) - fb'(y) and b(y) satisfies the fourth order ODE b""b2 + 12A 26 + 3A(A + y)b"'b - 6A66" + 4A6' 2 - 8 A 2 ( A + y)b' (4.27) +2X2(A + y ) 2 6 " + —X(A + y)b"b' = 0 where capital letters represent arbitrary constants. Remark 1: One can show that cases 1-4 can be obtained f rom case 5 by considering various l i m i t i n g values of various constants. Thus there are actual ly two dist ict cases: either a' = 0 (cases 1,2,3,4,5) or a' ^ 0 (case 6). Remark 2: Cases 1-5 lead to solvable Abe l ODEs (4.2) th rough a t ransformat ion (4.1). For subcases 2, 4, 5, 6, the result ing ODEs are not solvable by the method of "Abe l invar iant" described in Kamke [14], page 26. Moreover, none of these ODEs are l isted among the 15 solvable ODEs of fo rm (4.2) whose solutions are given in Kamke. 5 A point symmetry analysis reveals that the O D E (4.27) admits two symmetries: n = („ + A) | + s»£, n = (»+ A? A + » ( , + A) | 5These 15 ODEs are numbers 36, 37, 40, 41, 42, 43, 45, 47, 48, 111, 145, 146, 147, 169, 185 from Chapter 1 of Kamke 83 w i t h [ l i , ! ^ ] = l 2 - 6 Using canonical coordinates for Y2, -3, s(r) = {y + Af y + A leads to a 3rd order O D E for z(r) = s'(r) t ha t inheri ts the symmetry Y\ = r^. Comput ing the the canonical coordinates of Yi we obta in the transformations v(q) = lnr,q = z,w(q) = v'{q) t ha t lead to a second order O D E w"wq2 - 3q2w'2 + 3g 2 (Xq - 1) w2w' - lOqww' + q{9Xq - 10)w3 - 2q2 {Xq - l ) 2 w4. (4.28) A search for po in t symmetries of this O D E reveals a symmetry Y 3 = (q(Xq - 1)) ~ + w{\ - 3Ag) whose canonical coordinates ^ 0 ( A g - l )2 ^ = l n ( ^ - i ) lead to the Abe l equation (4.24) for u(t) = p'(t). Thus the two ODEs (4.23) and (4.27) are connected th rough a sequence of (non-local) t ransformations! I t is not immediately obvious whether these two ODEs are also connected by a point t ransformat ion. Sett ing A = 0 in (4.27), we obtain: Proposition 4.3.3 The ODE y" = (b2 + 36 3 y) y' - (b0 + hy + b2y2 + 6 3 y 3 ) b3 (4.29) admits symmetries X = and dx Y = y— + (b0 + biy + b2y3 + 6 3 y 3 ) — . with [X, Y) = 0. 6 I t is interesting to note that Y2 satisfies the ansatz for a variational symmetry that was developed in Chapter 3 (see Example 3.3.8) even though neither the ODE (4.27) nor its solved form is self-adjoint. 84 Remark: One can show that the O D E (4.29) admits an eight-parameter symmetry group and is l inearizable. I n summary, the O D E (4.19) admits at least two symmetries i f f f(y),g(y) are given either by Theorem 4.3.2 or by Theorem 4.3.1. The subcases 2,4,5,6 of Theorem 4.3.2 lead to families of A b e l ODEs whose solut ion is not given in Kamke [14]. 4.3.2 Classification of integrating factors of (4.19) I n th is section we w i l l f ind al l cases for which (4.19) admits an integrat ing factor w of the fo rm Note tha t (4.19) admits a symmetry Thus i f (4.19) admits an integrat ing of the fo rm (4.30) then by Theorem 4.2.3 i t must also admi t an integrat ing factor of the fo rm w = eXx (a(y) + yib(y)) for some A, a(y),b(y). The determining equations E(wG) = 0 then s impl i fy to a system of ODEs for f(y),g{y),a(y),b(y): b' = 0, 2a' + Xb - 2fb = 0, 2a" + 2A6' - fb' - bf = 0, g'a + a'g - X (bg + fa) + X2a = 0. Th is system reduces fur ther, w i t h two cases possible. Case 1: I f b =fi 0, one can assume w i thou t loss of generality that b = 1, leading to the fol lowing result. Proposition 4.3.4 The ODE w = a{x,y) +y1b{x,y). (4.30) (4.31) admits an integrating factor of the form wi = eXx (a{y)+yi) (4.32) iffa(y),g(y) satisfy 2X(g - aa') + A 2 a + 2(a'g + g'a) = 0. (4.33) 85 If A ^ 0 then the conservation law corresponding to w\ is given by P i = S(w, - y i , G)/\ = eXx (yia(y) + \y\ + ja(y)g(yh) . (4.34) Sett ing A = 0 and then solving (4.33) leads to Corollary 4.3.5 The ODE admits an integrating factor v" - A^v1 + g(y) = 0 (4.35) » 2 = ^ + , , (4.36) The corresponding conservation law is given by P 2 = 4r<T2(y) + f 9(t)dt + A - ^ + l-y\ + Ax. (4.37) Case 2: I f b = 0 then a' = 0 and so one can assume, w i thout loss of generality, tha t a = 1, which leads to the fol lowing two proposit ions: Proposition 4.3.6 The ODE y"+(\g'(y) + *)y' + 9(y) = o (4.38) with A / 0 admits an integrating factor w3 = eXx. (4.39) The corresponding conservation law is given by P 3 = S(w, -yu G)/X = eXx(yi + jg(y)). (4.40) Proposition 4.3.7 The ODE y" + f(y)y' + A = 0 (4.41) admits an integrating factor 1. The corresponding conservation law is y' + fy f(t)dt + Ay. 86 Note tha t the O D E P 3 = 0 inherits the symmetry ^ and thus leads to a part icular solut ion of a_ dx (4.38). Using the t ransformat ion (4.1) one thus obtains: Corol lary 4 .3 .8 The Abel ODE y' = g{x)y* + ( l y ' ( x ) + \)y2 (4.42) admits a particular solution V = —^r- (4-43) I n summary, we have the fol lowing theorem: T h e o r e m 4 .3 .9 The ODE (4-19) admits an integrating factor of the form (4-30) iff the ODE is given by one of (4.31), (4.35), (4.38), (441). One can also combine the various proposit ions above to obta in cases for which (4.19) admits two funct ional ly independent conservation laws. For instance i f -Am = \s'[y)+X (444) then the O D E (4.35) admits conservation laws (4.37) and (4.40). Furthermore these conserva-t ion laws are funct ional ly independent whenever A ^  0. The solut ion of (4.44) is given by 9(y) = \ ( - A y -C+ ( (Ay + Cf + 1A) ^  (4.45) where C is an arb i t rary constant. As an example, tak ing C = 0, A = 1, A = 1 we get g(y) = \{-y + (y2 + ±)1/2)-Using j ( y 2 + 4 ) V 2 d y = I y ( y 2 + 4 ) i / 2 + 2 l n { y + {y2 + 4 ) i / 2 ) the conservation laws (4.37), (4.40) become: * - ( - , + ( ; + 4 ) v ^ - ^+ ^ 2 + 4 > i / 2 + i ' f a + b 2 + 4 ) i / 2 ' + - y + ( ^ 4 ) ^ 4 ^ + - -87 Ps = ex (yi-\y + \{y2 + ±)1'2). El im ina t i ng y\ f rom the system P 2 = Ci,Pz = C3 one obtains a general solut ion to the O D E «"wrw) y'+\(-«+<"2+4»"2) - »• <4-46> Th is O D E does not admit any point symmetries other than J j . The corresponding A b e l O D E is 4.4 Discussion I n th is chapter we have shown how the methods of previous chapters can be appl ied to find sub-families of solvable ODEs f rom a given fami ly of ODEs. We have considered Emden-Fowler fami ly of equations as well as the fami ly (4.19). Any solvable case of (4.19) leads to a solvable Abe l equation (4.2) using transformations (4.1). For b o t h families, we have obtained several new solvable cases not l isted in the standard reference by Kamke [14] or any other reference i n the bibl iography. W h e n the fami ly under consideration admits a symmetry, one can use i t to generate ansatzes for seeking integrat ing factors and symmetries. Unl ike the symmetry ansatzes studied i n [9], our symmetry ansatz does not require a pr ior i knowledge of the Lie algebra structure of the O D E i n question. We have appl ied such ansatzes to (4.19) which admits a point symmetry £ . As a result, we found al l cases for which (4.19) admits either another point symmetry or an integrat ing factor l inear in y'. The ansatz given in Theorem 4.2.2 w i t h a ^ 0, i f successful, leads to an integrat ing factor whose corresponding conservation law can be found without quadrature using that theorem. The u t i l i t y of th is is demonstrated in Proposi t ion 4.3.4 where the integrat ing factor w\ was found depending on an arb i t rary funct ion. Nevertheless, the corresponding conservation law 88 P i was found w i thou t quadrature. I n addi t ion one can always f ind a part icular solut ion of the resul t ing conservation law when a ^ 0 and the O D E is of order two. F ind ing an integrat ing factor is a task no more dif f icult than finding a symmetry. However once found, a reduct ion of order using a symmetry requires finding canonical coordinates. To do this in general, one must solve an auxi l iary O D E (see Section 1.1.3). 7 By contrast, an integrat ing factor always leads to a reduct ion through a quadrature, w i thou t ever having to solve any addi t ional ODEs. As well , an integrat ing factor reduct ion is a reduct ion in the original variables unl ike a symmetry reduct ion which requires a change of variables involv ing a dif ferential subst i tu t ion. 7 I f an n-th order ODE admits a solvable Lie Algebra of n symmetries then there is an algorithm by due to Lie that can find a general solution using quadratures only, never having to solve any additional ODEs (see Stephani [21], Chapter 9.3) 89 Chapter 5 Conclusions and future work 5.1 Conclusions I n this thesis we have examined the connections between conservation laws and symmetries, b o t h for self-adjoint and non self-adjoint ODEs. The goal was to gain a better understanding of how to combine symmetry methods w i t h the method of conservation laws to obta in results not obtainable by either method separately. I n Chapter 1 we have reviewed symmetry methods and how to use symmetries to ob ta in reduc-t ion of order of ODEs. I n Chapter 2 we have discussed how to f ind conservation laws. I n the absence of symmetries, one can look for conservation laws direct ly by seeking solutions of a linear P D E (2.2). The s i tuat ion is more interesting when a symmetry of an O D E is known. Lemma 2.1.4 shows tha t i f X is a symmetry generator and P is a conservation law of an O D E G = 0 then XP is also a conservation law of G = 0. Conversely, i t is shown (Lemma 2.1.5) that i f Po is any conservation law of G = 0 then there exists a conservation law P w i t h XP = Po- Since zero is a ( t r iv ia l ) conservation law, this leads to an ansatz: seek a conservation law P satisfying XP — 0. Such an ansatz is equivalent to the method of differential invariants for point symmetries, bu t works for general L ie-Backlund symmetries as well. I f such a P is found, one can also seek a conservation law Q w i t h XQ = 1. As shown in Section 2.1.4, for second order ODEs, Q can be found using quadratures only. Instead of looking for conservation laws, one can seek integrat ing factors tha t characterise 90 them. A n integrat ing factor w of an O D E G = 0 can be found by seeking a par t icu lar solut ion of the determining equation E(wG) = 0 (see Theorem 2.2.13). Given an integrat ing factor, the corresponding conservation law can be computed by a quadrature using Theorem 2.2.14. W h e n in add i t ion to an integrat ing factor one also knows a symmetry, i t is sometimes possible to compute a non- t r iv ia l conservation law without quadrature. I n part icular , given a conservation law P whose integrat ing factor is w, and given a symmetry X of G = 0, the conservation law XP can be obtained f rom w, X and G w i thout any quadrature, as shown i n Theorem 2.3.2. I n Chapter 3 we discussed self-adjoint ODEs. A n Euler-Lagrange equation is self-adjoint and an integrat ing factor of a self-adjoint O D E is also a symmetry of i t . Conversely, a variational symmetry w r i t t en i n evolut ionary fo rm is an integrat ing factor. Thus there is a one-to-one correspondence between integrat ing factors and var iat ional symmetries of self-adjoint ODEs. Given a var iat ional symmetry, one can use Noether's Theorem 3.2.9 to f ind the corresponding conservation law. Moreover, the result ing conservation law admits the var iat ional symmetry tha t was used to find i t . Thus a two-fold reduct ion of order is possible using a single var iat ional symmetry : one reduct ion in or iginal variables and one symmetry reduct ion. I n general, one can ask how many reductions are possible i f r var iat ional symmetries are known. The answer is 2r i f and only i f the conservation laws corresponding to variat ional symmetries are funct ional ly independent, and the var iat ional symmetries fo rm an abelian Lie algebra. More generally, th is question is answered in Theorem 3.3.4. I n Example 3.3.3, three var iat ional symmetries tha t f o rm a simple Lie Algebra are used to achieve a four-fold reduct ion of order. I n i ts fu l l generality, Noether's Theorem requires a quadrature to obta in a conservation law. However when more then one symmetry is known, i t is sometimes possible to obta in a conser-vat ion law w i thou t quadrature. I n part icular, a commutator of two var iat ional symmetries is shown in Theorem 3.3.1 to be a variat ional symmetry. Furthermore, a conservation law corre-sponding to such a commutator can always be obtained w i thou t any integrat ion, as shown i n Theorem 3.3.1. 91 A commutator of a var iat ional symmetry and a non-variat ional symmetry may also be a vari-at ional symmetry. I n part icular we show that for a self-adjoint O D E of the fo rm a commutator of a scaling symmetry and a variat ional symmetry is always a var iat ional symme-try. I n most cases the conservation law corresponding to such a commutator can be obtained w i thou t any integrat ion (see discussion after Theorem 3.3.10). More generally, a commutator of a var iat ional symmetry w and a non-variat ional point symmetry v need not be a var iat ional symmetry. However, there is an expression u(v,w) given by (3.20) which results in a var iat ional symmetry. W h e n v is also var iat ional , u(v,w) = [v,w]. For a self-adjoint O D E (5.1) i t is possible to te l l when a given point symmetry is var iat ional , w i thou t using G. One merely needs to check i f the symmetry satisfies condit ions (3.16) of Theorem 3.13. Th is provides an ansatz for looking for var iat ional symmetries: i f G has a var iat ional point symmetry X, then i t must be of the fo rm (3.19): Thus the determining equations for var iat ional point symmetries are immediately reduced f rom an overdetermined system of PDEs to an overdetermined system of ODEs. Other consequences are given i n Example 3.3.9. I n Chapter 4 we have shown how the methods of previous chapters can be appl ied to f ind sub-families of solvable ODEs f rom a given fami ly of ODEs. We have classified point symmetries of the Emden-Fowler O D E as well as i ts integrat ing factors linear in y'. We have identif ied two cases for which G admits an integrat ing factor. I n the first case, the corresponding conservation law is invariant under the scaling symmetry admi t ted by G and hence we were able to f ind a fu l l solut ion of G = 0. I n the second case the result ing conservation law does not admi t a scaling symmetry i n general, G = y n - g(x,y,yi,...,yn-i) (5.1) G = y" — Axnymy'1 = 0 (5.2) 92 bu t a specific solut ion was nevertheless found. I n bo th cases symmetry methods alone fai l to produce a solut ion or a reduct ion of order in the same variables since bo th cases do not admi t po in t symmetries other than the scaling symmetry. Together, symmetry and integrat ing factor classification lead to eight new solvable cases of Emden-Fowler equation not found i n Kamke [14] or any other l i terature in bibl iography. These cases are: (Z ,m,n) = ( 0 , m , 3 - m ) , ( 1 , 1 , - 1 ) , ( l , m , - 1 ) , ( 1 , - 1 / 2 , - 2 ) . and the four cases obtained f rom the above using the t ransformat ion (4.4). We have developed ansatzes that use known symmetries to find new symmetries or integrat ing factors. The ansatz for a symmetry using a known symmetry (Theorem 4.2.5) reduces by one the number of independent variables in the determining equations for symmetries, whi le in t roduc ing an ext ra constant parameter A (see Example 4.2.6). Thus for point symmetries, the determin ing equations are reduced f rom an overdetermined system of PDEs to an overdetermined system of ODEs. Th is ansatz is general enough in the fol lowing sense: i f an O D E admits more than one symmetry then i t must also admi t a symmetry which is different f rom the known symmetry and which satisfies the ansatz. Conversely, i f a symmetry is found using this ansatz and w i t h A ^ 0, then i t w i l l necessarily be different f rom the known symmetry. The ansatz for integrat ing factors using a known symmetry (Theorems 4.2.2 and 4.2.3) also introduces an addi t ional parameter a , whi le reducing by one the number of independent var i -ables. I f an integrat ing factor is found using this ansatz w i t h a ^ 0, then the corresponding conservation law can be found w i thou t any quadrature. Moreover in such part icular solut ion to the O D E depending on an arb i t rary constant can then be found i f the O D E is of order two. Thus such an ansatz w i t h a ^ 0 leads to b o t h a conservation law and a par t icu lar solut ion for ODEs of order two. We have appl ied these ansatzes to the equation G = y"-f(y)y'-g(y) 93 (5.3) which admits a point symmetry ^ and which is equivalent to an A b e l O D E (4.2) using a t ransformat ion (4.1). Any solvable case of G thus leads to a solvable Abe l O D E . We found al l cases for which G admits either another point symmetry or an integrat ing factor l inear in y'. A symmetry classification of G resulted in four non- t r iv ia l solvable families (cases 2,4,5,6 of Theorem 4.3.2) of Abe l ODEs that are dist inct f rom al l solvable Abe l ODEs reported in Kamke. Each one of these families depends on several arb i t rary constants. A n integrat ing factor classification resulted in several cases for which a particular solut ion to G could be found. As a result, we have singled out a fami ly of Abe l equations (4.42) that depends on an arb i t ra ry funct ion. For this fami ly a part icular solut ion (4.43) is given. We have identi f ied a case for which G admits two funct ional ly independent conservation laws w i thou t admi t t i ng any point symmetry other then J j . Th is case is also not reported i n Kamke. 5.2 Future research Possible directions for fu ture work include: 1. Extension of ansatzes for the case where more than one symmetry is known. 2. W h e n is a commutator of two non-variat ional symmetries a var iat ional symmetry? 3. Exp la in the surpr is ing connection between the two fou r th order ODEs (4.23) and (4.27). Section 4.3. 4. Computa t ion of the solutions of the solvable Abe l ODEs found in Section 4.3. 5. Extension of the results to PDEs (especially those in Chapter 3). 6. App l i ca t ion of the ansatz (3.19) to f ind ODEs of the fo rm y" = f(x,y) tha t admi t a var iat ional symmetry. 94 Bibliography [1] Anco, S.C. k. B luman, G., (1998) Integrat ing factors and first integrals of ord inary differ-ent ial equations, Eur. J . A p p l . M a t h 9 no. 3, 245-259. [2] Anco, S.C. & B l u m a n , G., (1996) Der ivat ion of conservation laws f r o m nonlocal symmetries of dif ferential equations, , J . M a t h . Phys. 37 no. 5, 2361-2375. [3] Anco, S.C. 8z B luman, G., (1997) Direct Construct ion a lgor i thm of Conservation Laws f rom Fie ld Equat ion, , Phys. Rev. Let t . 78 no. 15, 2869-2873. [4] Anderson R.L, K u m e i S, Wu l fman C.E., (1972) General ization of the concept of invariance of dif ferential equations. Results of applications to some Schrddinger equations, Phys. Rev. Let t . 28 no. 15, 988-991. [5] A rno ld , V . L , (1991) Ordinary Differential Equations, Springer-Verlag. [6] Bessel-Hagen, (1921) Uber die Erhaltungssatze det Elekt rodynamik. , M a t h . A n n . 84, 258-276. [7] B l u m a n G. & K u m e i S., Symmetries and Differential Equations, Springer-Verlag, New York. Repr inted w i t h corrections, Springer, New York, 1996. [8] Boul ier F., Lazard D., Ol l iv ier F. and Pet i tot M. , (1995) Representation for the radical of a f in i te ly generated differential ideal, Proc. ISSAC 1995, A C M Press 158-166. [9] Bou l ton , A. , (1993) New Symmetries f rom old: Exp lo i t ing Lie Algebra Structure, M.Sc. Thesis, Universi ty of Br i t i sh Columbia. [10] Boyer, T . H . , (1967) Continuous symmetries and conserved currents, A n n . Phys. 42, 445-466. [11] Cheb-Terrab S.E, Duarte L.G.S. and da M o t a L.A.C.P., (1998) Computer A lgebra Solving of Second Order ODEs Using Symmetry Methods, Comput . Phys. Comm. 108 no. 1, 90 -114. [12] Gonzales-Lopez, A. , (1988) Symmetry and integrabi l i ty by quadratures of ord inary dif-ferential equations, Phys.Lett . A 133, no. 4-5, 190-194. [13] Hereman, W. , (1995) Chapter 13 i n vol. 3 of the C R C Handbook of L ie Group Analysis of Dif ferent ial Equations, Ed. : N.H.Ibragimov, CRC Press, Boca Raton, F lor ida. [14] Kamke, E., (1947) Differentialgleichungen, N.Y. Chelsea Pub l . Co. [15] Noether, E., (1918) Invariante Variationsprobleme, Nachr. Kdn ig . Gesell. Wissen. Got t ingen, Math. -Phys. K l . , 235-257. [16] Olver P.J., (1993) Applications of Lie Groups to Differential Equations, Springer-Verlag, New York. 95 [17] Reid G.J., (1995) A lgor i thms for reducing a system of PDEs to standard fo rm, determin ing the dimension of its solut ion space and calculating its Taylor series solut ion, Euro. J. A p p l . M a t h . 2, 2315-2327. [18] Reid G.J. , Lisle I.G., Bou l ton A. and W i t t k o p f A .D . , (1992) A lgor i thmic determinat ion of commutat ion relations for Lie symmetry algebras of PDEs, Proc. ISSAC 1992, A C M Press, New York. [19] Reid G.J. , W i t t k o p f A . D . and Bou l ton A. , (1996) Reduct ion of systems of nonlinear par t ia l dif ferential equations to simpli f ied involut ive forms, . Eur. J . A p p l . M a t h . 7 604-635. [20] Sheftel, M.B. , (1997) A Course in group analysis of differential equations - in Russian, Universi ty of St.Petersburg of Economics and Finance Press. [21] Stephani H., (1989) Differential equations: their solution using symmetries, Cambridge Universi ty Press. [22] Zaitsev, V .F . , (1988) On discrete-group analysis of ordinary dif ferential equations, Soviet M a t h . Dok l . 37 no. 2 403-406. 96 

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