A FIELD THEORETICAL TREATMENT OF MAGNETOHYDRODYNAMICS by MELVIN GILBERT CALKIN B.Sc, Dalhousie University, 1957 M.Sc, Dalhousie University, 195# A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of PHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1961 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study. I further agree that permission f o r extensive copying of t h i s thesis f o r scholarly purposes may granted by the Head of my Department or by his be representatives. It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Physics The University of B r i t i s h Columbia, Vancouver S, Canada. v Date October, 1961 FACULTY OF GRADUATE STUDIES PROGRAMME OF THE FINAL ORAL EXAMINATION FOR T H E DEGREE OF D O C T O R OF PHILOSOPHY of MELVIN GILBERT CALKIN B Sc Dalhousie, 1957 M Sc Dalhousie, 1958 SATURDAY, NOVEMBER 18th, 1961, A T 9:30 A . M . IN ROOM 301, PHYSICS BUILDING COMMITTEE IN CHARGE Chairman F H SOWARD F A KAEMPFFER W OPECHOWSKI F L CURZON L de SOBRINO External Examiner C T V R A SWANSON E HULL J OKULITCH W STEWART W B THOMPSON University of California at San Diego, La Jolla, California G R A D U A T E STUDIES A FIELD THEORETICAL T R E A T M E N T OF M A G N ETOH YDRODY N AMICS Field of Study Theoretical Physics ABSTRACT The action principle of field theory, usually applied only to linear theories, such as electrodynamics, is shown to be useful in the nonlinear theory of magnetohydrodynamics The well-known conservation laws are derived by utilization of invariance properties of the appropirate lagrangian In addition, a set of new conserva- tion laws, corresponding to generalizations of the Helmholtz vortex theorem in hydrodynamics, are obtained by exploiting a certain type of gauge mvanance of the magnetohydrodynamical action integral A number of formal simplifications are introduced which suggest themselves through the action principle Three problems which illustrate the resulting formulation of magnetohydrodynamics are discussed Electromagnetic Theory ..G Theory of Measurements Elementary Quantum Mechanics .A M Crooker .F A Theory of Relativity Kaempffer . P Rastall Group Theoretical Methods . Advanced Quantum Mechanics M Volkoff W. Opechowski F A Kaempffer Related Studies Differential Equations Modern Algebra Modern Geometry . C A Swanson . J H Lindsay B Chang ABSTRACT The action p r i n c i p l e of f i e l d theory, usually applied only to l i n e a r theories, such as electrodynamics, i s shown to be useful i n the nonlinear theory of magnetohydrodynamics. The well-known conservation laws are derived by u t i l i z a t i o n of invariance properties of the appropriate lagrangian. In addition, a set of new conservation laws, corresponding to generalizations of the Helmholtz vortex theorem i n hydrodynamics, are obtained by exploiting a certain type of gauge invariance of the magnetohydrodynamical action i n t e g r a l . A number of formal s i m p l i f i c a t i o n s are introduced which suggest themselves through the action p r i n c i p l e . Three problems which i l l u s t r a t e the r e s u l t i n g formulation of magnetohydrodynamics are discussed. Mathematical details including a number of new vector i d e n t i t i e s and proofs of formulae following from a theorem by Clebsch are relegated to appendices. -111- TABLE OF CONTENTS PAGE Introduction 1 Chapters 1 A Transformation of the Magnetohydrodynamical Equations 1.1 The Usual Magnetohydrodynamical 4 Equations . . . . . . . . 1.2 A Transformation of the Electromagnetic Equations 1.3 10 A Transformation of the Hydrodynamical Equation 2 3 4 Classical Field 16 Theory 2.1 The Concept of a C l a s s i c a l F i e l d ... 24 2.2 The F i e l d Equations 26 2.3 The Conservation Laws 27 The Action P r i n c i p l e i n Magnetohydrodynamics 3.1 Construction of the Lagrangian 3.2 The F i e l d Equations 36 3.3 The Conservation Laws 37 3.3A The Space - Time Imparlances 3.3B The Gauge Invariances .... .... 33 37 42 Illustrations 47 4.1 4$ Magnetohydrostatics PAGE Chapters 4 I l l u s t r a t i o n s (continued) 4.2 Alfven Waves 51 4.3 Discussion of a Particular Steady State Solution 55 Appendices A B C Some Useful Vector Theorems A.l Theorem 62 A. 2 Corollary 64 Clebsch's Theorem B. l Theorem 67 B. 2 The Gauge Transformations 6$ Calculations f o r Chapter 3 C. l The Lagrangian and i t s Derivatives . . 72 C.2 The F i e l d Equations 74 C.3 The Conservation Laws 75 C.3A The Space - Time Invariances . . . . 75 C.3B The Gauge Invariances 87 Bibliography 94 -V- TABLE OF FIGURES Figure I The Forces on the Representative P a r t i c l e s . Figure I I The Change i n the Flux of P Through a Moving Surface Z i s due to both the Change i n the Vector F i e l d and the Motion of the Surface. Figure I I I The I n f i n i t e s i m a l Surface dlo- Used f o r the Measurement of P Moves with the F l u i d from the Point x„ at t = o to the Point x at t , Figure IV The Lines of ^ are the Lines of Intersection of the Surfaces ^= constant and Figure V To construct the Surface rr)'=<K, constant. f o r the Trans- formation 1.3.27, one lays out Vectors from the Surface ^ = <k Figure VI and Joins the End Points. The Orientation of the Amplitude of the Vectors f o r an Alfven Wave (a) £ B p i n plane of B (b) SBp- perpendicular to the plane of 6 0 Figure VII The Three C y l i n d r i c a l Coordinates. Figure VIII The Current, Magnetic F i e l d , and Velocity D i s t r i b u t i o n i n the case u"' Figure IX i « B = i 8 and k and k . B ( r ) l Representative Surfaces «)•=. constant and 5 = constant i n the case t f = -p=- B = B(V) 1 - . f . -vi- Figure X The Motion of a Representative Surface constant, i n the case and —= > o »— _ _> tr=-p=. BB =,7== 9* ~ ~ ^= B(r)4.~ ACKNOWLEDGMENTS The author wishes to thank his supervisor, Prof. F.A. Kaempffer, f o r helpful discussions and c r i t i c i s m during the research and writing of t h i s t h e s i s . The author also wishes to thank the members of his committee: Prof. W. Opechowski, Dr. L. de Sobrino, and Dr. F.L. Curzon f o r advice during the writing of t h i s thesis. The author extends his thanks to Dr. P. R a s t a l l f o r his interest and help during the author's stay at the University of B r i t i s h Columbia. F i n a n c i a l assistance i n the form of a Studentship from the National Research Council i s g r a t e f u l l y acknowledged. -J_- INTRODUCTION In recent years magnetohydrodynamics has become one of the prominent f i e l d s of research i n t h e o r e t i c a l physics. The motivation f o r t h i s development has been partly supplied by current i n t e r e s t i n problems posed by the attempt to contain fusion reactions. These reactions occur i n nature, f o r example i n the i n t e r i o r of stars, i n the plasma state of matter. Since the plasma i s , i n certain approximations, an example of a magnetohydrodynamical system, large numbers of s p e c i f i c problems i n plasma physics have been solved using the magnetohydrodynamical description. An example of another magnetohydrodynamical system i s the magnetic pumping of a l i q u i d metal through the cooling pipes of a nuclear reactor. Curiously enough, the a v a i l a b i l i t y of general f i e l d t h e o r e t i c a l methods, which have been developed i n quite another branch of physics, namely quantum f i e l d theory, seems to have been l a r g e l y overlooked by workers i n magnetohydrodynamics. In such a formulation, the physical laws governing the f i e l d are summarized into a single v a r i a t i o n a l or action p r i n c i p l e , from which may be derived by application of standard techniques the f i e l d equations and conservation laws. Action p r i n c i p l e s f o r the electromagnetic and hydrodynamical f i e l d s separately have been known f o r many years and have been extensively used i n the development of quantum electrodynamics and quantum hydrodynamics. In the present -2- work i t i s shown that the f i e l d equations of magnetohydrodynamics, which i s the electromagnetic f i e l d coupled by means of currents i n the conducting f l u i d to the hydrodynamical f i e l d , are also derivable from an action p r i n c i p l e . The introduction of an action p r i n c i p l e i s made possible through the use of appropriate f i e l d variables, which are suggested by those chosen i n the case of the electromagnetic and hydrodynamical f i e l d s . In electro- dynamics the appropriate variables are not the e l e c t r i c and magnetic f i e l d strengths but the electromagnetic potentials. In addition to these variables i t i s found to be convenient to represent the source of the electromagnetic f i e l d i n terms of a single vector function, the p o l a r i z a t i o n . In hydrodynamics one introduces i n place of the v e l o c i t y the hydrodynamical potentials, f i r s t discussed by Clebsch. A generalization of Clebsch s representation w i l l be used i n T the treatment of the magnetohydrodynamical f i e l d . Once the proper f i e l d variables have been decided upon, i t i s a simple matter to write down an action p r i n c i p l e from which the f i e l d equations may be derived. The conservation laws follox^ from invariance properties of the action p r i n c i p l e . into two classes: These invariance properties f a l l the space-time invariances from which a r i s e the conservation laws of momentum, energy, angular momentum, and center of mass, and the gauge invariances, that i s , the transformations among the f i e l d variables which leave the physical observables unchanged. These -3- l a t t e r transformations lead i n the case of magnetohydrodynamics, amongst other things, to conservation of mass, a generalization of Helmholtz*s vortex theorem of hydrodynamics, and the conservation of the volume integrals of fl-B and y-6 where fl i s the electromagnetic vector p o t e n t i a l , 8 i s the magnetic induction, and <r i s the f l u i d v e l o c i t y . The r e s u l t i n g formulation of magnetohydrodynamics i l l u s t r a t e d by applying i t to three simple problems. is In the f i r s t example, the r e s t r i c t i o n on the variables imposed by the condition that the f l u i d v e l o c i t y vanishes i s discussed, while i n the second, Alfven waves i n an unbounded medium through which passes an i n i t i a l l y constant magnetic are studied. field F i n a l l y , a time independent solution i n which the v e l o c i t y i s everywhere p a r a l l e l to the magnetic f i e l d i s treated. CHAPTER 1 A TRANSFORMATION 1.1 OF THE MAGNETOHYDRODYNAMICAL EQUATIONS The Usual Magnetohydrodynamical Equations. The motion of a conducting f l u i d i n an electromagnetic f i e l d and an external non-electromagnetic conservative f i e l d i s described by the equations of electromagnetism coupled with the equations of hydrodynamics. To avoid problems introduced by d i s s i p a t i v e effects, i n the following assume the f l u i d to be i n v i s c i d and i n f i n i t e l y conducting. In addition assume either that the f l u i d i s incompressible, or that the motion i s i s e n t r o p i c . The equations describing such a system are: Maxwell's equations i n a medium moving with a v e l o c i t y small compared with that of l i g h t ry^ £ = (Panofsky and P h i l l i p s , 1955, p. 147) 3S 1.1.1 bt 1.1.2 V-B^O V*/-t I = J + £<£ -h \ Mo/ V-(K E) 0 1 2t (K £) 0 1.1.3 = E 1.1.4 the Ohm's law r e l a t i o n f o r i n f i n i t e conductivity § + if*B = o 1.1.5 the dynamical equation of hydrodynamics modified by the additional Lorentz force term -5- = at -r- 67xy-)x tr = -^/TT , + -!- Iir/ - + § ) 4 1.1.6 + J - J< B where Tr0=-p. 1.1.7 for the case of an incompressible f l u i d and 7T(=j"^ 1.1.8 for the case of isentropic motion, and the equation of continuity & •*• V.(?<r) = O 1.1.9 which, i n the case of an incompressible f l u i d reduces to = O 1.1.10 The following notation has been used: E i s the e l e c t r i c f i e l d strength, B the magnetic induction, J the current density i n a coordinate frame which moves with the f l u i d , e the charge density, /*„ and K p e r m i t t i v i t y of the f l u i d 0 the permeability and the (assumed constant and equal to those of free space), cr the f l u i d v e l o c i t y , p the mass density, p the pressure, and the potential per unit mass of any external non-electromagnetic conservative f i e l d . R.M.K.S. units w i l l be used through out t h i s work. To make the above d e f i n i t i o n s more precise, consider a metal consisting of neutral atoms, electrons, and ions i n various states of i o n i z a t i o n . To make the formalism applicable to the most general s i t u a t i o n of t h i s type, -6- introduce N types of p a r t i c l e s , denoting by mi the mass of the i t h type of p a r t i c l e , <^ the charge of the t t h type of c particle, n the number of t t h type p a r t i c l e s per unit volume t at a space-time point (*,t), and u" the average v e l o c i t y of 6 the i.th type of p a r t i c l e at a space-time ooint ( x , t ) . The charge density £ i s then given by £ = Z ^ i %'<• l . l . l l 1=1 the t o t a l current J" by T IT ~ Z.^^L iTc 1.1.12 the mass density p by e ri = 2^ ^L^l 1.1.13 i-=r 1 and the mass v e l o c i t y <s (or v e l o c i t y of the center of mass) by pir = 2l ""<: i tfi L = I 1.1.14 m The current J i s defined by J" = J " - £U" = T (<£i 1.1.15 and i s the current with respect to the center of mass. With these d e f i n i t i o n s , the Maxwell equations 1 . 1 . 1 1 . 1 . 4 are exact i n the n o n - r e l a t i v i s t i c l i m i t . However, the Ohm s law r e l a t i o n 1 . 1 . 5 and the dynamical equation 1 . 1 . 6 f are only v a l i d provided certain conditions are f u l f i l l e d . To i l l u s t r a t e the type of conditions, consider a system consisting of only two types of p a r t i c l e s and approximate Figure I - to follov; page 6 The Forces on the Representative P a r t i c l e s -7- t h i s system i n the following manner: the volume occupied by the gas i s divided up into macroscopically small and microscopically large volumes; i n each volume a l l the particles of type 1 (2) are replaced by a single p a r t i c l e having a charge <?, (6?^) equal to the sum of the charges of type 1 (2) p a r t i c l e s , a mass M, (M ) equal to the sum of z the masses of tvpe 1 (2) p a r t i c l e s , and a v e l o c i t y equal to the average v e l o c i t y of type 1 (2) <r, (ir^) particles; p a r t i c l e mechanics i s applied to these two representative particles. C o l l i s i o n s between p a r t i c l e s of the same type do not change the average momentum of t h i s type p a r t i c l e . However, c o l l i s i o n s between p a r t i c l e s of different types do change the average momentum of each type. s h a l l be approximated This effect by a drag force. The charge density £ , the t o t a l current J/ , the mass T density p, the v e l o c i t y of the center of mass vr, and the current with respect to the center of mass J" are given by 1.1.16 1.1.17 1.1.18 1.1.19 1.1.20 where V i s the volume under consideration. The equations of motion are (Figure I) 1.1.21 where I?, (D^) i s the drag force exerted by the representative p a r t i c l e 2 (1) on p a r t i c l e 1 (2). In order to approximate the drag force £ > consider c o l l i s i o n s between the two types of p a r t i c l e s . At each c o l l i s i o n with a type 2 p a r t i c l e , a particular type 1 p a r t i c l e loses on the average the momentum (assuming i s o t r o p i c m, + m scattering) a Let the mean time between c o l l i s i o n s of a particular type 1 p a r t i c l e with any type 2 p a r t i c l e be ^, . Then the effect of c o l l i s i o n s of type 1 p a r t i c l e s with type 2 particles may be approximated by a drag force p, on the representative particle 1 = - M ' m »- (\£t- y a ) * T ' 1.1.23 Similarly one finds the drag force 5^ on the representative particle 2 p i = -_mA (V^-if, ) t"' = - 2 , The two c o l l i s i o n times 1.1.24 and r are related by *± _ ^ a . Upon substituting these expressions into 1.1.21 and 1.1.22, these two equations may be combined to give two l i n e a r l y independent equations (provided Q,M^~ Q^M,*0) only the macroscopic quantities y and J . equation of motion of the center of mass involving The f i r s t i s the -y- Q r <*t = € E + X.*B = £ f e r<B)+ J x 8 - -r - - - + l 1.1.25 - The second i s a generalization of Ohm's law " — ) j 1.1.26 The interesting case i s the one i n which the two types of p a r t i c l e s are electrons having charge -e and mass me, and ions having charge He and mass mL. In order to allow f o r -n electrons and -n ions per unit volume, take e t Q,= -T) eV 1.1.27 M, = -n m \/ 1.1.28 e e e Qx = ^ 2eV 1.1.29 L M^tiiffljV Then, assuming m 1.1.30 e « , 1.1.26 becomes (Cowling, 1957, p. 100; Spitzer, 1956, p. 21) *I = ^Cf+ir x ?)dt m e — m ~ CTx? - 1.1.31 JV-' e Let <» be the frequency at which J changes, and oo = g 6 m c e the cyclotron frequency of the electrons. «*e«' i.e.: Then, provided 1.1.32 the current changes slowly i n comparison with the time between ion-electron c o l l i s i o n s , and 1.1.33 i.e.: the electrons are not free to s p i r a l around the l i n e s of §, the terms with J r "' and e_ j g x ott me ~ ~ are negligible i n comparison . Equation 1.1.31 then reduces to the usual Ohm s T e law r e l a t i o n J> cr (£+i£* 6) 1.1.34 where <r= r m e 1.1.35 e i s the conductivity. In many situations i t i s possible, f o r mathematical convenience, to replace the usual Ohm s law 1.1.34 by the T Ohm s law f o r i n f i n i t e conductivity 1.1.5. I*et L be a T length representative of the dimensions of the magnetic f i e l d , and ir a v e l o c i t y comparable with the v e l o c i t i e s present. Then, provided <3-»_! 1.1.36 1.1.34 may be replaced by 1.1.5 (Cowling, 1957, p. 6 ) . This condition must"not v i o l a t e the condition implied by 1.1.33, namely ar« 1.2 Tk^ 6 A Transformation of the Electromagnetic 1.1.37 Equations. The description of the source of the electromagnetic f i e l d i n terms of J and e i s not convenient f o r many purposes because J" and £ are not independent but must s a t i s f y the equation of continuity, obtained by combining the divergence of 1.1.3 with + £ IT)-*- V.(j 1.1.4 £J = dt O 1.2.1 In the present case i t i s much more convenient to use of the four quantities J and £ subject to the instead constraint 1.2.1, three quantities P , the p o l a r i z a t i o n , defined by J fLf = + K7*(P* if)+ (v- P)lf 1.2.2 1.2.3 £ = -V.P which w i l l take care of 1.2.1 between £ and J,£ identically. The r e l a t i o n i s formally the same as that between the d i e l e c t r i c p o l a r i z a t i o n and charge (Panofsky and 1955, P. Phillips, 147). More insight into the meaning of P i s obtained i f one integrates 1.2.2 over an open s u r f a c e ^ which moves with the f l u i d i . e . : always consists of the same f l u i d elements Jj.otff- - J" |J£ +• v*(p* if) +• (V.P) tfj- do- The l e f t hand side of 1.2.4 i s simply the current through the surface Z, which i n turn i s the rate at which conduction charge <? -nd. passes through Z , Co St COTldt. ( Qcot>&. i s the t o t a l charge which passes through £ i n the d i r e c t i o n of the normal TI ). (Panofsky and P h i l l i p s , 1955, The r i g h t hand side can be shown p. 144-147) to be the rate 1.2.4 Figure I I - to follow page 11 The Change i n the Flux of P Through a Moving Surface X. Is Due to Both the Change i n the Vector F i e l d and the Motion of the Surface at which the f l u x of P through Z i s changing bearing i n mind that t h i s change i n the f l u x of P arises from both the changes of the vector f i e l d P and the motion of the surface Z (Figure I I ) . Equation 1.2.2 i s then equivalent to 1.2.5 S i m i l a r l y , integrate 1.2.3 a closed surface Z over a volume JCI bounded by c 1.2.6 The l e f t hand side i s the net excess charge <? ex jfL. i n the volume The r i g h t hand side may be transformed using Gauss's theorem into a surface i n t e g r a l over Z c 1.2.7 It i s apparent from the r e l a t i o n s defining the polari z a t i o n f , equations 1.2.2 and 1.2.3, that P may be chosen to be an a r b i t r a r y function of space at any time, say t = o. The p o l a r i z a t i o n f o r a l l future times w i l l then be uniquely given. For example, one may choose P to be zero everywhere at t= o . The p o l a r i z a t i o n at a p a r t i c u l a r f l u i d point can then i n p r i n c i p l e be measured f o r a l l future times by the following method: at t = o one constructs an i n f i n i t e s i m a l element of surface near the point X„ i n the f l u i d , and as time increases, allows t h i s surface element to move with Figure I I I - to follow page 12 The I n f i n i t e s i m a l Surface d<r Used f o r the Measurement of P Moves with the F l u i d from the Point x 6 at t = o to the Point * at t -13- th e f l u i d ; one measures the amount of conduction charge which passes through the surface as i t moves; suppose that at time t the surface i s i n the neighbourhood of the point X, and has an area d<r(t) and normal -n(t) (Figure I I I ) ; using equation 1.2.5 one then has Second. * £(*,t)-T>(t) Since 5c5 t. , con< d<r(t) rift) , and d<r(t) are known, t h i s equation enables one to f i n d the magnitude of P i n the d i r e c t i o n -n/tr) at the point x at the time t . By choosing three surfaces, the normals of which are independent at time t , the complete p o l a r i z a t i o n vector Pfe,t) may then be determined. Because the p o l a r i z a t i o n P can be chosen to be an a r b i t r a r y function of space at t = 0 , i t i s determined only up to a gauge transformation where A s a t i s f i e s -L- -I- ^ f A x i f ) = o 1.2.9 and the i n i t i a l condition V ' ± m 0 1.2.10 Proceeding as above one can show that 1.2.9 and 1.2.10 imply that the flux of A through any open surface Z which moves with the f l u i d i s constant & \ t • * * • < > 1.2.11 Since the f l u x i s a measure of the number of l i n e s of A -14- passing through Z, 1 . 2 . 1 1 implies that the l i n e s of the gauge function A must move with the f l u i d . One frequently describes the s i t u a t i o n by saying that the l i n e s of the vector are "frozen" i n the f l u i d . In addition, because of 1 . 2 . 1 0 , the f l u x of A through any closed surface £ c vanishes 1.2.12 <b A -do- = 0 Z Thus there are no sources or sinks of the gauge function A. The magnetic induction B s a t i s f i e s an equation of the form 1 . 2 . 9 , as may be seen i f E i s eliminated from 1 . 1 . 1 by means of 1 . 1 . 5 , and further, because of 1 . 1 . 2 the i n i t i a l condition 1 . 2 . 1 0 i s met. frozen i n the f l u i d . Thus the magnetic f i e l d lines are This fact has played a very important part i n the development of magnetohydrodynamics (Alfven, 1950) In the present work i t provides a possible choice f o r the gauge function, namely 1.2.13 £ = <* B where <x i s an a r b i t r a r y constant. Using the representation 1 . 2 . 2 and 1 . 2 . 3 f o r the current and charge, the source equations of the electromagnetic f i e l d beome *\7x / — 6 - Pxcf) = V.(«A£+ P) - O ^-(K E-t-P) 0 1.2.14 1.2.15 The remaining Maxwell equations 1 . 1 . 1 and 1 . 1 . 2 serve to define the e l e c t r i c and magnetic f i e l d s i n terms of the electromagnetic potentials E = - v<t> at 1.2.16 - = v'xfl 1.2.17 6 and # being the vector and scalar potentials respectively. The potentials are determined only up to a gauge transformation fl'= fl-«7 1.2.18 X <j> -» <f>' = <f> - itb 1.2.19 X being an a r b i t r a r y function of the space coordinates and the time. Expressed i n terms of the potentials, the source equations 1.2.14, 1.2.15 are V* |~_L _ u-J = P x V- [K (-V# - O ^j«*(- t |£ j + p j = v ~- ^ | J+ 1.2.20 P J 1.2.21 O 1.2.20 and 1.2.21 together with the Ohm's law r e l a t i o n 1.1.5, which i n terms of the potentials becomes - V6 - i f + if* (v*Pi) = O 3t 1.2.22 - and suitable boundary conditions, specify the electromagnetic f i e l d i n an i n f i n i t e l y conducting medium. I t i s these equations which w i l l be derived from a v a r i a t i o n a l p r i n c i p l e i n Chapter 3. -xu- 1.3 A Transformation of the Hydrodynamical Equation. As was the case with the electromagnetic equations, equation 1 . 1 . 6 the hydrodynamical i s not i n a suitable form to be obtained from a v a r i a t i o n a l p r i n c i p l e . Here another advantage of expressing J i n terms of P becomes obvious, f o r by using a vector identity (Appendix A.2) and the fact that the l i n e s of 6 move with the f l u i d , i . e . : B s a t i s f i e s 1.2.9 express 1 . 1 . 6 and 1 . 2 . 1 0 , as a f i r s t step one can i n the form On taking the c u r l , 1 . 3 . 1 leads to the generalized vortex equation ^ £ +• V*(£«u~) 1.3.2 = O where the generalized v o r t i c i t y 5 =7«[ir^^r] has been introduced. 1.3.3 In the case of ordinary hydrodynamics, §=o , 1.3.2 implies that the f l u x of £ through any open surface reduces to the ordinary v o r t i c i t y V**r . Equation Z which moves with the f l u i d i s constant In other words, the l i n e s of £ are frozen i n the f l u i d . •X =o Further, since f - i d e n t i c a l l y , there are no sources or sinks of ^ (Section 1.2). The c i r c u l a t i o n theorem i s obtained as i n ordinary hydrodynamics by using Stokes' theorem to change the surface i n t e g r a l 1.3.4 into a l i n e i n t e g r a l around the closed curve P bounding Z i ^ ! r * i , f j . j 8 r .0 1.3.5 The only r e s t r i c t i o n s on P are that i t be closed and that i t move with the f l u i d . One may, f o r example, choose P to be a closed l i n e of 6, say P . G The second term i n 1.3.5 vanishes and one i s l e f t with (De, 1957) Al y.d = r o 1.3.6 To make the hydrodynamical equation 1.3.1 suitable, as a second step one introduces the generalized hydrodynamical or Clebsch potentials x, 07 , 5 defined by (Appendix B.l; Lamb, 1932, p. 248; and Ito, 1953) U-+ J.. Bxf ^ - v ^ - t - ^ v\ 1.3.7 These are p a r t i c u l a r l y convenient, because one can always v i s u a l i z e the l i n e s of the generalized v o r t i c i t y g = v*(tr+± §*pj = v-?*VS 1.3.8 as the l i n e s of intersection of the surfaces y= constant and 5= constant (Figure IV). The generalized Clebsch potentials reduce to the ordinary Clebsch potentials when the magnetic induction B vanishes. -18- As was the case with the electromagnetic potentials, the Clebsch potentials are determined only up to gauge transformations. I f P as well as the physical variables are s p e c i f i e d , the l e f t hand side of 1 . 3 . 7 i s completely determined. The transformations of the Clebsch potentials are then just the "ordinary" gauge transformations (see 1 . 3 . 1 # , 1.3.19, and 1 . 3 . 2 0 below). However, i t has been shown i n Section 1.2 that the p o l a r i z a t i o n i s not completely specified and may be transformed 1.3.9 r - f ' - e+A where the l i n e s of A move with the f l u i d and 1 . 2 . 1 0 ) . In general, i f f (equations 1 . 2 . 9 i s so transformed, the Clebsch potentials must also be transformed ix-*x', *}-*")', i f the physical variables i f , § , p are to remain One reads from equation 1 . 3 . 7 that X', unchanged. , fe' must s a t i s f y -V(V-*)+ «i'V*'-*t = j §* A 1.3.10 These combined transformations of the polarization and the Clebsch potentials w i l l be called "generalized" gauge transformations. The i n f i n i t e s i m a l generalized gauge transformations have the form SA 1.3.11 1.3.12 1.3.13 1.3.14 -19- and s a t i s f y + $y - Vy = ±. B*SA 1.3.15 where So. i s a small constant, and So.G, = -S% + S $ 1.3.16 V To obtain additional conservation laws, the following p a r t i c u l a r choices of S>]\ are u s e f u l : i. I f P as well as the physical variables remains unchanged, SA and the right hand side of 1.3.15 vanish, and one i s l e f t with v Stx.Gf +• %«! v$ - Vy = o 1.3.17 The vector equation 1.3.17 may be regarded as a system of three l i n e a r inhomogeneous algebraic equations i n the two unknowns § 7 , 5 5 . minant In order that a solution exist the determust vanish. That i s , G must be a function afx, <i,z) only of ft) , $ , t ; the equation 1.3.17 has f o r a solution the ordinary i n f i n i t e s i m a l gauge transformations S v = - So. *5 1.3.is S§=So,£S SX= Sa/n ^ 1.3.19 _ 5OLQ 1.3.20 The corresponding f i n i t e transformations are just the canoni c a l transformations (Appendix B.2; It8, 1953). ii. Since the l i n e s of § move with the f l u i d , one may- choose S A proportional to B : 1.3.21 SA=S>«8 where Soc i s an i n f i n i t e s i m a l constant. For t h i s choice the right hand side of 1.3.15 again vanishes and the most general solution has the form 1.3.18, 1 . 3 . 1 9 , iii. 1.3.20. Since the lines of £ move with the f l u i d , one may choose 6A proportional to 1.3.22 SA= $pg = Sf VyxVS where &p i s an i n f i n i t e s i m a l constant. For t h i s choice 1 . 3 . 1 5 becomes V So.£ + (b^ + i£ e . v y + £& s.v^v v =o 1.3.23 with a solution 8X = -H<nB-V$ 1.3.24 e = e 1.3.25 e.Vm " 7 1.3.26 SS = - £ £ e.VS f The most general solution of 1 . 3 . 2 3 d i f f e r s from t h i s particular solution only by an ordinary gauge transformation 1.3.18, 1 . 3 . 1 9 , 1.3.20. The transformations given by 1 . 3 . 2 5 and 1 . 3 . 2 6 can be more r e a d i l y interpreted i f one uses Taylor's theorem to write them i n the form Figure V - to follow page 20 To Construct the Surface ^'=<x t f o r the Transformation 1.3.27, One Lays Out Vectors ^ 6 from the Surface ^ = a. and Joins the End Points -21- o?'(x,t) = <>?(x- *f e,t) 1.3.27 $'(x,t)= $(*-^§,t) 1.3.28 To obtain «j' and §' at a space-time point (x,t) one replaces the value of Ss and § at t h i s space-time point by t h e i r values a distance T 6 away on a l i n e of the magnetic induction B. Given a surface = constant = a. (say), one can construct the surface nj'ea. as follows: from the surface >y=a. draw vectors ^ B ; the surface consisting of the end points of these vectors i s ^' = a. (Figure V). S i m i l a r l y , from the surface £ = constant = £> (say), one can construct the surface The transformations £> . 1.3.27 and 1.3.28 thus correspond to a displacement of the surfaces ry= constant and 5= constant along the l i n e s of B. Under the two previous generalized gauge transformations 1.3.18-1.3.20 and 1.3.21, the generalized v o r t i c i t y £ does not change. On the other hand, under the generalized gauge transformation 1.3.22, 1.3.24, 1.3.25, 1.3.26 the generalized v o r t i c i t y transforms as follows: fe* "* « = to + S P 1.3.29 &*±<) Since the l i n e s of both the untransformed and transformed generalized v o r t i c i t y move with the f l u i d , one infers from 1.3.29 that the l i n e s of v «(j 1.3.30 must also move with the f l u i d (Appendix A.2). It remains to obtain the f i e l d equations s a t i s f i e d by the Clebsch potentials. Since the lines of £ are the l i n e s -d2- of i n t e r s e c t i o n of the surfaces 7 = constant and S= constant, and since i t has already been shown that the l i n e s of ^ move with the f l u i d , one should be able to choose ^ , § i n such a way that the surfaces 07= constant, ^= constant also move with the f l u i d . (Lamb, 1 9 3 2 , How t h i s may be done w i l l now be shown p. 2 4 S ) . Substituting 1 . 3 . 7 into 1 . 3 . 1 one obtains + rlL H# v§ 5 * - v = o 1 . 3 . 3 1 1 . 3 . 3 2 1 . 3 . 3 3 where j i = ' n - o l - ^ ( u - / i + # - ? x 5 | and — at = ^ - at + u-.^ i s the substantial derivative. equation £7 — 1 . 3 . 1 7 , Using the reasoning following one obtains the solution to 1 . 3 . 3 1 - = 1 = . 3 . 3 4 I . 3 . 3 5 ^Or where II i s a function only of 7 , 5 , and t . Apart from the fact that the l e f t hand sides of and 1 . 3 . 3 5 contain the substantial derivatives ^ 1 . 3 . 3 4 i n place of the ordinary time derivative -j|, these equations have the form of Hamilton's equations: 5 , ^ ,-0_ playing the roles of "coordinate", "conjugate respectively. momentum", and "HamiItonian" Further, the ordinary gauge transformations are i d e n t i c a l with the canonical transformations B.2). (Appendix The amilton-Jacobi theorem states that within the H sets of functions related by canonical transformations i t i s possible to f i n d a set f o r which the transformed vanishes (Goldstein, 1 9 5 0 ) . "Hamiltonian" In these variables, which henceforth w i l l be denoted simply b y ^ , *) , 5 1 . 3 . 3 6 1 . 3 . 3 7 1 . 3 . 3 8 " The f i r s t of these equations i s a generalization of the Bernoulli theorem of hydrodynamics, while the last two equations are a generalization of the Helmholtz vortex theorem. equation 1.1.9 Together they are equivalent to the usual dynamical 1 . 1 . 6 , and with the conservation of mass equation determine the motion of an i n f i n i t e l y conducting f l u i d i n an electromagnetic f i e l d . It i s these equations which xiri.ll be derived from a v a r i a t i o n a l principle i n Chapter 3. -24- CHAPTER 2 CLASSICAL FIELD THEORY 2.1 The Concept of a C l a s s i c a l F i e l d . It i s the purpose of t h i s chapter to review briefly- some of the ideas of c l a s s i c a l f i e l d theory. No attempt w i l l be made to derive the r e s u l t s from f i r s t p r i n c i p l e s as t h i s has already been well treated i n the l i t e r a t u r e . (The theorem which shows how to construct the conservation laws from the symmetry transformations of the action princ i p l e i s due to Noether, 1918. The notation and ideas i n t h i s chapter are based on the review a r t i c l e by H i l l , 1951, and on Weyl-, 1922). The concept of a f i e l d as a physical entity represents one of the most far-reaching ideas i n physics. The concept i s most e a s i l y grasped using the electromagnetic f i e l d as an example. Two another which may ways: charged p a r t i c l e s exert a force on one be considered to arise i n two distinct either the "action at a distance" picture, whereby the f i r s t p a r t i c l e d i r e c t l y exerts a force on the second p a r t i c l e located some distance away; or the " f i e l d " picture i n which the presence of the f i r s t p a r t i c l e gives r i s e to an electromagnetic f i e l d which then exerts a force on a second p a r t i c l e . In adopting the second alternative one passes from "action at a distance" to " i n f i n i t e s i m a l action". The f i e l d has physical r e a l i t y because on demand one can measure i t s value at any given point i n space and at -o- any time, namely by introducing a test p a r t i c l e having a charge s u f f i c i e n t l y small that i t does not a f f e c t the o r i g i n a l f i e l d appreciably and by measuring the forces on it. However, the f i e l d concept goes even further than t h i s . The f i e l d i s considered to exist even though there i s no measuring apparatus present to detect i t . For example, consider a l i g h t bulb burning i n a completely sealed box. One assumes that the box i s f i l l e d with l i g h t . However, t h i s cannot be v e r i f i e d u n t i l some measuring device i s introduced. For some purposes i t i s useful to widen the concept of a f i e l d to include quantities which, although not d i r e c t l y observable themselves, give r i s e to observables. Such i s the case i n electromagnetic theory i n which one introduces i n place of the physically observable e l e c t r i c and magnetic f i e l d strengths the unobservable electromagnetic potentials. There i s also the case of hydrodynamics i n which one i n t r o duces i n place of the observable v e l o c i t y the unobservable Clebsch potentials. Mathematically, i t makes no difference whether or not the f i e l d functions are observable. In order to treat the most general possible case, denote a l l the f i e l d functions, observable or not, by *Y" (oc Also, since i t i s as easy to work with r\ coordinates as i t i s with the four physical coordinates, namely, the three space coordinates and the time, denote the coordinates by ^ ( k - . ^ , ,-n). 2.2 The F i e l d Equations. One obtains the f i e l d functions by solving a system of p a r t i a l d i f f e r e n t i a l equations, the f i e l d equations, subject to certain boundary or i n i t i a l conditions. physical f i e l d equations are obtainable or action p r i n c i p l e . "lagrangian" Most , from a v a r i a t i o n a l That i s , i t i s possible to f i n d a function £ of the f i e l d functions, t h e i r f i r s t derivatives with respect to the coordinates (the r e s t r i c t i o n to f i r s t derivatives, although not necessary, w i l l be assumed i n the following), and the coordinates such that the f i e l d equations r e s u l t from the following prescription: one constructs the action i n t e g r a l I over a volume of coordinate s p a c e i l f o r which d(x) i s the i n f i n i t e s i m a l element 2.2.1 and requires that t h i s integral be stationary for small changes i n the state functions which vanish on the boundary of the volume. A necessary condition for the action i n t e g r a l to be stationary i s that the f i e l d functions s a t i s f y the Euler-Lagrange equations 2.2.2 These N equations are the f i e l d equations. 2.3 The Conservation Laws. By means of the functional relations v«^ '«(x<) r = F*(r(x),*) 2 , 3 , 1 one can introduce new coordinates x' and new f i e l d functions ** . One demands that the action be numerically unchanged // under t h i s transformation. This requires that — '.*') *<*' = *(r, Z?,x)d(x) } ^ f ' / V ; 2 In t h i s equation the unprimed variables ^ . 3 . , and X on the r i g h t hand side are expressed i n terms of the primed variables by means of the functional transformation 2 . 3 . 1 . In general, these new coordinates x' and new f i e l d functions ^ ' w i l l bear l i t t l e resemblance to the o r i g i n a l coordinates x and f i e l d functions Y, * v /# I f , however, X' and describe another possible motion of the system then the transformation i s a symmetry transformation and the f i e l d i s invariant with respect to t h i s symmetry transformation. In order that the transformation 2 . 3 . 1 be a symmetry transformation, x' and must s a t i s f y a system of f i e l d equations having the same functional form as the o r i g i n a l f i e l d equations. lagrangian This w i l l be true provided the transformed , defined by 2 . 3 * ' ) + • . 2 can be written , ^ 2 . 3 where f& i s some set of TI functions of the coordinates X' . 3 2 -28- and the f i e l d functions t " . I f the functions J l * vanish, then <£ i s form invariant. The symmetry transformations f a l l into two classes: (i) the space-time transformations, and ( i i ) the gauge transformations. Any physical f i e l d must be invariant with respect to the space-time transformations, i . e . : a displacement of the o r i g i n of the cartesian space axes, a displacement of the time o r i g i n , a rotation of the cartesian space axes, and a transformation to a cartesian frame moving with a uniform v e l o c i t y with respect to the o r i g i n a l frame. Because of the homogeneity of space and time, the choice of the o r i g i n of one's cartesian space axes and time o r i g i n i s immaterial; because of the isotropy of space, the orient- ation of one's cartesian space axes i s also immaterial. F i n a l l y , because of the p r i n c i p l e of r e l a t i v i t y , moving frames are equally good observing frames. a l l uniformly The choice of a particular coordinate frame i s solely one of physical or mathematical convenience. Gauge invariances arise when the f i e l d varia bles are not completely determined on s p e c i f i c a t i o n of the physical observables. The theory w i l l be gauge invariant because any physical p r e d i c t i o n must not depend on a p a r t i c u l a r choice of gauge. Bearing i n mind the difference between the two types of transformations, one may treat them both using the same formalism. For transformations a r i s i n g continuously -29- from the i d e n t i t y transformation, i t i s s u f f i c i e n t to study the i n f i n i t e s i m a l transformation = 4 " * f x ) + 6r"(x) 2 . 3 . 4 the f i n i t e form being obtained by i t e r a t i o n . I t can be shown by use of the action principle that with each i n f i n i t e s i m a l symmetry transformation one can associate a d i f f e r e n t i a l conservation equation *?*}8K'+*L Ajfes -2L. k ax ft d 8v«+sn. kl=o 2 . 3 . 5 *h This theorem i s usually referred to as "Noether's theorem". In physics the coordinates X^ are the time t and the three cartesian space coordinates X, <-(,£. Eouation 2 . 3 . 5 2 . 3 . becomes *f +• V.S = o 6 where the "density" cr associated with the symmetry transformation 2 . 3 . 4 i s at at at and the "current density" S i s s= -1^. a^n" i ^ ^ i t + / * sx - l £ SX.<7<K«)-<S<K"+-SJ2 at v, - a*7f« ~ / av^" On integrating 2 . 3 . 6 2 . 3 . 8 over a volumeSL bounded by a closed surface Z. and using Gauss's theorem one obtains the i n t e g r a l conservation law i - f cr d(x) = -i S-d<r 2.3.9 Note that the "density" cr and the "current density" S are not uniquely determined. The conservation law 2.3.6 remains unchanged i f one adds to cr any quantity of the form V- o. , provided one subtracts |^ from S. This non-uniqueness can often be used to simplify the expressions f o r cr and S. In p a r t i c u l a r , i f cr can be put i n the form V- b , then 2.3.6 i s not a true conservation law, because the conserved quantity can be made to vanish. The d i s t i n c t i o n between the space-time invariances and the gauge invariances also appears i n the conservation laws which result from them. The former lead to the conser- vation laws of momentum, energy, angular momentum, and center of mass. On the other hand the l a t t e r may or may not lead to a conservation law. To i l l u s t r a t e t h i s difference i n the conservation laws a r i s i n g from space-time and gauge invariances, consider the well-known cases of the electromagnetic f i e l d and the hydrodynamical f i e l d . As a t y p i c a l space-time transformation, take the i n f i n i t e s i m a l displacement of the time o r i g i n t - + t ' = t 2.3.10 + it Both the electromagnetic and hydrodynamical f i e l d s are invariant under t h i s transformation, which i n both cases leads to the conservation of energy theorem. electromagnetic f i e l d this i s For the -31- 2.3.11 whereas, f o r the hydrodynamical f i e l d t h i s i s 2.3.12 v. dcr In both these conservation laws the conserved namely the t o t a l energy i n the volume Si, observable. quantity, i s a physical This w i l l be true for a l l conservation laws r e s u l t i n g from space-time transformations. In contrast, now consider the gauge transformations. I f one considers the electromagnetic potentials as the f i e l d variables, the electromagnetic f i e l d i s invariant under the gauge transformation 1.2.18, 1.2.19. S i m i l a r l y , i f one considers the density and Clebsch potentials to be the f i e l d variables, the hydrodynamical f i e l d i s invariant under the ordinary gauge transformation of the Clebsch potentials, the i n f i n i t e s i m a l form of which i s given by 1.3.18-1.3.20. Gauge invariance of the source free electromagnetic does not lead to a conservation law. That i s , the field conserved quantity can be made to vanish by adding a divergence expres sion to the "density" <r. On the other hand, gauge invariance of the hydrodynamical f i e l d leads to the conservation law 2.3.13 where Cf - i s the generating function of the i n f i n i t e s i m a l gauge transformation. By choosing , equation 2.3.13 leads to the conservation of mass equation, and the conserved quantity i s a physical observable, the mass of f l u i d i n the volumell. In general, however, the conserved quantity- w i l l not be a physical observable, since i t s value w i l l depend on the p a r t i c u l a r choice of gauge one adopts f o r the Clebsch potentials. -33- GHAPTER 3 THE ACTION PRINCIPLE IN MAGNETOHYDRODYNAMICS 3.1 Construction of the Lagrangian. To apply the techniques of Chapter 2 to the s p e c i f i c case of magnetohydrodynamics one must f i r s t obtain the appropriate lagrangian. Since no general technique f o r i t s construction i s a v a i l a b l e , one must resort to a "cut and try" procedure. There are, however, known results which make t h i s task simpler; i n p a r t i c u l a r , laerangians f o r the electromagnetic (Panofsky and P h i l l i p s , 1955, hydrodynamical (Bateman, 1932, P. 369) p. I64 and Ito, 1953) are well-known. One only has to combine these two and fields lagrangians i n a suitable manner to obtain a lagrangian f o r the magnetohydrodynamical f i e l d . In order to discuss some d e f i n i t e case, i n t h i s chapter assume that the motion of the f l u i d i s i s e n t r o p i c . Further, assume that the external conservative f i e l d $ vanishes. With minor modifications one may also treat, i n addition to electromagnetic i n t e r a c t i o n s , interactions with a s p e c i f i e d external non-electromagnetic f i e l d , or with a self-produced g r a v i t a t i o n a l f i e l d . The following discussion i l l u s t r a t e s the method by which a lagrangian suitable f o r magnetohydrodynamics may be constructed. For the case of the electromagnetic f i e l d i n free space a lagrangian i s 3.1.1 €m where £ and B are expressed i n terms of the vector and scalar potentials A and <i> by means of 1.2.16 and 1.2.17. of jCen with respect tofland 0 Variation gives the Maxwell source equations 1.1.3 and 1.1.4 with the current and charge densities J and £ set equal to zero. In an i n f i n i t e l y conducting l i q u i d metal moving with a v e l o c i t y y, the Ohm's law r e l a t i o n 1.1.5 must be added to the Maxwell source equations 1.1.3 and 1.1.4. This additional constraint on the potentials can be incorporated into the v a r i a t i o n a l p r i n c i p l e by adding to £ , the term en £•(£•*• y*§) where P i s a lagrange m u l t i p l i e r 3.1.2 Variation of s?e mt with respect tofl, ^ , and P then gives the Maxwell source equations 1.2.20, 1.2.21, and the Ohm's law r e l a t i o n 1.2.22 provided P i s interpreted as the polarization. In addition to providing the correct electromagnetic equations, a suitable action p r i n c i p l e for magnetohydrodynamics must also provide the correct hydrodynamic equations. Considering u- to be an independent variable (as was done in 3.1.2) a lagrangian which gives Clebsch's transformed hydrodynamical equations i s 3.1.3 where f , ^ » 7 , ^ , and \r are a l l varied independently. -35- Adding 3.1.2 and 3.1.3 one obtains a lagrangian for the magnetohydrodynamical f i e l d , v a r i a t i o n with respect "to J3 > 0 > £ > p , ^ , " ? , ^ and hydrodynarnic giving the proper electromagnetic f i e l d equations, and variation with respect to cf giving the d e f i n i t i o n of the generalized Clebsch potentials £'= 1.3.7 «?\Ef-±2- isf+*P.(E+ a./*. ? " The-transformations S fa4->-\>rf-(&de} b>t ' Dt 2. ~ j p* 3.1.4 r J discussed i n Chapter 1 were o r i g i n a l l y discovered using t h i s lagrangian. For some purposes, however, i t i s preferable that only the f i e l d variables in the action p r i n c i p l e . 6 , 0 , P , p , ; f , 7 , and £ appear In ordinary hydrodynamics the appropriate lagrangian i n t h i s case i s where i f i s an abbreviation f o r -Vf+yVS and 5 are varied and only p , T( , independently. This suggests that i n magnetohydrodynamics one t r y £ = *?ie-j x --L- 2. \e\ + x P.£+p["^3'-/wil--L ly/1Lat at 2. where £, 8 , and cf are expressed Cp. J p3 df7 J 3.1.6 i n terms of the electro- magnetic potentials, the Clebsch potentials, and the p o l a r i z a t i o n by means of 1.2.16, 1.2.17, and 1.3.7. < 3 , ^ , £ , p iT( 1 7 i 5 are to be varied independently. Only 3.2 The F i e l d Equations. To v e r i f y that it as given by 3.1.6 i s a suitable 2.2.2 lagrangian f o r the magnetohydrodynamical f i e l d , use to f i n d the v a r i a t i o n of £. with respect to A , <f> , P , p , (Appendix C.2) X t - y , ^ = ± (K E [ £ 1 -hP)- V*/-L B - P x o - ) = O 0 « V.(K E+P) =O 0 C<JP =£"-»- tfxS «- 2| M ? Dt N 3.2.1 3.2.2 = o 3.2.3 = o 3.2.4 =o f 3.2.6 = o 3 . 2 . 7 2 . These equations are equivalent to the f i e l d equations of magnetohydrodynamics: i . e . : equations 3 . 2 . 1 and 3 . 2 . 2 are the source equations of the electromagnetic f i e l d and 1 . 2 . 2 1 ; 3.2.3 i s the Ohm's law r e l a t i o n when combined with 3 . 2 . 6 , 1 . 2 . 2 2 ; . 2 0 3.2.4, i s the generalized Bernoulli equation 1 . 3 . 3 6 ; 3 . 2 . 5 i s the eauation of continuity 3.2.6 1 1 . 1 . 9 ; and 3 . 2 . 7 when combined with 3 . 2 . 5 are the generalized Helmholtz vortex equations 1 . 3 . 3 7 and 1 . 3 . 3 # . Thus the lagrangian it as given by 3 . 1 . 6 i s a suitable laeran^ian for the magnetohydrodynamical f i e l d . 3.3 The Conservation Laws. In t h i s section the conservation laws for magneto- hydrodynamics a r i s i n g from the ordinary space-time invariances of the action p r i n c i p l e , and from the various gauge invariances introduced i n Chapter 1 are discussed. a symmetry transformation, one can immediately Given write down the corresponding conservation law, using the techniques of c l a s s i c a l f i e l d theory (Chapter 2 ) . In general, the conser- vation laws found d i r e c t l y i n t h i s manner must be rewritten i n order to bring them to a more conventional form. Since the calculations involved are lengthy, only the f i n a l form of the conservation laws are given i n t h i s section. The d e t a i l s of the calculations can be found i n Appendix C.3. The space-time transformations: a displacement o r i g i n of the space axes, a displacement of the of the time o r i g i n , a rotation of the space axes, and a transformation to a frame moving with a uniform v e l o c i t y with respect to the o r i g i n a l frame are treated f i r s t . 3.3A The Space-Time Invariances i. The action p r i n c i p l e i s invariant with respect to an i n f i n i t e s i m a l displacement of the o r i g i n of the space axes x -» x' = x +- sx where X i s the vector having f o r components the three cartesian coordinates and £X an i n f i n i t e s i m a l constant vector giving the magnitude and d i r e c t i o n of the 3.3.1 38- displacement. The lagrangian i s form invariant. The i n t e g r a l conservation law which r e s u l t s from t h i s invariance property i s the conservation of momentum theorem of magnetohydrodynamics 2. f (?<S+ K ExB).&X a d(x) +• 'is lE{ -n - K(,EE••rr +• Jx I8l -n - JL 8 z B-T\ ). 6x do- 3.3.2 Dropping the a r b i t r a r y constant vector &x J the term following the time derivative i s the sum of the momentums of the f l u i d and the electromagnetic f i e l d i n the volume IL. The f i r s t term i n the surface integral is the rate at which f l u i d momentum i s carried across the surface Z into the volume H; the second term i s the t o t a l force exerted on the f l u i d i n Si by outside hydrodynamic pressure; the remaining terms are the i n t e g r a l of the Maxwell stress tensor over Z, ii. The action p r i n c i p l e i s invariant with respect to an i n f i n i t e s i m a l displacement of the time o r i g i n "fc-»"t=t + S't / 3.3.3 where St i s an i n f i n i t e s i m a l constant. The lagrangian i s form invariant. The i n t e g r a l conservation law which r e s u l t s from t h i s invariance property i s the conservation of energy theorem of magnetohydrodynamics -39- ^ ^ i ^ ' ~ ' = l ~ " - + f j " ^ U--T] "f- - i - §X - 8 -Tj J <St 3 . 3 . 4 fli<3- Dropping the a r b i t r a r y constant -St , the term following the time derivative i s the sum fluid, of the k i n e t i c energy of the the i n t e r n a l energy of the f l u i d , and the energy of electromagnetic f i e l d i n the volume SI, the The f i r s t term i n the surface i n t e g r a l i s the rate at which k i n e t i c energy is carried into i l through Z; on writing f<*£ = j> is 3 . 3 . 5 dp + f> the sum the second term, as can be seen of the rate at which external hydrodynamic pressures do work on the volume of f l u i d and the rate at which i n t e r n a l energy i s carried into XL through £ ; the t h i r d term, being the i n t e g r a l of the Poynting vector over the surface Z, i s the rate at which electromagnetic f i e l d energy i s carried into Si through Z. iii. The action p r i n c i p l e i s invariant with respect to an i n f i n i t e s i m a l rotation of the space axes x->x/= x + s 3 . 3 . 6 # where b& i s an i n f i n i t e s i m a l constant vector lying i n the direction of the axis of the rotation and equal to the angle of the rotation. A and P transform according to The having a magnitude vector state functions -4U- 5 fl' = g + fl* 3.3.7 f -* £ ' « P+ P«Sg 3.3.8 The lagrangian i s form invariant. The i n t e g r a l conservation law which results from t h i s invariance property i s the conservation of angular momentum theorem of magnetohydrodynamics — f Cf at H" + K £ x S ) x X . 8 0 e etfx) = — ^ ^p <J" tf-TJ + P"0 z + !<«|E| TI-K f£.Ti X 6 (S^-n-lBS.-nlxX.Se d<r 3.3.9 Dropping the a r b i t r a r y constant vector SB, the term following the time derivative i s the sum of the angular momentums of the f l u i d and electromagnetic f i e l d i n the volume SI, The f i r s t term following the surface i n t e g r a l i s the rate at which f l u i d angular momentum i s carried into SL through 21; the second term i s the torque exerted by outside hydrodynamic pressures on the volume of f l u i d i n i l ; the remaining terms are the i n t e g r a l of the cross product of the Maxwell stress tensor with X over Z. iv. In magnetohydrodynamics one often neglects the charge convection current £<s and the displacement current ^.(K0E) i n the source equation 1.1.3, the reason being IT* that these terms are of order —- other terms i n the equation. This can be accomplished K i n comparison with the through the action p r i n c i p l e by omitting the term — iEl 2- -41- in the lagrangian, or formally by setting K = O . The 0 modified action p r i n c i p l e i s then invariant with respect to the i n f i n i t e s i m a l g a l i l e a n transformation (transformation to a frame moving with a uniform v e l o c i t y with respect to the o r i g i n a l frame) x/= x + Scf t 3.3.10 0 where Sy" i s an i n f i n i t e s i m a l constant vector giving the 0 r e l a t i v e v e l o c i t y between the two frames. The Glebsch and electromagnetic scalar potentials transform according to 3 <t> + Stf -B - 3 - n 3.3.12 4 The modified lagrangian i s form invariant. The integral conservation law which results from t h i s invariance property i s the center of mass theorem of magnetohydrodynami c s. )• S±r0 dM i . f ( at-fX ? Z + _L IB I11 -f} a/*. — B e.-n t ). Sir d<r /*• ~ " ' "" 3.3.13 The volume i n t e g r a l may be written t f f u- dfx)- f ? X dfx) SL J SL J = t f? i r J _n. d(x) - (pdM •in. SrC~ = (momentum) t — (mass)* (radius vector of center of mass) 3.3.H -42- In those situations i n which the surface i n t e g r a l vanishesj for example, i f y-.-n = o 3.3.15 B.-n = o 3.3.16 and P + — (S/ * = 3 constant 3.3.17 over Z, 3.3.13 becomes (momentum) t — 3.3.18 (mass) * (radius vector of center of mass) = constant Further, i f these conditions are f u l f i l l e d , the conservation of momentum theorem gives, neglecting the charge convection and displacement currents momentum = constant 3.3.19 Also i t w i l l be shown that mass = constant 3.3.20 Then, 3.3.1$ implies that the center of mass moves with a constant 3.3B i. velocity. The Gauge Invariances The action p r i n c i p l e i s invariant with respect to an i n f i n i t e s i m a l gauge transformation of the electromagnetic potentials -43- fl ^ - * fl'= -» A'- fl + V&\ _ A 3 . 3 . 2 1 3 . 3 . 2 2 at where 8A i s an i n f i n i t e s i m a l function of the space coordinates and the time. The lagrangian i s form invariant. The invariance property 3 . 3 . 2 1 , 3 . 3 . 2 does not give 2 a true conservation law, because the conserved quantity may be made to vanish by adding a divergence (Section 2 . 3 ) . expression This i s not surprising, since usually i n discussing the electromagnetic f i e l d with sources, invariance under the transformation 3 . 3 . 2 1 conservation of charge equation and 1 . 2 . 3 1 . 3 . . 2 2 leads to the In our case, however, 1 . 2 . 1 has been s a t i s f i e d i d e n t i c a l l y by introducing the polarization. ii. The action principle i s invariant with respect to an i n f i n i t e s i m a l ordinary gauge transformation of the Clebsch potentials np-t^'^yf S S o . + aS * S Zo. - v 3 . 3 . 2 3 . 3 . 2 4 3 . 3 . 2 5 by where Q i s an arbitrary function of ^ and and 8 a. i s a small constant. The lagrangian i s form invariant. The i n t e g r a l conservation law which results from t h i s 3 -44- invariance property i s - ~ ^ S<xp § d(x) = ^ &tKf> Q <S T! 3.3.26 d<T Set Q=-l and cancel the small constant So. on both Equation 3.3.26 becomes the conservation of mass sides. theorem ± f? d(x) = -<£ -u--TJ da- 3.3.27 e The term following the time derivative i s the mass of the f l u i d i n the volume SI. The surface i n t e g r a l i s the rate at which mass i s carried into SI through Z. I f v.r\ vanishes, the mass of the f l u i d i n SI i s constant. Set Gi = -y and Q = -t. Equation 3.3.26 gives an alternate formulation of the generalized Helmholtz vortex theorem (equations 1.3.37 and 1.3.38) a. j p op at cJL(x) = — <D tpy i£ TJ dLa~ J L J " pS d(*) = - £ p5 tr.-T) d<r iii. 3.3.28 3.3.29 The action p r i n c i p l e i s invariant with respect to an i n f i n i t e s i m a l gauge transformation of the p o l a r i z a t i o n P-*P'=P + S<*B 3.3.30 The lagrangian transforms according to 3.3.31 The integral conservation law which r e s u l t s from t h i s invariance property i s (Woltjer, 195$) i-f ctfx) =-<£ i£5 (E*f\-t\ + <f> B-r>) olcr fl-B atj^ z. J z 2 . v - _ 3.3.32 - The i n t e g r a l conservation law 3.3.32 i s e s s e n t i a l l y another way of saying that e l e c t r i c and magnetic f i e l d s £ and 6 are perpendicular i n an i n f i n i t e l y conducting f l u i d . iv. The action principle i s invariant with respect to an i n f i n i t e s i m a l gauge transformation of the polarization and the Clebsch potentials p_* p'= p S(9 S 3.3.33 + X~*x'= ?(y / & ? l£ /» S-v? 3.3.34 = y - |£ B.Vnj 3.3.35 - i£ B-v^ 3.3.36 The lagrangian transforms according to + i ? <V-§) The integral conservation law which results from t h i s invariance property i s (Woltjer, 1958) j | J ~ f u-. 6 Ji(x) S =-j £ § x Lf.-n |f*p + JL /u-|*J § -n J d<r 3.3.38 -46- One now asks whether t h i s set of conservation laws i s complete. To be more precise, i s i t possible to f i n d more independent conserved quantities which are constructed only from the f i e l d quantities A , and t h e i r derivatives? , P , p , ^ , ^ 7 , and 5" It i s possible to generalize some of the preceeding r e s u l t s . For example, Sp i n 3.3.38 can be an a r b i t r a r y function of ny and §. However, the author has not been able to find any more e s s e n t i a l l y d i f f e r e n t conservation laitfs. I t i s f e l t that further conservation laws, i f they exist, must arise from the symmetry transformations resulting from a more complete solution of the generalized i n f i n i t e s i m a l gauge transformation, equation 1.3.15. -47- CHAPTSR L ILLUSTRATIONS It i s not the purpose of t h i s thesis to solve s p e c i f i c problems i n magnetohydrodynamics, nor to present a scheme whereby certain classes of problems can be solved. The purpose i s to show how general theorems applicable to any magnetohydrodynamic system, such as conservation laws, follow from a single p r i n c i p l e , the action p r i n c i p l e . To formulate the action p r i n c i p l e , a new way of writing the magnetohydrodynamic equations i n terms of the electromagnetic potentials, the p o l a r i z a t i o n , and the Clebsch potentials has been developed. Whether or not t h i s reformulation of magnetohydrodynamics w i l l prove useful for solving s p e c i f i c problems has not been determined. At the moment, i t seems that no great s i m p l i f i c a - t i o n can be achieved by using t h i s formulation. On the other hand, i t i s worth-while to conclude t h i s thesis by setting up and solving a few simple problems i n magnetohydrodynamics using this new formulation. No new problems w i l l be solved, no new results w i l l be obtained; the purpose of this chapter being merely to show what the new variables^look "like i n certain s p e c i f i c cases. In the f i r s t section the general magnetohydrostatic problem, that i s , the problem i n which the f l u i d v e l o c i t y vanishes, i s treated. It i s shown that the only non-vanishing Clebsch potential i s the scalar potential X •> a n c * that, at any instant the surfaces n( constant are the surfaces s = - i f o— constant. In the second section, small amplitude oxcillations, with a s p e c i f i c application to Alfven waves, are discussed. Here again, one finds that the only non-vanishing Clebsch potential i s the scalar potential, so that the perturbed system i s " i r r o t a t i o n a l " , i n the sense that the generalized v o r t i c i t y ^ vanishes. In the f i n a l section, a situation i n which a l l three Clebsch potentials are non-vanishing i s set up and solved. This example i s based on an exact stationary solution of the magnetohydrodynamic equations i n which the f l u i d v e l o c i t y i s everywhere p a r a l l e l to the magnetic f i e l d and proportional to i t (Chandrasekhar 1956). 4.1 Magnetohydrostatics. In the next section a problem involving small amplitude o s c i l l a t i o n s of an incompressible f l u i d about a steady state i n which the f l u i d v e l o c i t y vr vanishes w i l l be treated. Before discussing t h i s problem, i t i s worth-while to consider what r e s t r i c t i o n s are imposed on the currents, magnetic J f i e l d s , and Clebsch potentials by the condition that the f l u i d be i n steady state. These r e s t r i c t i o n s are discussed using the representation of the velocity i n terms of the Clebsch potentials. If the f l u i d v e l o c i t y sT vanishes, the e l e c t r i c f i e l d E i n the f l u i d also vanishes because of Ohm's law, equation 1.1.5. Faraday's law, equation 1.1.1, then shows that the magnetic induction B i s independent of time. The charge density £ i n the f l u i d vanishes because of the source equation 1 . 1 . 4 The current density J~ i n the f l u i d i s . related to the magnetic induction B through the source equation ^«^T - 1 r s ) . 1 . 3 , which i n t h i s case reduces to j = 4 . 1 . 1 The polarization P can be found from the defining equations 1 . 2 . 2 and 1 . 2 . 3 which simplify to £ £ = Jfx) 4 . 1 . 2 and These equations have the solution ?<X,t)= J(6)t + Pjx) The polarization at t = o kil.b f Pjx), i s r e s t r i c t e d by the condition In the case of vanishing f l u i d velocity the hydrodynamical equations, that i s , the Bernoulli equation and the Helmholtz vortex theorem, equations 1 . 3 at . 3 = # *y = o g =o . 3 . 3 7 . 3 . 3 6 and reduce to , f 1 1 + $ 4 . 1 . 6 4 . 1 . 7 4 . 1 . 8 , These equations can be integrated immediately to give X-(f + $)t +*„(*) 4.1.9 in which ^ f x ) i s the scalar Clebsch potential at t = o , and Wfc) 4.1.10 4.1.11 F i n a l l y the condition that the f l u i d v e l o c i t y vanishes becomes, i n terms of the Clebsch potentials, £_\7fp+p $J +- J* E|Jt -VXo +7^5 + 1 P,x6 = O 4.1.12 If one d i f f e r e n t i a t e s equation 4.1.12 with respect to t one obtains the usual magnetohydrostatic equation (Cowling, 1957, p. 24), Jx S = vfp + f i ) 4.1.13 which shows that the electromagnetic Lorentz force T« 6 i s balanced by the hydrostatic pressure gradients i n the f l u i d . If one substitutes this r e s u l t back into 4.1.12, one obtains the following r e s t r i c t i o n on the i n i t i a l values of the scalar Clebsch potential X° , the polarization P , and the 0 values of the " r o t a t i o n a l " Clebsch potentials 7 and 5 If one compares this result to equation 1.3.10 which gives the generalized gauge transformations of the polarization and Clebsch potentials, i t i s apparent that one can always choose a gauge i n which x° , 7 , ^ , a n d £*<> H & vanish. In t h i s gauge the surfaces X~ constant w i l l , at any -51- instant, coincide with the surfaces "f' + i? = constant. As time increases, the surfaces X- constant move i n the d i r e c t i o n - (~f v + &), a p a r t i c u l a r surface moving i n such a way that (•^-t-§)t 4.2 remains constant. Alfven Waves. In t h i s section the so-called Alfven waves i n an incom- pressible, unbounded f l u i d are discussed (Alfven, 1950, p.76). For s i m p l i c i t y , the charge convection and displacement currents are neglected i n the Maxwell source equation. Before r e s t r i c t i n g ourselves to the s p e c i f i c case of Alfven waves, the general theory of small amplitude o s c i l l a t i o n s about an i n i t i a l state i n which the magnetic f i e l d i n the f l u i d i s a constant S 0 i s treated. The analysis i n the preceding section shows that i n this i n i t i a l state the polarization P and Clebsch potentials x>") > a n d ^ vanish. The i n i t i a l pressure i n the f l u i d i s given by p=p»-p<§. To discuss small amplitude o s c i l l a t i o n s about t h i s i n i t i a l state one applies a perturbation to the system and denotes by SS r , SP, of the magnetic f i e l d B , By , and S£ the small deviations F i n the f l u i d , the polarization P, and the Clebsch potentials X» 7 J values. anc * £ from t h e i r initial Neglecting squares and higher powers of these small quantities, one finds that & § , SP , &y , $y , and F s a t i s f y the following equations: i f ^ F = V*(h<s* B0) = (S.-V)$£ 4.2.1 4.2.2 7. SB 4.2.3 at 4.2.4 e 4.2.5 £1*7 = o at 4 . 2 . 6 at V. Sir = o 4 . 2 . 7 i n which the f l u i d v e l o c i t y Sy i s given by S i r = _ <7S* _ J- 8 a x SP 4 . 2 . 8 —o The Helmholtz vortex theorem, equations 4.2.5 and 4.2.6, immediately shows that hy and S i are independent of time. Since $y and S S are zero i n i t i a l l y , they must remain zero f o r a l l time. This means that to f i r s t order i n small quantities, the system i s " i r r o t a t i o n a l " , i n the sense that the generalized v o r t i c i t y ^ vanishes. In order to solve the remaining equations, the simplest procedure i s to use the source equation 4.2.3 and the Bernoulli equation 4.2.4 to eliminate the variables SP and from the Faraday equation 4.2.1 and the equation of continuity 4.2.7. We s h a l l then be l e f t with a system of four equations involving only S § and Sp, F To accomplish t h i s , one d i f f e r e n t i a t e s the Faraday equation 4.2.1 with respect to the time, obtaining -53- = -(e.r)[r *& + J- B x a ] 4.2.9 The second step follows from the r e l a t i o n giving the v e l o c i t y i n terms of the scalar Clebsch potential and p o l a r i z a t i o n , equation 4.2.8. Next, one uses the Bernoulli equation 4.2.4 and the source equation 4.2.3 to replace d&X and asf at at i n equation 4.2.9 by Sp and p i/*. S§ ) . / F After some rearrangement equation 4.2.9 becomes The variables &P and S^f can be eliminated from the equation of continuity 4.2.7 using the same technique: one d i f f e r e n t i a t e s equation 4.2.7 with respect to the time, uses the Bernoulli equation 4.2.4, the source equation 4.2.3, and f i n a l l y the equation 4.2.2 f o r the divergence of S B p. . As a r e s u l t , one obtains The modified Faraday equation 4.2.10 and modified equation of continuity 4.2.11, together with equation 4.2.2, form an independent 6 B and S p . F system and may be solved f o r Substituting the solution into the source equation 4.2.8 enables one to solve thpse equations f o r the remaining variables & P , &% and S y . To i l l u s t r a t e the method, we look f o r solutions having the form of plane waves propagating i n an unbounded medium i n the d i r e c t i o n -j=- with a frequency w. That i s , we look f o r solutions of 4.2.10 and 4.2.11 i n which 5 6 and F L (k x - t) Sp have the space-time dependence e , the constant K being the propagation vector of the wave. assumed space-time dependence of S8F For this and Sp the modified equation of continuity 4.2.11 gives 2>P + J- e • s e c = o 4.2.12 In order to f i n d the dispersion r e l a t i o n , or r e l a t i o n between the frequency w and propagation vector k, we subs t i t u t e this value of Sp + 6 .$8 0 , together with the F assumed space-time dependence of & § , into the modified f Faraday equation 4.2.10, obtaining <o-= (e .kf /*<>? 4.2.13 a F i n a l l y , the equation 4.2.2 f o r the divergence of S S shows that the perturbation i n the magnetic f i e l d SS F F is perpendicular to the propagation vector fe- ^F ° s i.e.: 4.2.14 = the waves are transverse. There are two independent p o l a r i z a t i o n directions f o r the wave. I f the propagation vector fe i s not p a r a l l e l to the i n i t i a l magnetic f i e l d B , 0 these may be taken to be: vectors B and K, or (2) 6 6 0 the vectors B 0 (1) SB F F j . the plane of the n perpendicular to the plane of and k. A l l the variables can now be expressed i n terms of the perturbation i n the magnetic f i e l d &§ : F the perturbation Figure VI - to follow page 54 The Orientation of the Amplitude of the Vectors f o r an Alfven Wave (a) 68F (b) & § F i n plane of B perpendicular 0 and k to the plane of B 0 and k -5 5- i n the pressure Sp i s given by equation 4.2.12; the polari z a t i o n by the source equation 4.2.3 SP = — k* SB SF 4.2.15 the scalar Clebsch potential by the Bernoulli equation 4.2.4 and equation 4.2.12 &"X=. -. L_ B .SB 0 4.2.16 F and the f l u i d v e l o c i t y S y by the r e l a t i o n between the v e l o c i t y and the Clebsch potential and p o l a r i z a t i o n 4.2.$ Sir = _ g»-h § Bp. 4.2.17 The motion of the f l u i d i n the magnetic f i e l d B an e l e c t r i c field&E a generates which can be determined by Ohm's law 5E = §a* 8v = - J L d i S xSS 0 4.2.18 F The orientation of the amplitude of the vectors f o r the two modes of propagation i s shown i n Figure VI. 4.3 Discussion of a Particular Steady State Solution. In this section a s i t u a t i o n i n which the " r o t a t i o n a l " Clebsch potentials y and I do not vanish i s discussed. A p a r t i c u l a r time independent hydrodynamical solution of the magneto- equations f o r an incompressible, i n v i s c i d , and i n f i n i t e l y conducting f l u i d i s given by (Chandrasekhar, 1956) - = 7== ^ 4.3.1 Figure VII - to follow page 55 The Three C y l i n d r i c a l Coordinates -56- p + _ i _ (6| = p a 4 e The magnetic induction 8 must s a t i s f y the condition * § - . 3 . 2 1.1.2: o 4 . 3 . 3 Otherwise, S can be a completely a r b i t r a r y function of the space coordinates. The e l e c t r i c f i e l d £ i n the f l u i d vanishes because of Ohm's law 1.1.5. Then, because of the source equation 1.1.4, also vanishes. The current density J i s determined by the source equation the charge density 6 i n the f l u i d 1.1.3, which i n t h i s case reduces to 4 ?=jroV*§ . 3 . 4 In the remainder of t h i s section, an example of t h i s Let r , <p , and EL be the class of solutions w i l l be treated. three c y l i n d r i c a l coordinates, and l , r lf, and l z the unit vectors i n the r , <p , and 2 directions (see Figure VII). Suppose that the magnetic f i e l d has the form The v e l o c i t y y; i s given by 4 . 3 . 1 i « * 6 - ) l 4 f where u"(W i s the magnitude of the v e l o c i t y . . 3 . 6 The pressure p i s determined by 4 . 3 . 2 F i n a l l y , the current density T which w i l l produce this magnetic f i e l d i s given by the source equation 4 . 3 . 4 , which / Figure VIII - to follow page 56 The Current, Magnetic F i e l d , and VelocityD i s t r i b u t i o n i n the Case sr = ~ 8 = B(V)1 -7 l- i n t h i s case becomes A(re) X= — 1- 4 B(V)=v£^ As an example suppose that «ir . 3 . 8 . The above solution can then be approximated i n the laboratory by sending a constant current J = a. ^ <o i through a g c i r c u l a r cylinder of mercury which i s rotating about i t s axis as a r i g i d body at an angular v e l o c i t y <*> (Figure VIII). I t w i l l be i n s t r u c t i v e to see the form that the polari z a t i o n f and the generalized § take for the case Clebsch potentials X>7 » d a n B = B(r) %9 . The p o l a r i z a t i o n f i s defined, up to gauge trans- formations , by the equations £f V*(P*ir) = J = + at A (rB) 1 4 . 3 . 9 * oif and 7 , - p c 4 0 . 3 . 1 0 I t i s r e a d i l y v e r i f i e d that a p a r t i c u l a r solution of these equations for P i s P = Jt = — fi-S)l- 4 . 3 . 1 The l i n e s of P are straight l i n e s p a r a l l e l to the Z -axis. P increases l i n e a r l y with time. In order to calculate the generalized £ = V* (\r+ -^SXP), u+|§*f . vorticity one must f i r s t evaluate the expression Substituting expression one obtains f from equation 4 . 3 . 1 1 into t h i s 1 if + Sx p — — is 1^ •+• —I— Slif».6)l, 4 . 3 . 1 2 The generalized v o r t i c i t y £ i s then obtained taking the c u r l of both sides of equation g . v-x^ |exfj= + 4 . 3 . 1 2 . One finds that v.ir = J- 4 . 3 . 1 3 In t h i s case the generalized v o r t i c i t y reduces to the ordinary hydrodynamical v o r t i c i t y . The vortex l i n e s are straight l i n e s p a r a l l e l to the 2. - a x i s . The generalized Clebsch potentials 17 and ^ which determine the " r o t a t i o n a l " motion may now be calculated, up to gauge transformations, from the r e l a t i o n giving £ as the l i n e s of intersection of the surfaces 7 = constant =constant (equation = ' 3 7 = ~ -L A<W) f dr and 1.3.8) 4.3.14 1. ~* I t i s reasonable to assume that a solution of t h i s equation exists i n which y depends only on r and t , and £ depends only on r , <p and t : ry = fy (r, $ = $ ( r 4 t) i ? ) . 3 . 1 5 4.3.16 t ) I t w i l l now be shown that a solution of this form can i n fact be found. For the assumed dependence of y and £ on the coordinates, equation *?} dr 21 d<p = 4 . 3 . 1 4 reduces to J*(W) 4 . 3 . 1 . 3 . 1 7 dr In order to determine £ one d i f f e r e n t i a t e s equation with respect to <p and obtains 4 7 3^7 dr =o 4.3.18 3f=»- I f fy? does not vanish, or i n other words, i f v r i s not a constant, 5 must have the form & - a(r t)f t + t6-,t) 4.3.19 i n which a. andfc>are two a r b i t r a r y functions. choose a. to be a constant, say 1. immediately One can However, one cannot eliminate the function b. On choosing a= I, the Clebsch potential £ reduces to S-p + tfot) 4.3.20 The other " r o t a t i o n a l " Clebsch pot e n t i a l nj can then be obtained from 4.3.17 npsrir+cte) 4.3.21 i n which c i s an a r b i t r a r y function. In addition to giving correctly the generalized vorticity 7 and I must be chosen so that the surfaces 7 = constant and £= constant move with the f l u i d 1.3.37 and 1.3.38). (equations For Clebsch potentials 07 and v e l o c i t y jf having the forms 4.3.21, 4.3.20 and 4 . 3 and . 6 , these conditions become 2>t _ _ j£ at at 4.3.22 r r and P a r t i c u l a r solutions of equations 4 . 3 . 2 2 and 4 . 3 . 2 3 are 2. X Figure IX - to follow page 59 Representative Surfaces 7=constant and 5= constant i n the Case tr = -—=. 8 = —= B{V)1, r 4 . 3 . 2 4 4 . 3 . 2 5 and c= O In t h i s p a r t i c u l a r choice of gauge, the " r o t a t i o n a l " Clebsch potentials 7 and £ are 4 . 3 . 2 6 4 . 3 . 2 7 = rir At t=o , the surfaces constant and 7 = constant, are the meridian planes <p= constant, and the cylinders constant (Figure IX). the surfaces velocity r= As time increases, every point of constant rotates with a l o c a l angular . The surfaces 7 = constant remain unchanged. For example, i f tr = «or, the pattern at t = o rotates about the 2 -axis with a constant angular v e l o c i t y «o. The remaining Clebsch p o t e n t i a l , the scalar potential X , can be found from the d e f i n i t i o n of the Clebsch pote n t i a l s i n terms of the v e l o c i t y y+ -j e«P =-vx i-rrjvf I f one uses the previously determined 4.3.28 expressions for*?, f and P this equation reduces to 4 . 3 . 2 9 4 . 3 . 3 0 which may be immediately integrated to give ^=-a t + l f(t) I _ I Figure X - to follow page 60 The Motion of a Representative Surface i n the Case tr= -~=z § - ^== B(V)lj, and X"constant, d£ > o -61- f(t) being an a r b i t r a r y function. This function can be determined from the Bernoulli equation 1 . 3 . 3 6 , which becomes dt p if «-Pa. 4 . 3 . 3 1 4 . 3 . 3 2 4 . 3 . 3 3 A solution to t h i s equation i s f-^ftt The scalar Clebsch potential i s then X~ -xr Jt z The surfaces ^ c o n s t a n t are the cylinders (Figure X). r= constant APPBNDIX A SOME USEFUL VECTOR THEOREMS A.l Theorem. Let a.«.[j7x(b* b , and c be three a r b i t r a r y vector f i e l d s . c ) ] •*-fcx[y* (c* g±)2 + <k* [?* = t x c v^.a. +• CKa.7.t +• fe* ^/T^' Then k)] S )3 A. 1.1 Proof: A.1.1 i s most e a s i l y proved using the techniques of tensor calculus. Denote by x*- (c = l , space coordinate, and by a. ,fc*", c L component of the vectors c 2, 3) the Cth cartesian the c t h cartesian b » £ respectively. Then the L th component of Ax[7*ffc«£)] i s &«Cv*(k«£)lL = fjip £ klm A * £ (trc*) t '' A 1 2 where :..k = 6^ ^ , =+/ i f ( i , j , k ) i s an even permutation of (1, 2, 3) = ° i f any two of L , j , or are equal = -t i f ( i. , i , k ) i s an odd permutation of (1, 2,3); A.1.3 Using the r e l a t i o n where c =i j =o & i f L »j i f t+j A.l.5 ops- one can write Permuting A , b , c c y c l i c l y , one obtains the analogous expressions •*">«• (ax*/ l>vn/ 4 m uW A.1.7 and + 6- imn 1 Adding A.1.6, A.1.7, and A.1.8 one obtains -"'•lax*/ - £. lax" / 1 -* mt lax "/ 1 -64- Using the r e l a t i o n the r i g h t hand side of A.1.9 becomes = [bxc V<x + C K O . P.fc f ax b 7-c + V(a.> £)} lL which completes the proof. A.2 Corollary. Let 's be any function i^rhich s a t i s f i e s £dt + V.(kc) = o Then + i - Ax/ ii? 4- ^ X ^ " £) + (V- t j c l J? - JL ~ at ft L bt » „ fcj - * [r*ft J ft . fejjx c * v[ • Proof: The l e f t hand side of A.2.2 i s fi ft * . f c j ] -65- A.2.3 + ^- i a x f y x (fox c)]+ bx f ^ x f c x f l L ^ + C x t ^ . a +• a c V . x k j By using A.1.1, A.2.3 may be written ± JL (*.* t=>) + J- [y* ('axb)Jxc + J- q.< y V. c + -L V£c..a.x\ '] 2 A. 2.4 which, using A.2.1, i s e a s i l y shown to be equal to the right hand side of A.2.2. In the present work the vector i d e n t i t y A.2.2 has two important applications: i. Set: a. = f , b = S , c = y- ,^5 = p . Then, because of the r e l a t i o n between P and J (equation 1.2.2), the facts that the l i n e s of 6 move with the f l u i d and B has no sources or sinks (B s a t i s f i e s 1.2.9, 1.2.10), and the equation of continuity (equation 1.1.9), the Lorentz force per unit mass -^rJxB can be expressed i n the form ± j*s «• A^lf»xgj-»-|JVx^. r*e)J*</ + c^^jr.^i. p* ejj A.2.5 was used i n Section 1.3 i n transforming the hydrodynamical equation 1.1.6 to the form 1.3.1 from which was extracted the generalized vortex theorem. ii. Set c = cr. Let a and b be any two vectors the l i n e s of which move with the f l u i d , i . e . : a. and t s a t i s f y 1.2.9. The l e f t hand side of A.2.2 vanishes. of the r i g h t hand side, one finds that Taking the c u r l satisfies A.2.5 -66- 1.2.9, i . e . : the l i n e s of also move with the fluid. Thus, i f one i s given any two independent vectors the l i n e s of which move with the f l u i d , one can always construct a t h i r d vector, the l i n e s of which also move with the f l u i d . In p a r t i c u l a r , from the magnetic induction B and the generalized v o r t i c i t y one can construct which move with the f l u i d . s*('"p" § 7 x , the l i n e s of -67- APPENDIX B CLEBSCH»S THEOREM B.l Theorem. Take any vector f i e l d a.. f i n d functions X , *) , £ Then, i t i s always possible to such that o ^ - ^ + ^ S . „ B.l.l and hence ""ft-<V** B.1.2 Proof: Take Then V. b = O identically. Integrate the d i f f e r e n t i a l equations f o r the lines of b, <** = ^ b x b^ = ^ B.l. 4 bg obtaining f . f x . i . Z . t ) = C, , ^^.1,2^)=^ B a # 5 Then, b can be expressed where ^ i s some function of x , tf , £ , and t . Since v. b = d , f , , fa.) of the combinations f, , vanishes, <j must be a function only , and t <2 = h r f , , f , t ) B.L7 fc and Take nj *= (f,,f*,t) B.l.9 *«*ft,fx,0 B.l.10 Then P«xv*S = ^LsJL? Choose 7 , £ | ^ 4 l Vf,xVf B . l . 11 z such that = k<V*.*> B.l.12 Then b = ^ x ^ B.l.13 and 7«(A-^?) = 0 B.l.15 from which follows B . l . l . B.2 The Gauge Transformations. For unique. a given vector A , the functions y , § are not Take another set of functions y', + Then y ^s' such that B.2. where ^ = B.2.3 Take the c u r l of B.2.2 Take the dot product of both sides of B.2.4 with V^ f o = vt'-(v*jxv*)= *($'.'yf }sJ , B.2.6 afx^z) Hence V - V f ^ M ) B.2.7 r - f c ' ^ * , * ) B.2.8 between the four functions A?' , i.e.: ,17 there are two functional r e l a t i o n s , so that only two of the four functions can be considered independent. Further ' ae^,?) ' 7 So that aCV, S') = 4 B.2.10 To i l l u s t r a t e the method f o r obtaining e x p l i c i t expressions f o r the gauge transformations, consider to be the independent variables i n the following. B.2.2 i s a set of three l i n e a r , algebraic equations i n -70- th e two unknowns nj' , 07 . d(F,,vanishes, a ( X . C ^ E For a solution, the determinant i.e.: F ( i s a function only of ) £, £', and t . B.2.2 becomes Take the cross product of B.2.11 with V$' and One obtains m= i f B.2.12 7'=- i f a*' B.2.13 bt and also •X'=r~ i B.2.14 F S i m i l a r l y by considering (I, 7 ' ) , (7, 5' ), or (-7, 07' ) to be the independent pairs of variables, one obtains "7= ifas B.2.15 £'= i f * - B.2.16 by' X'= X- *i —^ B.2.17 where F = fk^^'.t) z B.2.18 or £ = - ifs /»' = _ i f s a*' 7 B.2.19 B.2.20 -71- r'=r-f>+7- B.2.21 F 3 dip where F 3 = F 3 ^J '.*) B.2.22 6 or fe » - B.2.23 2- * B.2.24 F r ' « * - f « B.2.25 where F = Fi^Vit) + B.2.26 These transformations are just the canonical transformations (Goldstein, 1957, Chapter £ ) . -72- APPENDIX G CALCULATIONS FOR CHAPTER 3 C.l The Lagrangian and i t s Derivatives. In t h i s appendix the f i e l d equations and conservation laws which were stated without proof i n Chapter 3 w i l l be obtained from the lagrangian £ = 2L° |£| - _1_ a |s|*+ pE+ — — - (wi*- f-pk) ola] C.l.l The e l e c t r i c f i e l d § and the magnetic induction B are expressed i n terms of the electromagnetic potentials fl and <t> by means of equations 1.2.16 and 1.2.17, and the v e l o c i t y y i s expressed i n terms of the vector potential A, the p o l a r i z a t i o n P, and the Clebsch potentials ^ , and by means of equation 1.3.7. One must f i r s t obtain the derivatives of Jl with respect to: the f i e l d variables 5 , ^,7, and § , and the derivatives of the f i e l d variables with respect to the time t and the cartesian space coordinates x, ^ , and 2. derivatives are tabulated on page 73. These -7JS- y ^' . x bi 0< H. V O o 1 Wi ° bi Q-i 1 «oi (L| 1 + U i i O O O O «" 1L o ^ ° 0 ° 1 mi J O f> •*s 5 ^ 0 1 0-i 0-, uii UJ» Uli ^ £° ^ Y Y Q-l A) + + + o o o o o "f <^ i <^|*A l l /TJ O O O O Ml bt (01 b\ CQi 3i -|W Oil UJI UJI IV,,-H + + + ^1 4 ° ^ 01 ft O ^ ' AI A 1 ^ oT of ^ or or a!" ^ * ^ -74- C.2 The F i e l d Equations. Since £ does not contain any of the derivatives of the f i e l d functions P, p , or 7?, the Euler-Lagrange equations r e s u l t i n g from variation of P, p , and ^ may be read d i r e c t l y from the column W = — i n the table on page 73. £ + if* § = 0 dp =0 L J P T>t ' T>t G.2.1 C«2 • 2 «• The Euler-Lagrange equation r e s u l t i n g from a v a r i a t i o n of a component of fl, say A , i s x Equation G.2.4 and the two analagous equations obtained by varying fl^ and ft can be combined to give the vector equation 2 CeT = = at ~ + f)-^x/j-g-Pxy)-o / C.2.5 The Euler-Lagrange equation r e s u l t i n g from a variation of <f> is - + f*) = 0 C.2.6 The Euler-Lagrange equation r e s u l t i n g from a variation of ^ i s - - l£ - V.(eir) atAnd = O C.2.7 l a s t l y , the Euler-Lagrange equation r e s u l t i n g from a v a r i a t i o n of £ i s = C.3 ± ( ^ ) - r V . ( ^ V ; ) = o The Conservation Laws. For any given symmetry transformation a r i s i n g contin- uously from the i d e n t i t y one can immediately write a cor- responding conservation law if + at v.s = o c '3.1 The appropriate "density" cr and the "current density" S are found by using equations 2.3.7 and 2.3.8. In most cases, the density and current density found d i r e c t l y i n t h i s way must be rewritten i n order to bring the conservation law to the form quoted i n Chapter 3. of t h i s section to show how t h i s may C.3A i. I t i s the purpose be done. The Space-Time Invariances The lagrangian i s form invariant under the transformation * _ » x > X + Sx From 2.3.7, the "density" function <? i s c > 3 > 2 I f one uses the r e l a t i o n (&x~v)R = SxSx+^Sx.fl) C.3.4 and the d e f i n i t i o n of the Clebsch potentials, i . e . : o- becomes K £ X § ) . S X + (K E <r = 0 0 Similarly, + P)-V(SX-Q) C.3.6 from 2.3.8, the "current" density S i s I f one uses the r e l a t i o n C.3.4 and the d e f i n i t i o n of the Clebsch potentials C.3.5,S S = JCSK + becomes 8-£>* IT) - + V(Sx..fi)*[-L — B x f B x S x ) + (p* P) SX-VjzJ + p y tf. SX +- a" 8x P . S X y ) x ^ S x Sx ) C.3.8 I f one uses the relations % . f t ) « (j- S- PxifJ= ^x^sx.fl^- S-Pxu-jJ " S ^ ^ f e § - f x ^ C.3.9' P C.3.10 and ffxLfJx ('Sx6)<)t cf e < P . <5x = - y - x B - S x one obtains -77- S = it 8x + Vx^SK.ft^j- B - P SX(BKSX)- -_L ITKB.SK * ~ ' ~ S +(K*E P * (jT-~- V -) x p) C.3.11 If one uses the d e f i n i t i o n of the e l e c t r i c f i e l d E i n terms of the vector and scalar potentials ft and 0, equation 1.2.16, to set v>J = - £ - i f ~~ C.3.12 at in C.3.11, and substitutes f o r the lagrangian 2L- IE I*"-Ja. one ~ | 8 / S P Y £ - + ^ 6 ) + p f e ' - » 5 ! +JL rl*--fedpl ti 2.^0 ' " ~ L i t/ a t 2- Jf"- J C.3.13 obtains + JSs/£•{*• S"X - K„ £" E - S X - J- B«(B* SX) - — j e / * sx + f • (§+ sf* - §) +f ) i at •+ V* F&K • A L ~ ~ f C&x.fi) " (E +• jx - e)• s x . f i s?x ~ sx e-p< ir) I /*. ~ ~/ / J _ 8 - Px u-)] ~ ~ "/J C.3.H If one uses the r e l a t i o n s Bx(e* and s*) = •? S-sx - lei* sx C.3.15 -y»- f. + y* s ) sx - f (§ + u-* 8 )-sx = (e + ^ e j ^ s x x p ) c . 3 . 1 6 one obtains -K.eF.5X + i**/£"/*• <5X + _L (Sx.fl) - Sx.fl at - ( « . £ • + £»)-2. - ISI* sx -_L 6 e.sx A B - P K I T ) Ip~ I ^ „ C . 3 . 1 7 The f i e l d equations may now be used to simplify the expressions f o r o- and S. I f one uses Gauss's law, equation the density cr becomes C . 2 . 6 , cr = (pxr + K E< &).&X + 0 f&xfl(V„£ + f ) ] I f one uses the Bernoulli equation law r e l a t i o n S=p urd". C.3.18 C . 2 . 2 , the Ohm's and Ampere's lav/ C . 2 . 5 , S becomes G . 2 . 1 , Sx + pSx + il« fEp-SX - K ££-.6X + J- IS| SX - -L S8-5X a 0 + "iff -'5 r*[sx-Q^ S- f«ifjj c . 3 . 1 9 I f o and S are substituted into the equation of - continuity, C . 3 . 1 , one sees that the second term i n cr w i l l cancel the second from the l a s t term i n S. Further, the divergence of the l a s t term i n S vanishes identicall}?-. One thus obtains the conservation of momentum equation 3 . 3 . 2 . ii. The lagrangian i s form invariant under the trans- formation t - * t ' = t + St C.3.20 From 2.3.7, the density function cr i s *-[' * ^ p - ft * <(-f ) +•> Tt)] st c '3 21 Substituting the lagrangian/, from C . l . l , into C.3.21, and using the d e f i n i t i o n of the e l e c t r i c f i e l d i n terms of the vector and scalar potentials 1.2.16 to set ^ = -E-Vt C.3.22 i n C.3.21, one obtains cr= - j^-pfcr/*- + eJ|L JL? + 2S? l § f W * . § +f)-v*]st + C.3.23 S i m i l a r l y , from 2.3.8, the current density S i s Substituting ^ from C.3.22 into C.3.24, and using the relation *2< at 7 at = £ ? T „ £f + /cr/^ + JL p.( *e) e i t ' i>T c.3.25 u one obtains 5 = ^ K^-i- 6 - fx u-j - J _ at - i f P. (V« 8)]st e Ex § - (p K r ( a>t if) * £" 7»t 2. - / C.3.26 -ou- I f one uses the relations 6- PxV-j = V<t>«(±- P x i T ^ J L 6- P x i f j J - ^ B-fx vj c.3.27 and -r/Px-y)x£ - cr P-fvx B ) = f£ + o-xfi)*(>* LT) C.3.28 S becomes S - - f f t<** + f f ^ - 7 H * t "*) - £ 6" § 1 + (E + xfxS)* (is* P) - CKoE + f ) ^ ^x / j . S - Px i f ) - \^«" / at L (/<• ~ a " /J C.3.29 Using the argument following C.3.17, one obtains the conservation of energy theorem 3.3.4. iii. The lagrangian i s form invariant under the transformation * — *'= * + x «66> fl'= A + C.3.30 ftxS© C.3.31 P-P^P^PxSe C.3.32 From 2.3.7, the density function cr i s cr = (K E+ 0 P).(xxi6 V8 ~ ) •*- p (**$$)• (~V?(+<r} v$ ) C.3.33 I f one uses the r e l a t i o n XxSd.7fl-fl«S5 = v(x*83'Pi) + fx fxx£0) C 3 34 -Kl- and the d e f i n i t i o n of the Clebsch potentials o- = (?if + K EKB)-XKSB C . 3 . 5 one obtains t (« E + P)-v(x*&0-fl) 0 A S i m i l a r l y , from 2.3.8, the current density S i s P ) XXS0-?# + £u- XK63 • (-Vx + yVS) * I f one uses the r e l a t i o n Clebsch potentials C . 3 . 3 5 C . 3 . 3 6 and the d e f i n i t i o n of the C . 3 . 3 4 C . 3 . 5 one obtains S = £ x > S £ + V(x*&8'ft)*/— B - Px u") - _L g x / s x ( x * S £ ) ) + py ?-x*8§ f y fe«*£)-fBnP) C . 3 . 3 7 I f one uses the r e l a t i o n s V(x*6d.R ) x ^JL B - fx crj = Vx£x*8§.ft (j- B-PKO-JJ and Cf x vf)x (i<x^) )+ o- Cexf S becomes ( S=£xx&$+ -X*S0.ft (x«55) - ~(a;*8).(xx$d)P C . 3 . 3 9 6 - f x vrjj Vx^XxSg-R V7x/i_ 1 g _ f x if ) - - L 6x "~ i •* / r° ~ - t/xB.x>6© P + fK.£ f ) x * S £ + ('gxfXxSfl)) - - V<f> + ? tflT-XxSS C . 3 . 4 0 from C . 3 . 1 2 and £ from C . 3 . 1 3 into C . 3 . 4 O Substituting one obtains + .*? /£/*- X « 6 0 2 . - - - KeEE-XxhO - - - — / « BxfBx fxxS0)) . - - - - - _L /sI*" X*S0 + f• - o-xg) XxS© - P + P) CF+yxe).XxS© (x*S0-rl) - Xx at SQ.fi ~ (y + 7» §-fx /_L B - P X L T ) l/<<, ~ / ujj C.3.41 I f one uses the r e l a t i o n s 6* and (B*(x*se)) = e s.x-sa - p. f e + u-x B ) X x S f - f = (si^xtSS if< B).xx<Sa + i/xB)x ftxxSgjxf ) S becomes + JS / e / x x S d -K £"F-XxSd -f- — /e/* XxS0 - i s B.X*$0 x ft - c.3.42 (naE + P ) A f V x S e . f l ) - X x S ^ . f l ^7x/-L S - P X Y ) C.3.43 -so- using the argument following C.3.17 one obtains the conservation of angular momentum theorem 3.3.9. iv. modified -_L I t i s s l i g h t l y more d i f f i c u l t to see that the lagrangian i s r + p.g " +ef&- ** L« v a ' at i s form invariant under the 4.-Li r( --ted?'] a. i J^ J C.3.45 transformation X-* x'= x + &y t C.3.46 &g.X C.3.47 «5 + 5 tr0-fl C.3.48 0 o <z> To show t h i s , one uses 2.3.2 to write down the transformed lagrangian A l l the unprimed f i e l d functions and coordinates on the r i g h t hand side of C.3.49 must be expressed i n terms of the primed f i e l d functions and coordinates. The r e l a t i o n s between the derivatives with respect to the unprimed and primed coordinates are V =V' C.3.50 and °- 3 The r e l a t i o n s between the unprimed and primed magnetic induction, e l e c t r i c f i e l d , and v e l o c i t y are then 5 1 B = <7* A = <7'xft'= B' C.3.52 e = -v*0 - AS at at' and TT = - VJ( 7 7^ + Y P* 6 = -V'(T('+8*U-£') = tr'- Sir +»7' v'$'+f' ?'*§' C.3.54 0 Substituting these expressions into C.3.47, the transformed lagrangian >£' becomes JC"/V' \ £T' *')= - J - |§7* ax' / ay*0 = ^ f ^ ; 0 , x ' j + f'so;. , + P ' §"'+ 6r'+ P ' iv.xfi 7 ^ v - y ^ ' s B ' X P ' J C3.55 the l a s t step following from the d e f i n i t i o n of the Clebsch potentials, C.3.5. Thus £ i s form invariant. From 2.3.7, the density function cr i s -85- C.3.56 I f one uses the r e l a t i o n s ^ t - ^ f l = v(h&t-A)+ §x5if t C.3.57 e and the d e f i n i t i o n of the Clebsch potentials C.3.5, cr becomes cr = p fVt - * ) • *if«, + P.?YsLr t. ft) 0 C.3.58 From 2.3.8, the current density S i s C3.59 I f one uses the r e l a t i o n C.3.57 and the d e f i n i t i o n of the Clebsch potentials C.3.5, S becomes - P S<J 0 ft a- ^ y - * )- 5o- + tr Stf^t 0 • §x P C.3.60 I f one uses the relations C.3.61 and ( f x o-)x (s* Sc/ t) +• </s</;t-gxP = _ (VxS).^if 0 *)f 0 one obtains C.3.62 f + P S V . t - ^ - P Sifo-0 '(^KB)-(^t) + e i r ^t-x:).6 r i C.3.63 o Substituting Vt> from C.3.12 and £ from C.3.13 (with K = o ), 0 one obtains - P - (S^t-R ) - Z**t. 6 V*/*!- B-P< cr) I f one uses the r e l a t i o n s Bx ("s, 5ir t) - 3 B. S<fat 0 /§l*<5u-0fc - C.3.65 and P. ('£•+ u-x B) S&t- P = fe+ax8).$Lr t 0 P) (E+s*8)K(&}Sat* S becomes + JL (B( x - i s S.SV t 6<Sat 0 + ( f + i f x B ) x ^ r t x P) / i 0 C.3.66 •or - p l (&ir t- ft ) e -Sir.t A B- Pxtr) Using the argument following C.3.17 one obtains the center of mass theorem 3.3.13. C.3B The Gauge Invariances Since the vector and scalar potentials fl and <f> i. appear i n the lagrangian only i n the gauge invariant combinations £ and 6, the lagrangian i s form invariant under the gauge transformation ft-*fi'= From <r= - < K 0 C.3.68 2.3.7, £ + the density function cr i s P^.^SX C.3.70 By use of the Gauss's law equation C.2.6, cr can be written o- = - V. (K § + P)S\ Q C.3.71 The discussion i n Section 2.3 then shows that the gauge transformation C.3.68 and C.3.69 does not lead to a true conservation law. ii. Under the ordinary gauge transformation of the Clebsch potentials C.3.72 — 5 t u u — + So. C.3.73 - SaQ the lagrangian transforms into C.3.74 , where • ' ( « £ > * ' ) C3.75 Thus .£ i s form invariant under t h i s gauge transformation. The density function cr i s given by cr — - 6o-p Cf C.3.76 and the current density S by S — — Sep Cf jS The conservation law i s 3.3.26. iii. Under the gauge transformation the lagrangian £ transforms into £', where To show that the transformation C.3.78 leaves the action p r i n c i p l e invariant, one must write C3.77 - S « f . B = &oc(v<fi-V*fi C.3.80 +• | | - ^ f l j i n the form l ^ t + ^.H at C.3.81 ~~ How t h i s may be done w i l l now be shown. The f i r s t term on the r i g h t hand side of C.3.SO can be written, dropping the factor Soc (<f> Vx B)= V-(<t>B) C.3.82 The second term on the r i g h t hand side can be rewritten as follows: consider £ (fi.B) = £ (fi. Vxft) = ±3 .VxR +• at at - " at fl'Vx(l§) I at/ C.3.83 and Adding these two i d e n t i t i e s one obtains f o r the second term on the r i g h t hand side of C.3.80, dropping the factor Soc, iftB-BJ + f - f c e . g ) C.3.85 I f one uses the d e f i n i t i o n of the e l e c t r i c f i e l d E i n terms of the vector and scalar potentials to set i i = - E-VgS at i n the second term of C.3.85, one obtains C.3.86 C.3.87 Using these r e s u l t s , one can write f o r the transformed lagrangian ^|^Ce-S) + ^ ^ 9 + * f ) J £'fv\ g\x'j = C.3.88 Since the lagrangian does not contain derivatives of the transformed variable P, the conservation law 3.3.32 can be read d i r e c t l y from C.3.88. iv. Under the gauge transformation P - P ' « P + hpVy* C.3.89 C.3.90 C.3.91 C.3.92 the lagrangian £ transforms into £', where dX' ' / 2. 2^ 0 C.3.93 -VI- To show that the transformation C.3.89-C.3.92 leaves the action p r i n c i p l e invariant, one must write - S s / § . $ y x 7 £ + §.cy e-7¥ i 7 ' ) / C.3.94 / i n the form + v'. I I C.3.95 How t h i s may be done w i l l now be shown. The f i r s t term of C.3.94 can be written, dropping the factor - S £ the second step following from the d e f i n i t i o n of B and £ i n terms of the vector and scalar potentials ft and & . Dropping the f a c t o r - S ^ , C.3.94 becomes V.(m'rS'*£) 7 -' -<n'VS'- / = v % ( V ^ *fi"• i f+ e-Vm' bt ~ il - §v$' ' dt )~~ < iV dt V I c . 3 . 9 the l a s t step following from the d e f i n i t i o n of B i n terms of the vector p o t e n t i a l ft. C.3.94 has now been cast i n the appropriate form. However, a few further transformations w i l l be carried out to bring C.3.94 into a more suitable form. I f one uses the d e f i n i t i o n of the Clebsch potentials C.3.5, the second 7 -92- term i n C dt " ~ becomes . 3 . 9 7 - * ' 1 at ' v «~ Substituting - [ ^ j W - ^ 3 t h i s r e s u l t back into +7 c . 3 . 9 8 + '<7!r') + C . 3 . 9 7 , one obtains £6r-§)} C 3 . 9 9 I f one uses the d e f i n i t i o n of the Clebsch potentials C . 3 . 5 and the r e l a t i o n at ' a t C . 3 . 9 9 7 2>t ~ e~~ ~ C . 3 . 1 0 0 becomes 1 L -<V- Dt [E*V - -- f + ± ? Ex(BxP) + ~ cfxS-P 6 '*>*)*] C . 3 . 1 0 1 F i n a l l y , i f one uses the r e l a t i o n Cifx § ) . f C . 3 . 1 0 1 s = <V*s)x Cs« P ) becomes C . 3 . 1 0 2 -93- + C.3.103 The transformed lagrangian £j i s then given by From 2.3.7, the density <r i s <5- = y-§ C.3.105 and from 2.3.^, the current density S i s S = E , i r + i ( e 4 if*8)*(B.xe)+ ($f'-"?'^'+ W*)§ C.3.106 I f one uses the Ohm's law r e l a t i o n C.2.6 and the Bernoulli equation C.2.2 S becomes s = e* sr+ ^J-^p + j- t«i*-) £ The conservation law i s then 3.3.38. c.3.107 -94- BIBLIOGRAPHY Alfven, H., Cosmical Electrodynamics. Oxford, 1950. Bateman, H., P a r t i a l D i f f e r e n t i a l Equations of Mathematical Physics. Cambridge, 1932. Chandrasekhar, S., Proc. Nat. Acad. S c i . U.S.A., 42, 273, (1956). Cowling, T.G., Magnetohydrodynamics, New York, 1957. De, J . , Naturwissenschaften, 44, 256, (1957). Goldstein, H., C l a s s i c a l Mechanics. Reading, Mass., 1950. H i l l , S.L., Rev. Mod. Phys., 23, 253, (1951). It6, H., Progr. Theor. Phys., 9, 117, (1953). Lamb, H., Hydrodynamics, 6th ed., Cambridge, 1932. Noether, E., Nachr. k g l . Ges. Wiss. Gottingen, 235, (1918). Panofsky, W.K.H. and P h i l l i p s , M., C l a s s i c a l E l e c t r i c i t y and Magnetism. Reading, Mass., 1955. Spitzer, L., Physics of F u l l y Ionized Gases, New York, 1956. Weyl, H., Trans. Henry L. Brose, Space - Time - Matter. 4th ed., Methuen and Co. Ltd., 1922, (Dover Publications, Inc., 1950). Woltjer, L., Proc. Nat. Acad. S c i . U.S.A., 44, 833, (1953).
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A field theoretical treatment of magnetohydrodynamics Calkin, , Melvin Gilbert 1961
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Title | A field theoretical treatment of magnetohydrodynamics |
Creator |
Calkin, , Melvin Gilbert |
Publisher | University of British Columbia |
Date Issued | 1961 |
Description | The action principle of field theory, usually applied only to linear theories, such as electrodynamics, is shown to be useful in the nonlinear theory of magnetohydrodynamics The well-known conservation laws are derived by utilization of invariance properties of the appropirate lagrangian In addition, a set of new conservation laws, corresponding to generalizations of the Helmholtz vortex theorem in hydrodynamics, are obtained by exploiting a certain type of gauge invariance of the magnetohydrodynamical action integral A number of formal simplifications are introduced which suggest themselves through the action principle Three problems which illustrate the resulting formulation of magnetohydrodynamics are discussed |
Subject |
Magnetohydrodynamics. Mathematical physics |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-11-07 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085385 |
URI | http://hdl.handle.net/2429/38797 |
Degree |
Doctor of Philosophy - PhD |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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