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On the equations of motion of mechanical systems subject to nonlinear nonholonomic constraints 1960
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Title | On the equations of motion of mechanical systems subject to nonlinear nonholonomic constraints |
Creator |
Ghori, Qamaruddin Khan |
Publisher | University of British Columbia |
Date Created | 2011-12-09 |
Date Issued | 2011-12-09 |
Date | 1960 |
Description | Suppose q₁,q₂,…,qn are the generalised coordinates of a mechanical system moving with constraints expressed by r non-integrable equations (r〈n) (1) [equation omitted] where the dots denote differentiation with respect to the time t, and fα are nonlinear in the q’s. The equations (1) are said to represent nonlinear nonholonomic constraints and the system moving with such constraints is called nonlinear nonholonomic. From a purely analytical point of view, the author has obtained the equations of motion for a nonlinear nonholonomic mechanical system in many a different form. The importance of these forms lies in their simplicity and novelty. Some of these forms are deduced from the principle of d'Alembert-Lagrange using the definition of virtual (possible) displacements due to Četaev [ll] . The others are obtained as a result of certain transformations. Moreover, these different forms of equations of motion are written either in terms of the generalised coordinates or in terms of nonlinear nonholonomic coordinates introduced by V.S. Novoselov [23]. These forms involve the energy of acceleration of the system or the kinetic energy or some new functions depending upon the kinetic energy of the system. Two of these new functions, denoted by R (Sec. 2.3) and K (Sec. 2.4), can be identified, to a certain approximation, with the energy of acceleration of the system and the Gaussian constraint, respectively. An alternative proof (Sec.2.5) is given to the fact that, if virtual displacements are defined in the sense of N.G. Četaev [ll], the two fundamental principles of analytical dynamics - the principle of d'Alembert-Lagrange and the principle of least constraint of Gauss -are consistent. If the1 constraints are rheonomic but linear, a generalisation of the classical theorem of Poisson is obtained in terms of quasi-coordinates and the generalised Poisson's brackets introduced by V.V. Dobronravov [17]. The advantage of the various novel forms for the equations of motion is illustrated by solving a few problems. |
Subject |
Mathematics -- Problems, Exercises, Etc. Generalized Spaces |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2011-12-09 |
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 | 10.14288/1.0080604 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/39610 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0080604/source |
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, ON THE EQUATIONS.OF MOTION O P MECHANICAL SYSTEMS SUBJECT TO NONLINEAR NONHOLONOMIC CONSTRAINTS by QAMARUDDIN KHAN GHORI A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY In the Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1960. In presenting t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n permission. Department of \j\^J^Jj^iy^AA^J^S The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 3, Canada. GRADUATE STUDIES Field of Study: Applied Mathematics Non-linear Mechanics E. Leimanis Projective Geometry S. W. Nash Computational Methods F. M. C. Goodspeed Other Studies: Elementary Quantum Mechanics F. A. Kaempffer Theoretical Mechanics W. Opechowski Electricity and Magnetism R. Barrie Outside Interest: Islamic Religion. 3% Pttttarsttg of ^rtttsh (Eoiambta Faculty of Graduate Studies P R O G R A M M E O F T H E FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY of QAMARUDDIN K H A N GHORI B.Sc. (Hons) Sind University, Pakistan 1950 M.Sc. Karachi University, Pakistan 1952 IN ROOM 225, BUCHANAN BUILDING THURSDAY, APRIL 28, 1960 AT 3 P.M. COMMITTEE IN CHARGE DR. W. OPECHOWSKI, Chairman E. LEIMANIS REV. T. J. H A N R A H A N D. DERRY J. GRINDLAY M. MARCUS P. RASTALL C. SWANSON R. HOWARD External Examiner: DR. W. KAPLAN University of Michigan, Ann Arbor, Mich. ON T H E EOUATIONS OF MOTION OF MECHANICAL SYSTEMS SUBJECT TO NONLINEAR NONHOLONOMIC CONSTRAINTS ABSTRACT The author has obtained the equations of motion for a nonlinear nonholonomic mechanical system in many a different form. The importance of these forms lies in their simplicity and novelty. Some of these forms are deduced from the principle of d'Alembert- Lagrange using the definition of virtual (possible) displacements due to N. G. Cetaev (Izv. Kazan, Fiz.-Mat Obsc.6 (1933), no. 3, 68-71), The others are obtained as a result of certain transformations. More- over, these different forms of equations of motion are written either in terms of the generalised coordinates or in terms of nonlinear non- holonomic coordinates introduced by V. S. Novoselov (Leningrad. Gos Univ. Ucenye Zap. 217. Ser. Mat. Nauk 31 (1957), 50-83). These forms involve the energy of acceleration of the system or the kinetic energy or some new functions depending upon the kinetic energy of the system. Two of these new functions, denoted by R & K, can be identified to a certain approximation, with the energy of acceleration of the system and the Gaussian constraint, respectively. An alternative proof is given to the fact that, if virtual displace- ments are defined in the sense of N. G. Cetaev, the two fundamental principles of analytical dynamics—the principle of d'Alembert- Lagrange and the principle of least constraint of Gauss—are con- sistent. If the constraints are rheomonic but linear, a generalisation of the classical theorem of Poisson is obtained in terms of quasi- coordinates and the generalised Poisson's brackets introduced by V. V. Dobronravov (C R. (Doklady) Akad. Sci. U.R.S.S. (N.S.) 44 (1944), 221-234). The advantage of the various novel forms for the equations of motion is illustrated by solving a few problems. - i i - A B S T R A G T Suppose <l^,q.2» • • •» cl n a r e g e n e r a l i s e d coord- inates of a mechanical system moving with constraints expressed by r non-integrable equations ( r ^ n ) where the dots denote d i f f e r e n t i a t i o n with respect to the time t , and f are nonlinear In the q f s . The equat- ions (1) are said to represent nonlinear nonholonomic constraints and the system moving with such constraints i s c a l l e d nonlinear nonholonomic. Prom a purely a n a l y t i c a l point of view, the author has obtained the equations of motion f o r a non- l i n e a r nonholonomic mechanical system i n many a d i f f e r e n t form. The importance of these forms l i e s i n t h e i r s i m p l i c i t y and novelty. Some of these forms are deduced from the p r i n c i p l e of d'Alembert-Lagrange using the d e f i n i t i o n of v i r t u a l (possible) displacements due to Cetaev [ l l ] . The others are obtained as a r e s u l t of cert a i n transformations. Moreover, these d i f f e r e n t forms of equations of motion are written either i n terms of the generalised coordinates or In terms of nonlinear non- holonomic coordinates introduced by V.S. Novoselov [25] . - i i i - These forms involve the energy of acceleration of the system or the k i n e t i c energy or some new functions depending upon the k i n e t i c energy of the system. Two of these new functions, denoted by R (Sec. 2 . 3 ) and K (Sec. 2.4), can be i d e n t i f i e d , to a certain approximation, with the energy of acceleration of the system and the Gaussian constraint, respectively. An a l t e r n a t i v e proof (Sec.2 . 5 ) i s given to the fact that, i f v i r t u a l displacements are defined i n the sense of N.G. Cetaev [ l l ] , the two fundamental princ- i p l e s of a n a l y t i c a l dynamics - the p r i n c i p l e of d'Alembert- Lagrange and the p r i n c i p l e of least constraint of Gauss - are consistent. I f the1 constraints are rheonomic but l i n e a r , a generalisation of the c l a s s i c a l theorem of Poisson i s obtained i n terms of quasi-coordinates and the generalised Poisson's brackets introduced by V.V. Dobronravov [17") . The advantage of the various novel forms f o r the equations of motion i s i l l u s t r a t e d by solving a few problems. - i v - T A B L E O P C O N T E N T S INTRODUCTION. Chapter 1. Chapter 2. 2.1. 2 • 2 • 2.3. 2.4. 2*5. 2.6. 2.7. 2.8. 2.9. 2.10. 2.11. Chapter 3. 3.1. NOTATIONS AND DEFINITIONS. EQUATIONS OF MOTION AND THEIR TRANSFORMATIONS. Some General Considerations. The General Equations of Appell A New Form for the Equations of Motion. A Transformation of the Equat- ions of Motion. The Function K and the Gaussian Constraint. Legrangian Form f o r the Equat- ions of Motion. Another Transformation f o r the Equations of Motion. A Novel Form f o r the Equations of Motions. Another Novel Form f o r the Equations of Motion. Transition from Equations of Motion with Lagrange's M u l t i p l i e r s to Equations Free from Them. The Equations of Motion i n the Form of Determinants. APPLICATIONS. Some General Considerations. Page, 1 8 15 15 15 21 26 27 31 35 38 42 44 49 53 53 3.2. 3.3. 3.4. Chapter 4. 4.1. 4.2. 4.3. BIBLIOGRAPHY. Motion of a System of Two Wheels and Their Axle on a Horizontal Plane. Motion of a Heavy C i r c u l a r Disc on a Horizontal Plane Motion of a Heavy B a l l on a Fixed Horizontal Plane. QUASI-COORDINATES OR NON- HOLONOMIC COORDINATES. Some General Considerations. Poisson's Theorem i n Linear Non-holonomic Coordinates. Appell's Equation i n Non- linea r.Non-holonomic Coordinates. - v i - Acknowledgements, The author wishes to express his thanks to Dr. E Leimanis f o r suggesting the topic of t h i s t h e s i s , and f o r his h e l p f u l discussion during i t s preparation. He gladly acknowledges his indebted ness to the National Research Council of Canada whose f i n a n c i a l assistance has made thi s study possible. - 1 - INTRODUCTION. Let q , q g, ..., q^ be the generalised coordinates of a mechanical system subject to constraints expressed by r nonintegrable equations of the type (^^n) (1) ^ ( t ^ l » 3 2 » ' « ' > % ; ^ q 2 , . . . , q n ) ^0 ( <*=l,2,...,r ), where the q's are the derivatives of the q's with respect to the time t and +• are nonlinear i n the q's. The equat- ions (1) are said to represent nonlinear nonholonomic con- s t r a i n t s , i However, i f the equations (1) reduce to non- integrable P f a f f i a n equations, the constraints are r e f e r r - ed to as l i n e a r non-holonomic. The problem of mechanical systems moving with non- l i n e a r nonholonomic constraints i s an acute problem of anal- y t i c a l dynamics. The idea of such constraints originated with Appell [3j4;5^j , Delassus [l2;15;16] and t h e i r contemp oraries who, i n an attempt to deduce the fundamental princ- i p l e s of a n a l y t i c a l dynamics f o r such,systems from the dynamics of systems moving with l i n e a r constraints, were confronted with two serious problems. F i r s t , the r e a l e x i s t - ence of such constraints was not known. Secondly, considering such constraints, from a purely a n a l y t i c a l point of view, the two fundamental p r i n c i p l e s of a n a l y t i c a l dynamics - the p r i n c i p l e of d'Alembert-Lagrange and the p r i n c i p l e of le a s t constraint of Gauss - appeared to be inconsistent. Though the f i r s t problem i s s t i l l open, the second has been d i s - cussed by N.G. Cetaev f i l l . I n 1933 he offered a new d e f i n i t i o n of a v i r t u a l (possible) displacement f o r such systems. As i t should be expected, his d e f i n i t i o n embraces the usual d e f i n i t i o n of such displacements f o r systems which are holonomic or move with l i n e a r nonholonomic constraints. In 1948 G.S. Pogosov [ 2 6 j found the equations of motion for a nonlinear nonholonomic system i n the form (2) = Q. - (L Q U=IA.~,*-A**+\,T4^,'*), where s ' i s the energy of acceleration of the system c a l - culated on the basis of the equations of constraint ( 1 ) , Q's are the generalised forces and . =. r - • These «*- 2% equations which are e s s e n t i a l l y Appell*s equations were deduced from the p r i n c i p l e of le a s t constraint of Gauss by a long and complicated method. In 1957 V.S. Novoselov £ 2 2 ; 2 3 } 2 4 J started a series of papers on nonlinear nonholonomic systems. One of his papers [22 j | contains a variety of r e s u l t s deduced from the equations of motion involving undetermined m u l t i p l i e r s of Lagrange. Another paper £ 2 3 d e a l s exhaustively with the various forms of the equations of motion i n nonlinear quasi-coordinates or nonlinear nonholonomic coordinates. He obtains several Important and i n t e r e s t i n g r e s u l t s . The quintessence of his researches i s the generalisation of the c l a s s i c a l results f o r l i n e a r nonholonomic systems to non- l i n e a r nonholonomic systems. The present thesis i s concerned with nonlinear non- holonomic mechanical systems from a u n i f i e d point of view 0 The s t a r t i n g point of these considerations i s a synthesis of - 3 - the d i f f e r e n t i a l principle;: of d'Alembert-Lagrange and an idea of P. Woronetz [̂ 29̂ • According to Woronetz we con- sider i n the equations of constraint, a cert a i n number of velocity-parameters, equal i n number to the degrees of freedom of the system, as Independent.parameters. Through- out the discussion the i n d i c i a l and summation conventions are used. A b r i e f resume of the d i f f e r e n t aspects of the work i s given below: ( i ) The consistency of the p r i n c i p l e of d'Alembert- Lagrange and the p r i n c i p l e of Gauss, as proved by Cetaev jjLl^J , demands that the former p r i n c i p l e must lead, to Appell*s equations of motion. That i t Is so Is shown by fi n d i n g the equations of motion (Sec.2.2) ( 3 ) ^ § - = Qi ( Z = r + | , r + ^ , . . v n ) , where Q ' = Q . - a Q . The method applied i s easier and more di r e c t than that of Pogosov, [ 2 6 • Furthermore, i f S i s the function s' f o r the corresponding holonomic system, i t i s shown that the equations (3) can be written i n the symmetric form ( 4 ) \ o A *X " } (* = i , a , . . . , r , - A. = r+i,r+.*,.../n,;. Let T'be the k i n e t i c energy of the system calculated by taking the constraints Into account and l e t T̂ be the function T' considered as a function of q's and t only. The equations of motion are then obtained i n a new form (Sec. 2.3) - 4 - (5) I S - = Qi * + + .••,*,"), *\ where r\ = J_(f"'--3"Ti'). It i s also shown that R coincides with S as f a r as the terms i n q are concerned. In Sec. 2.4 the equations of motion are transformed to the form (6) = o •a=r+i,'*>*,...,tx,), 3 % where K' = R ' - QI <{,, - ^ Later on (See. 2.5) the funotion K i s i d e n t i f i e d , to a certain approximation, with the Gaussian constraint. In the same section an alte r n a t i v e proof f o r the consistency of the p r i n c i p l e of d'Alembert-Lagrange and the p r i n c i p l e of Gauss i s given. I f R and T are the functions R' and T' f o r the corresponding holonomic system, the i d e n t i t y (Sec.2.6) y i e l d s the equations of motion ( 7 ) i _ ^ - l I _ - a = a.(l_5I_2L^9 I f i n place of T the function T i s used, the - 5 - equations are transformed into the equations (Sec.2.7 ) (s> i - 2L' - +.-UL -i-!3L S Q'. oLt 3fy 3^. 2><fc «lt 3̂ , • Jfy V where T, i s what T becomes when considered as a function of the q's only and R, Is R regarded as a function of the q's only. With the help of the i d e n t i t y the equations are obtained i n a novel form: Again, by virtue of the I d e n t i t y A - 21 +£L =11. Cs^i,*,...^) the equations of motion assume another novel form (Sec.2.9) (10) % *H ( j » ^ + * , . . . , t u ) * * • where T 0 denotes T considered as a function of the q's only, and .T0 denotes T regarded as a function of the q's only. A certain transformation discussed i n Sec.2.6 allows the t r a n s i t i o n from the E q s . ( 7 ) to equations i n terms of Lagrange's m u l t i p l i e r s . The converse problem i s discussed - 6 - i n See.2.10. In Sec.2.11 the Eqs.(7) are put i n the form of determinants a l l of which are obtained according to a general scheme from an (r + l ) x n matrix. When the mechanical system i s holonomic or moving with l i n e a r nonholonomic constraints many results of other authors, notably I.Cenov £7;8;9;10^| and 1.1 .Metellcyn [21], follow as immediate c o r o l l a r i e s from the results of th i s chapter. ( i i ) Despite the fact that i n nature no mechanical system has so f a r been discovered which moves with non- l i n e a r nonholonomic constraints, i t i s sometimes possible to write i n an a r t i f i c i a l manner the equations of l i n e a r constraints i n a nonlinear form; Based on such consider- ations three well-known examples have been solved to support the general treatment of Chap#2. The c l a s s i c a l methods of solving these examples depend on the equations of motion i n terms of Lagrange's m u l t i p l i e r s . The methods used i n t h i s thesis completely avoid the use of such multiplierso ( i i i ) Sec* 4.2 deals with the generalisation of the c l a s s i c a l theorem of Polsson i n terms of l i n e a r quasi- coordlnates or l i n e a r nonholonomic coordinates so as to be applicable to systems moving with rheonomic constraints. To th i s end use has been made of the generalised Poisson's brackets Introduced by V.V.Dobronravov [ l 7 ^ . I f the con- s t r a i n t s are scleronomic the resu l t reduces to that of Dobronravov [ l 7 ] established i n 1944« Introducing nonlinear nonholonomic coordinates i n the manner of V.S.Novoselov £23^J , the author has obtained the general equations of Appell (Sec.4.3) i n the form f o r holonomic systems, and In the form (12) $4- = 9 ' ; ' (*=r+vf f o r nonlinear nonholonomic systems. The denote the k i n e t i c c h a r a c t e r i s t i c s . CHAPTER.I NOTATIONS AND DEFINITIONS 1.1 Consider a mechanical system consisting of N p a r t i c l e s , and denote by x y one of the three rectangular coordinates of any one p a r t i c l e of mass nr, • Further denote by the component of the resultant external force corresponding to x y . I f the mechanical system i s free to move, the motion of the system w i l l be governed by the Newtonian equations (1.1.1) 7 Y V ) 5 V 3 ^ V > (V-W..3*0i where the dots denote d i f f e r e n t i a t i o n with respect to the time t . In w riting the equations (1.1.1) as well as throughout our work we use the following Notations: ( i ) An index unrepeated implies a given range of values, and, when repeated i n a single term, summation over that range. ( i i ) As a derogation from t h i s rule, an index within parenthesis, although repeated i n a single term, w i l l not be an index of summation. With these notations the motion of a free mechan- i c a l system i s completely determined by the equations - 9 - (1.1.1) . On the. other hand, I f the motion of the mechanical system i s subject to some constraints expressed by r <̂ 3N equations of the type: (1.1.2) f*CV PX^...,X N-*^ the equations (1.1.1) are no longer v a l i d , and to obtain the equations of motion i t i s necessary to apply one of the two fundamental p r i n c i p l e s of a n a l y t i c a l dynamics, either the p r i n c i p l e of d'Alembert-Lagrange or the princ- i p l e of least constraint of Gauss. Although the p r i n c i p l e of lea s t constraint of Gauss i s the most general p r i n c i p l e , i t i s the formalism of the pr i n c i p l e of d'Alembert-Lagrange which i s mostly used i n a n a l y t i c a l mechanics and which we s h a l l apply i n most of our work. To formulate the p r i n c i p l e of least constraint of Gauss l e t us f i r s t define the term "constraint". D e f i n i t i o n 1. Let x be the acceleration of the p a r t i c l e of mass i n any kinemetically possible t r a j e c t o r y for which the coordinates and:velocities at the instant con- sidered are the same as i n some actual t r a j e c t o r y . The constraint i s defined by the function (1.1.3) <S(xv) = I ^ C ^ , ' ^ ) . ^ ' ? V ; 3 N , which i s of the second degree i n x^ • The following i s then the formulation of - 10 - K The p r i n c i p l e of le a s t constraint of Gauss - Of a l l the t r a j e c t o r i e s consistent with the constraints (which are supposed to do no work), the actual t r a j e c t o r y i s that which has the least constraint. The other p r i n c i p l e - the p r i n c i p l e of d'Alembert- Lagrange - i s the u n i f i c a t i o n of the p r i n c i p l e of d'Alembert and the p r i n c i p l e of v i r t u a l displacements. This combined p r i n c i p l e was given by Lagrange. That t h i s i s a d i f f e r e n t - i a l p r i n c i p l e can be seen from the formulation of The p r i n c i p l e of d'Alembert-Lagrange - For every system of v i r t u a l (possible) displacements /Sx̂ s a t i s f y i n g the cond- i t i o n s (1.1.4) the equation (1.1.5) ( V v " ^ v ) f c v = 0 ' T=I,Z,->.,3«, holds. 1.2. The mechanical system with which we deal i s of the most general type. I t may be subject to moving constraints, i n which case i t Is rheonomic: i f the constraints are f i x e d , i . e . independent of the time, i t i s scleronomic. The con- s t r a i n t s may be defined by non-integrable equations i n x^ , i n which case i t i s non-holonomlc; otherwise holonomlo. In the case of a non-holonomic system the constraints i f def- ined by non-integrable P f f a f i a n equations w i l l be c a l l e d l i n e a r : otherwise nonlinear. The rheonomic non-linear - 11 - non-holonomic system i s the most general, including the others as special or degenerate cases. The equations(1.1.2), supposed to be independent, allow us to express the rectangular coordinates x^ as functions of n = 3N-r independent parameter q^,q2»•••»^n» c a l l e d the generalised coordinates, and of the time t . Let the transformation equations be (1.2.1) \ = ̂ V V " ' ' ^ E * ^ ' \ ) > *«l,V.,rv. D i f f e r e n t i a t i n g (1.2.1) with respect to the time, we .get Substituting the value of x v and from (1.2.1) and (1.2.2) i n (1.1.2) we get the equations of constraint i n the following form: (1.2.3) i ^ V V ' ' , ^ ^ ^ * V i ^ i ^ % ' ^ = ° ? ^ Let us now assume that the equations (1.2.3) can be solved to obtain any r, say the f i r s t r, q's as functions of t, q's and the remaining q's. Then we s h a l l have relations of the form: d.2.4) i^U t j v V ' " V ^ 4 '--HĤ vV) In view of (1.2.4) the re l a t i o n s (1.2.2) take the form: 12 - (1.2.5) V * X M S > \ ) - V 1.3. So f a r as holonomic or l i n e a r non-holonomic systems are concerned, the p r i n c i p l e of d'Alembert-Lagrange and the p r i n c i p l e of least constraint of Gauss are found to be consistent. The question a r i s e s : "Can these p r i n c i p l e s be extended to nonlinear non-holonomic systems?" In an attempt to answer t h i s question Appell [3;4;5^ and Delassus |jL2;15;16 3 found that the p r i n c i p l e of least con- s t r a i n t of Gauss could be extended whereas the p r i n c i p l e of d'Alembert-Lagrange broke down. In other words, the two fundamental p r i n c i p l e s of a n a l y t i c a l dynamic showed an Inconsistency. In 1933 N.G.Cetaev [ l l ] considered the problem of nonlinear non-holonomic systems. In order to remove the Inconsistency between the two above-mentioned p r i n c i p l e s he proposed a new d e f i n i t i o n of a v i r t u a l (possible) displace- ment which can be expressed as follows: D e f i n i t i o n 2. S^y i s said to be a v i r t u a l displacement consistent with the constraints (1.2.5) provided that the r e l a t i o n s (1.3.1) = ̂ $\ ; (W+1,T>V^>V=»;V^ hold, where &q ± are i n f i n i t e l y small a r b i t r a r y q u a n t i t i e s . The constraints f o r which the relations (1.3.1) hold are c a l l e d constraints of the type of Cetaev. - 13 - A sa l i e n t feature of D e f i n i t i o n 2 of a v i r t u a l displacement i s the fact that i t contains as a special case the usually given d e f i n i t i o n of a v i r t u a l displacement f o r a holonomic or l i n e a r non-holonomic system. Moreover, the existence of •>< v i r t u a l displacements, s a t i s f y i n g the con- dit i o n s (1.3.1) has been shown by Cetaev [ l l ] • He also proved the r e l a t i o n (1.3.2) ^ V ^ ^ C ^ r H j . - In the above relations dx^ , dq^ denote the change i n • « x v , q I ' r e s P e c * l v e l y » a l ° n 8 the actual motion during an i n t e r v a l of time dt and bx^ , b q^ ref e r to the correspond- ing changes, during an i n t e r v a l of time it=dt, along any conceivable motion which i s consistent with the imposed con- s t r a i n t s . Prom (1.3.1) and (1.3.2) i t follows that S x , ̂ q^ can be taken proportion to dx^ - S x,dq^, k'q̂ * respectively. By virt u e of the relations (1.3.1) the constraints (1.1.2) impose the following conditions on the variations of the rectangular coordinates x v : (1.3.3) — ^ =0 (f< = t,V-,^V=l,^3N). The conditions (1.3.3) In the generalised coord- inates assume the form: (1.3.4) . ^ ~ H = ° («l = l,^»vrJS = »,A,...,Tv). That i s , the re l a t i o n s (1.3.3) and (1.3.4) are equivalent. As a consequence of D e f i n i t i o n 2 of a v i r t u a l - 14 - displacement i t becomes necessary to restate The p r i n c i p l e of d'Alembert-Lagrange f o r constraints of the Cetaev type: In the ease of i d e a l constraints f o r every system of v i r t u a l displacements ST s a t i s f y i n g the conditions IS. SXV=O, the equation j_. •. . • • (1.3.5) ( y - l ? V , 3 N ) holds. - 15 - CHAPTER 2 EQUATIONS OP MOTION AND THEIR TRANSFORMATIONS. 2.1. Some General Considerations. In t h i s chapter we s h a l l derive the equations of • i motion i n various forms. The mechanical system w i l l be assumed to be subject to nonlinear non-holonomic constraints of the Cetaev type. The derivation of the d i f f e r e n t forms of the equations of motion w i l l be ei t h e r centred around the a p p l i c a t i o n of the p r i n c i p l e of d'Alembert-Lagrange, as given by the equation (1.3.5), or based on some transform- ations. Moreover, the equations w i l l e i t h e r involve the k i n e t i c energy or the energy of acceleration or some function R or K to be defined l a t e r . Of the functionsR and K the former w i l l be shown to coincide, under cert a i n conditions, with the energy of acceleration and the l a t t e r , under the same conditions, with the Gaussian constraint defined by (1.1.3) 2.2. The General Equations of App e l l . As shown by N.G. Cetaev [ l l ] , on the basis of Def- i n i t i o n 2 of a v i r t u a l displacement, the two fundamental p r i n c i p l e s of a n a l y t i c a l dynamics - the p r i n c i p l e of d'Alembert-Lagrange and the p r i n c i p l e of least constraint of Gauss - are consistent. I t i s , therefore, necessary that the application of either p r i n c i p l e should lead to the same form of the equations of motion. We s h a l l deduce the - 16 - so-called equations of Appell from the p r i n c i p l e of d'Alembert-Lagrange. These equations were f i r s t obtained by Appell \p] , using the p r i n c i p l e of least constraint of ' Gauss. However, the p r i n c i p l e of d'Alembert-Lagrange f a i l e d to give them. Let us consider a mechanical system whose po s i t i o n Is characterized by n generalised coordinates Q^,^* * • • » qn» and assume that i t moves under the most general type of nonlinear non-holomic constraints of the type of Cetaev. Let these constraints be expressed by r Ijn equations: ( 2.2.1) U t 'VV--'Vv^---4)=f^ t 'V\>° Further, l e t us suppose that the functional matrix I k I i s of rank r . According to Woronetz [29] we can then choose, without loss of generality, the l a s t n-r q^ (I =r+l,r +2,...,n) as independent parameters and solve the system of equations (2.2.1) with respect to q , (ot=*l,2,. • •, r) Thus we obtain the following equations (2.8.2) \ = \&1{%>-%> W V - ' W 8 ^ ' V '%)• The equations of transformation from the set of rectang- u l a r coordinates (x , ) to the set of (q ) variables are ~v s (2.2.3) \ ~ \ & \ \ > " ^ = \ & % \ > ( V = \,Zr..,2>tt) . - 17 - D i f f e r e n t i a t i n g the equations (2.2.3) with respect ' to tjWe get (2.2.4) V " i t " ' Substituting from (2.2.2) i n the equations (2.2.4) we f i n d , by putting a dash to every function of the independ- ent velocity-parameters: (2.2 .5) \=*(|JVV"'Vit;W"i)5V where represents terms not containing q^. Prom (2.2 .5) i t follows that (2.2.6) 5 x v „ 3 x v According to the p r i n c i p l e of d'Alembert-Lagrange we have (frt x _ X ) £ x - O , V. (V) v v/ v ' whereSx̂ s a t i s f y the conditions (1.3.1). Hence we have (2.2.7) h^v-^T^V 0- Since <Sqi are Independent, (2.2.7) leads to the r e l a t i o n s : (2.2.8) Introduce the function S = J_ rrv x x , - 18 - c a l l e d the energy of acceleration of the system, and substitute i n S the expression f o r x from ( 2 . 2 . 5 ) . Then v S transforms into S which i s a function of 4-L»$J[ ( i =r + l , r + 2,.. .,n), q g (s = 1,2,...,n) and t . By virtu e of ( 2 . 2 . 6 ) we obtain ( 2 . 2 . 9 ) l i - = rrv x 2 ^ = * v * 111 . I f we put . / ( 2 . 2 . 1 0 ) Q[ = \ ^ Z > the e q u a t i o n ^ 2 . 2 . 8 ) , with the help of ( 2 . 2 . 9 ) and ( 2 . 2 . 1 0 ) reduce to the form ( 2 . 2 . 1 1 ) — = V ; >. U = r +|,^,...,-n). 7)% These are the general equations of Appell* Corollary 1 . In 1948 G.S. Pogosov [26] obtained the equat- ions of motion f o r nonlinear non-holonomic constraints of the Cetaev type, using the p r i n c i p l e of lea s t constraint of Gauss. These equations follow as an immediate c o r o l l a r y of the equations ( 2 . 2 . 1 1 ) . Prom the relations ( 2 . 2 . 4 ) we have i • - . I f we put - 19 - we obtain from (2.2.6) and (2.2.12) the relations ( 8 . 8 . 1 4 ) 22r .25f.22b:-* 2a. 3<fc >% Put ting (2.2.15) 9 ^ X ^ , ^ X 151, i t follows from (2.2.10) and (2.2.14) that Hence the equations of Appell take the form a s ' ( 2 . 2 . 1 6 ) i f = 9 r * ^ : . These are the equations of motion obtained by Pogosov. Corollary 2. I t i s possible to,write the equations (2.2.16) i n a symmetric form. To thi s end, a l l we have to do i s to use the function S i n place of s ' • We have (2.2.17) l i ' W H - ^ S ^ . Also from (2.2.2) i t follows, on d i f f e r e n t i a t i o n with respect to t, that Hence by virtue of (2.2.15) we get (2.2.18) i | t = 2$i = _ a . . F i n a l l y the equations (2.2.16), In view of (2.2.17) and (2.2.18), assume the symmetric form - 20 (2.2.19) 2 L _ Q t = * . / ! § . _ Q , V = l ^ ) . . . ) t ; i = T T + l / + ^ - ) Y V ) . Corollary 5, Suppose we define r parameters \ , ' 1 2 by means of the relations (2.2.20) I f . - 9 , = a. Then the equations (2.2.19) give (2.2.21) iS__$ - A A l f t : But from the equations (2.2.1) we have 2k 1%.+ i f £ = o / 1 or, by v i r t u e of (2.2.13) we get (2.2.22) 2 k = f t l f e . Using (2.2.22), the equations (2.2.21) become (2.2.23) = (p> = I A - ^ ^ = r + l ^ , . The n equations (2.2.20) together with (2.2.23) give the equations of motion i n terms of Lagrange's m u l t i p l i e r s . - 21 •- • • 2.3. A New Form fo r the Equations of Motion. Here we again obtain the equations of motion f o r the mechanical system treated In the previous section. These equations, In place of involving S, the energy of acceleration of the system, w i l l involve a new function R which depends on the k i n e t i c energy of the system. Further- more, we s h a l l investigate the r e l a t i o n s h i p between the functions S and R. Suppose the p o s i t i o n of the mechanical system Is defined by n generalised coordinates q^,q 2» • • • ari& l©t the nonlinear non-holonomic constraints of the type of Cetaev, imposed on the mechanical system, be defined by r ^n equations of the form (2.3.D {st'\V"^v^Ai"'^*o,. t- a ,^- r>- I f the functional matrix i s of rank r and the q's are suitably numbered, we have (2.3.8) ^ \ ( t i V V " ' V V , , V » ' " - ^ * t f i % i ^ (^ = i > ^ , ; . . , r 5 i = r + j , r + ^ l . . . , t v l - s = i,*,...,Tu) . Let the cartesian coordinates, x , i n terms of v the generalised coordinates be given by the following equations: D i f f e r e n t i a t i n g these transformation equations t h r i c e with respect to t, we get - 22 - -fx* 1 X y (2.3.3)< 17 btvfcV H n , v ^ Y * i J ^ / 4 ^ W » ^ J +term.s not containing ( e ( ( ^ s l , 2 y . . . , r ; = r + l , r + 2,... ,-n) Hence, using the notation (2.2.13), we have (2.3 .4) 3xT Let T be the k i n e t i c energy of the system which, with the help of the equations of constraint (2.3.2), i s transformed to T' . Then we have the following r e s u l t s : / . . . • / • . . . 41' . . . . . ... T = *«. v. x- , \ — m. x x , T = nv x x. +nv x x % i V > V (V) v y •> * — -' -* — - - Using (2.3.3) and (2.3 .4 ) , we f i n d (2.3.5.) 3 ^ + — - r Let us now introduce a function T , which i s the function T considered as a function of the q's and t only, i . e . f o r fixed values of the q's. In what follows we denote by a. the fixed value of q's, s =. 1,2,... ,n. With SO 5 t h i s i n view, corresponding to the expressions (2.3.3) we get the following expressions: - 23 - x - 3 * , "3* ~bx.. (2.3.6) < V -t 5 V \ +terms not containing q«_ Since the second of the relations (2.3.6) shows that i t follows from the relations (2.3.6) that (2.3.7) m , X Because of (2.3.7) the r e l a t i o n (2.3.5) becomes (2.3.8) I f ) v - * i Let us define a function R'as follows: K - !_ (T — 3 T The r e l a t i o n (2.3.8) then reduces to - 24 - (2.3.9) 2* ,^x,2*» ^ i . f ^ . - . * ) . sit a l i But from the p r i n c i p l e of d'Alembert-Lagrange we have or, the independence of5q i"s leads to the r e l a t i o n s (2 3 10) m x ( i = r + i , ^ ^ . " , ^ where , , ^ . 9j - X l i e . 4 " v a f e Prom (2.3.9) and (2.3.10) i t f o l l o w s that (2.3.11) 3R' ~ ' which are the r e q u i r e d equations of motion. Comparing the equations (2.2.11) and (2.3.11) we observe that both S and R s a t i s f y the same e q u a t i o n . In ot h e r words, the f u n c t i o n R c o i n c i d e s with the f u n c t i o n s' , the energy of a c c e l e r a t i o n of the mechanical system, as f a r as the terms i n q^ ( i =. r •+ l , r + 2, .. . ,n) are concerned, F u r t h e r , l e t R denote the f u n c t i o n R7 without t a k i n g i n t o c o n s i d e r a t i o n the equations of c o n s t r a i n t (2.3.1), i . e . without changing the dependent q i n t o q. * I and q i n t o q.. Then we have - 2 5 - (©1 = 1,̂ ...̂ ; i = r+i ,r y n , ) . In view of ( 2 . 2 . 1 8 ) the above r e l a t i o n s become ( 2 . 3 , 1 2 ) = - ***4- Also from ( 2 . 2 . 1 4 ) and ( 2 . 2 . 1 5 ) we have ( 2 . 3 . 1 3 ) By v i r t u e of ( 2 . 3 . 1 2 ) and ( 2 . 3 . 1 3 ) the equations of motion ( 2 . 3 . 1 1 ) assume the symmetric form ( 2 3 1 4 ) 1 1 - 9 . - A / 2 l ^ Q ( 2 . 3 . 1 4 ) ^ ^ o < ^ ^ ^ Comparing the equations ( 2 . 3 . 1 4 ) with ( 2 . 2 . 1 9 ) , we f i n d that R and S both s a t i s f y the same d i f f e r e n t i a l equations. Consequently the function R coincides with S as f a r as the terms i n q g (s = . 1 , 2 ,.. .,n) are concerned. Special Case. Let the l i n e a r non-holonomic constraints be of the form where A ^ , A^ are functions of Q-^qg,. • ̂ q ^ a n d *• Then 0(4. ^ «*- K Accordingly the general equations of motion ( 2 . 3 . 1 1 ) by - 26 - vi r t u e of (2.3 013) reduce to the following ones: (2.3.15) ^5- = ?i+h9 , ( c < = W V ^ ^ W v > The above equations f o r the l i n e a r non-holonomic systems were established by I.Cenov \_$~\ • 2.4. A Transformation of the Equationsof Motion. In the preceding section we obtained the equations of motion i n the symmetric form. (2.4.1) 2* -ft-a.(2£.^oV- U^...,r,U^^...,^. Let us define a function K as follows: (2.4.2, K - R - Q ^ - Q ^ . A. o(, Then by virtue of (2.2.18) we obtain and, consequently the equations' (2.4.1) reduce to the form (2.4.3) 2ii=° .,«.). Moreover, l e t K denote the function K when the equations of constraint (2.2.1) are taken into consider- a t i o n . Then K w i l l s a t i s f y the equation (2.4.4, K ' . R'-ty - f y , where q i s considered as a function of q (I=r+l,r+2,..,,n) 1 - 27 - Hence we again have US! = lJ?L _ Q. + a Q . By v i r t u e of (2.3.13) the equations of motion (2.3.11) then assume the form ( 2 . 4 . 5 ) ^ 7 - = ° (Uol.r+a.,.-.."-)- The equations of motion i n the form (2.4.5) show that the function K' assumes stationary values i n the actual motion when compared to any conceivable motion (consistent with the constraints), obtained by varying q^ i n K' • In the next section we s h a l l prove that the function K' I S a c t u a l l y a minimum along the actual motion of the mechanical system. 2.5. The Function K and the Gaussian Constraint. In order to show that of a l l t r a j e c t o r i e s consist- ent with the constraints, the actual trajectory i s that which has the least value of the function K , we s h a l l f i r s t prove that, as f a r as terms i n q^ are concerned, the function K' coincides with the Gaussian constraint defined by the equation (1.1.3) I f G denotes the Gaussian constraint, we have G = J_rrv (£i£ _ x Y _ _X X . tuumJk ru>t czmlai/alrva x. . = i _ rrv X x - 28 - The f i r s t term en the right-hand side i s the energy of acceleration s' obtained by taking constraints Into account. I f i n the second term we substitute f o r x i t s •v expression from the second of the relations (2.3.3), we get / G =S-X / + 2iM/)+terms not containing q. As remarked i n Sec.2.3 the function R coincides with S as f a r as terms i n q are concerned. Therefore, we can write 0 = R _ P ^ - ^ ^ •+ terras not containing q i _ K +terms not containing q- Thus the truth of the assertion i s proved. Next, to show the minimum property of K , we only have to prove that this property holds also f o r G. To establish this r e s u l t , l e t x be a t y p i c a l component of acceleration i n a t r a j e c t o r y under consider- ation (which i s supposed to be kinemetically possible but Is not necessarily the actual t r a j e c t o r y ) . Further, l e t x •vo be the corresponding component of acceleration i n the actual t r a j e c t o r y . We also assume that at the time t the coordinates, x v , and the v e l o c i t i e s , x , of the system are the same i n the considered,and the actual t r a j - ectory. Then, i f dx i s the change i n x v along the actual t r a j e c t o r y i n an i n t e r v a l of time dt, and Sx i s the change V •' ,4- along the considered trajectory i n an i n t e r v a l of t Ime S t = , we have - 29 - *• Sx_. " ^-x (2.5.1) **4r* » x w s * ' S't ' v o ©Lt Now, according to equation (1.3.2) a small d i s - placement of the system,Sx , which Is proportional to dx^ - Sx̂ , i s consistent with the equations of constraint, i . e . i t i s a v i r t u a l displacement. Hence the p r i n c i p l e of d'Alembert-Lagrange can be written i n the form / m x _X V i x - S*0 = °> or, by v i r t u e of (2.5.1), i n the form (2.5.2) (™- x - X V x _ x V -o. or, f i n a l l y i n the form _L>w /2^ _ x ^ - J L r r v _ X ^ = J_rrv (x _ x f . Since the terms i n the summation on the right-hand side are a l l p o s i t i v e , i t follows that which establishes the r e s u l t . Remark 1. For a l i n e a r non-holonomic system the mimimum property of the function K' was proved by I.Cenov \lOr\. Remark 2. As a consequence of the fact that the function K' coincides with the Gaussian constraint G as f a r as terms In q are concerned, i t follows that (2.5.5) =0. (i=T +l,l- +*,...,Tv). - 30 - These equations establish the stationary property of G. That the stationary property automatically leads to a minimum has already been proved above. Since the equat- ions (2.5.3) were deduced from the p r i n c i p l e of d'Alembert- Lagrange, the compatibility of t h i s p r i n c i p l e with the p r i n c i p l e of least constraint of Gauss i s i n d i r e c t l y established. The following i s an a l t e r n a t i v e but i n t e r e s t i n g approach of deducing the p r i n c i p l e of Gauss from the princ- i p l e of d fAlembert-Lagrange. Prom the equation (2.5.2) we have (2.5.4) (^*»-XjCV%o) s But since x̂ "~x±0 represents the change i n the acceler- ation causing a deviation In the t r a j e c t o r y of the p a r t i c l e , we can put . Sx _ x _x . v ~ v VO The equation (2.5.4) then reduces to fnv X _ X ) S'x-O. Since the forces applied are given and cannot be varied, the above equation may be rewritten as follows: X3L ) - O . This, however, means that JL_7W _L3 _ X This again establishes the stationary property of the Gaussian constraint f o r the actual motion. To prove that - S l - i t is a c t u a l l y a minimum we can proceed as before, 2,6. Lagrangian Form f o r the Equations of Motion. In Sec. 2.3. we obtained the equations of motion f o r a nonlinear non-holonomic system i n the form given by equations (2.3.14), i . e . (2 with where T i s the k i n e t i c energy of the system without taking Into consideration the constraints imposed on the system, and T„ i s the value of T f o r f i x e d values of the general- ise d v e l o c i t i e s q_ (s =1,2,...,n). Here our aim i s to transform the equations (2.6.1) so that they assume a form s i m i l a r to Lagrange's equations of motion;. To t h i s end we f i r s t prove the following Lemma: For R and T defined above, the i d e n t i t y dR (2.6.2) *% *% holds f o r s =.1,2,... ,n. Proof: We have r T = ± . (2.6,3) -rY\, X X T — x X j. Y T V x x - 32 - Moreover, since we get 2t (2.6.4) < Prom the relations (2.6.3) and (2.6.4) i t follows that - # m. x y 4. TA. x — — i - (*0 V ^ , T (V) v 3f s '14 or, ( 2 . 6 . 5 ) 1 % Now i f we denote the fixed value of q by q , then s so ' (2.6.6) 3% rso at l a + X =: where i n the f i r s t term on the right-hand side of the l a s t r e l a t i o n we have interchanged the repeated s u f f i x e s . For these expressions of x , x and x. we have •v v if - 33 - T" = — T Y V x i. , T = XX., -r . . . . . ... Hence, i n view of (2.6.6) we f i n d or, JL^ - m x J J : fL + _ 1_ q (2.6.7;) Thus (2 . 6 .5 ) with the help of (2.6.7) reduces to • • * - or, » 2r\ ?x (2.6.8) ^ - x v ^ ' Also we have - m x' +m x L- ??2L x i f*Xy . s But from the f i r s t two relations of (2.6 .4) i t follows that Hence we get (2.6.9) (S= l , ^ - , ^ ) . By virt u e of ( 2 . 6 . 8 ) and ( 2 . 6 . 9 ) the Lemma is. established. The above i s an independent proof of the lemma which, of course, can be e a s i l y established i f we recognise the fact that R coincides with S, the enrgy of acceler- ation, as f a r as the terms i n q are concerned, and make s use of the well-known r e s u l t l£ = f L i I _ H (s= .,*,•••>-). Let us now use the i d e n t i t y ( 2 . 6 . 2 ) to transform the equations of motion ( 2 . 6 . 1 ) . This leads us to the following form of the equations of motion ( 2 6 1 0 ) iL2T._2L._9. m a (<LIL __L_ (o< = i ^ i . i . , r i i = Y , + i , t l + * , . . V T f ^ ' . These are the Lagrangian equations of motion f o r the non- l i n e a r non-holonomic systems. Some Special Cases; Case I . I f the system i s holonomic with n degrees of free- dom, we have The equations ( 2 . 6 . 1 0 ) then reduce to the usual form: it y% }% s Case I I . I f the system i s l i n e a r nonholonomic, the con- s t r a i n t s are given by non-integrable equations of the type-, where A ^ , A are functions of q^qg, • • • »<ln a n d - 35 - In such a case we have With these values of a . the equations ( 2 . 6 . 1 0 ) become o< -C Case I I I . Let us define r parameter A ^ , ̂ i n the following manner: ( 2 . 6 . 1 1 ) A.?Z_Eo = \ i f f i - where are the equations of constraint. Then the equations ( 2 . 6 . 1 0 ) y i e l d J i t 2 % * " p * P ^ Using the relations ( 2 . 2 . 2 2 ) , the above becomes ( 2 . 6 . 1 2 ) i - T ^ - — ( f - » A . . ^ , i = T + l > Y ^ . . ^ . The equations ( 2 . 6 . 1 1 ) together with . , ( 2 . 6 . 1 2 ) represent the equations of motion i n terms of the Lagrangian m u l t i p l i e r s . 2 . 7 . Another Transformation f o r the Equations of Motion. In the l a s t section we considered the equations of motion i n the so-called Lagrangian form, involving the - 36 - k i n e t i c energy T. I t i s assumed that i n the expression f o r T no substitution has been made f o r the dependent v e l o c i t i e s i n terms of the independent ones, i . e . con- s t r a i n t s have not been taken into account. In the present4 section our aim i s to transform the above-mentioned equat- ions by changing T into T , i n which the dependent v e l o c i t i e s have been expressed i n terms of the Independent ones. Let us assume that the equations (2.6.10) can be thrown into the following form: (2.7.1) i - - ^ ! ' - — = Q - * Q (^i,v^i--r + W,.vu) <* *\ d\ < • < . < • where i s a corrective term to be determined l a t e r . By v i r t u e of equations (2.6.10) we obtain from (2.7.1) the following expression;'; f o r : (2.7.2) A = i - 2E -21 -o-Jl^X . But on using (2.2.18) we have (2.7.3) -^r and ( 2 7 4 ) i l = *L ( ^ ^ l A - ^ ^ ^ r + v ^ . . ' * - *\ By virtue of (2.7.3) and (2.7.4) we get from (2.7.2) - 3 7 - the following expression f o r : ( 2 . 7 . 5 ) ^ = + l t - ) . Let us now regard T as a function of q^ ( < * = 1 , 2 , r) only, and i n the sequel denote i t by T, . Then using ( 2 . 2 . 1 8 ) , we have ______ i l {2 7 6) ±ZL= - a , , < L * L _ * . _ _ L , and ( 2 . 7 . 7 ) 3T, _ 2T 1% 7 \ ^ 3<fo Using ( 2 . 6 . 2 ) , ( 2 . 7 . 6 ) and ( 2 . 7 . 7 ) , we get from ( 2 . 7 . 5 ) the following expression: r- D. - a 201 + I I I-A <L__L --L2IL o r . ( 2 . 7 . 8 ) D. _ _.fL___L B R F i n a l l y l e t us consider R a s a function of q ( * = l , 2 , . . . , r ) only and i n the sequel denote i t by R, - 38 - Then by v i r t u e of (2.2.18) we get ( 8. 7. 9 ) 2^ . - a ^ In view of (2.7.9) the f i n a l expression f o r D^, given by (2 . 7.8), becomes D UL L EL + l ^ i . With this expression f o r the equations (2.7.1) take the form ( 2 . 7 . 1 0 ) j L i ^ . i n V . i i i . f L ^ - dt * \ 2 \ 2 > % dt d\ ly. These are the required transformed equations. In the case of l i n e a r nonholonomic.systems the equations of motion i n the form (2.7.10) were established by I. Cenov [8] . 2.8. A Novel Form f o r the Equations of Motion. Once again we s h a l l transform the equations of motion (2.6.10) by means of an i d e n t i t y to be established below. This novel form of the equations of motion w i l l include as a special case the r e s u l t obtained by I. Oenov £ 7 } f o r holonomic systems. Let us f i r s t prove the following Lemma : I_f T denotes the k i n e t i c energy of a_ holonomic mechanical system, then the i d e n t i t y 3 9 - ( 2 . 8 . 1 ) holds f o r s = l > 2 ,...,n. Proof: We have ( 2 . 8 . 2 ) T = L T = l u x x Also we have f o r A , S = 1 , 2 , . . . , n the following expressions ( 2 . 8 . 3 ) . < T f l f e + I T X =. d% % + 4 s ^ ^ s V *%at V at* ' Now i n view of the re l a t i o n s ( 2 . 8 . 2 ) and ( 2 . 8 . 3 ) . we f i n d that ( 2 8 4 ) ~ ^ - *V X X 2**, and ( 2 . 8 . 5 ) . - ~ ^VZj)^—• Since from ( 2 . 8 . 4 ) and ( 2 . 8 . 5 ) we have - 40 - By virtue of (2.8*3) the above becomes i 21-21 = ^ i f i ^ i t apt \ay\rfc rutjj - _TTV x ax. • r r This proves the Lemma. Taking into consideration the i d e n t i t y (2.8.1) we have ( 8 . 8 . 6 ) v at a«ys / . t - a ^ Hence the equations of motion (2.6.10) assume the form (2.8.7) -21 - Q.a a.. /*i-2I .21 -<?Y ( r f s l , » r . / j i s f + l , T + » , - | * ) - Special Cases: Case I. I f the system i s holonomic with n degrees of free- dom, we have CL , - o f o r 0(=:1,2,... , r ; i « r * - l , r+2,.. .,n, In this case the equations (2.8.7) reduce to - 41 - a r e s u l t of I.Cenov \j] • Case I I . I f the system i s l i n e a r nonholonomic, the con- s t r a i n t s are given by non-integrable equations of the type: where A ^ , A^ depend only on Q-^*^* * • A N D Hence Consequently the equations of motion (2.8.7) take the form ZA.K - ZL-Q. 4 . UlUl - l l - Q \ . 0 (oi _ i, zy , v r ; *. a . f 4 I,T+*,. . •, TV) . Case I I I . Let us define r parameters ,Ao,...,/\ by J- __ I* means of the equations ,8.8.8, AJLIJ: - _ £ - <?, = ^ ( < , p . . . W ) , where i- are defined by the equations of constraint (2.2.1). Then from equations (2.3.8) we have (2.8.9) " > V ^ - Changing the order of summation on the right-hand side of equations (2.8.9) and making use of the r e l a t i o n s : 2 j> - CL _____ we can write the equations (2.8.9) i n the following form: - 42 - (2.8.10) *A2I.2L-Q. = > i k . The set of equations (2.8.8) together with (2.8.10) forms the equations of motion with r Lagrangian m u l t i p l i e r s . 2.9o Another Novel Form fo r the Equations of Motion. Here we s h a l l transform the equations of motion (2.8 . 7 ) so that they assume a very simple and novel form, •This i n t e r e s t i n g r e s u l t w i l l include as a special case a result obtained by I.Cenov [7] • To obtain the equations of motion i n the desired form l e t us f i r s t prove the following Lemma: If T denotes the k i n e t i c energy of a holonomic mechanioal system, then the i d e n t i t y (2.9.1) i - 2 1 + i t - * holds f o r s=l,2,...,n. •Proof: We have T = J_ -m. x x 2. w V v 7 (2.9.2) < where (2.9.3) J T f = w. x x (V; V V -TYV x x +-m- x x . X V X = 3* 2% (£,S= l , V - v ^ ) - - 43 - By vi r t u e of the relations (2.9.2) and (2.9,3) we have - * m x _ V + T r v X i i ( i _ _ U ^ v X ____ since How d i f f e r e n t i a t i n g the l a s t r e l a t i o n of (2.9.3) we get (2.9.4) * . S g i ^ ^ j . ^ ) + t e m 3 n o t o o n t a l n l n g 5 In view of the expressions f o r T from (2.9.2) and x from (2.9.4) we get _L_L _ A T * x. + m , x ___ 31̂ , 3% n s " S 3 ' S " _ ^ n v x ? _ k + 3 - i f i ^ + - ^ 0 On comparison of (A) and (B) the i d e n t i t y (2.9.1) follows. In section 2.8 we obtained the equations of motion i n the form - 44 or, With the help of the i d e n t i t y (2.9,1) this reduces to (2.9,6) Now l e t T̂ denote T considered as a function of q's only, and l e t T̂ denote T regarded as a function of q's only. Then we immediately have i 2 i = Z L -a. i l l 21 a iL By virtue of (2.9.7) and (2.9.8) the equations (2.9.6) take the form A 5_ _ 3 5- = Q . ( i = w + * > . . . , - ) > which are the equations of motion i n the desired form. 2.10. Transition from Equations of Motion with Lagrange's M u l t i p l i e r s to Equations Free from Them. In several previous sections we derived the equations (2.9.7) and (2.9.8) - 45 - of motion for a nonlinear non-holonomic system i n various forms which did not involve the undetermined m u l t i p l i e r s of Lagrange. Later, however, by means of a certain trans- formation we obtained from them the equations of motion containing the said m u l t i p l i e r s . Here we propose to con- side r the converse problem, I.e. the t r a n s i t i o n from the equations involving the undetermined m u l t i p l i e r s of Lagrange to those free from them. In the case of a l i n e a r non-holonomic system such a problem was solved by I . I . Metelicyn i n 1934. Let us s t a r t from the equations of motion obtained i n section 2.6, where (2.10.2) f*(t5VV=° (« = 'A-^> are the equations defining the nonlinear constraints. To obtain the equations (2.10.1) the following assumptions were made:- ( i ) The functional matrix »% i s of rank r . ( i i ) The functional determinant (2.10.3) - 46 - Under these assumptions i t i s possible to apply the imp- l i c i t function theorem i n order to obtain the following expressions f o r the dependent v e l o c i t i e s : ^tal,*,...,^i*-»<,*,»1,**»,-*'n'» On d i f f e r e n t i a t i n g the re l a t i o n s (2.10.2) with respect to q. we f i n d (2.10.4) In view of (2.10.3) we can solve the system of equations (2.10.4) f o r d % , obtaining the following expressions: LL - - a. (2.10.5.) a ^ * = _ a- .c_ 3(-f.>fi>- 3(f.>f* by Mg , (2.10.6) 3<K Denoting the left-hand side of the equations (2.10.1) we can rewrite them i n the form A »>* I f we introduce the notation / » . \ \\ we can rewrite the equations (2.10.6) i n the matrix form - 47 - (2.10.7) ' T V A = Let us now p a r t i t i o n the matrices i n (2.10.7) i n the following manner: f H_ If*... 2fe H« A _ \ >% * V irv, = B _ V 11 _ik . a-JL 1W - -IV, Then the equation (2.10.7) i s equivalent to the matrix equations (2.10.8) ] 1 B A _ TTTV Taking into consideration (2.10.3) we can eliminate between the equations (2.10.8) to y i e l d - 48 - (2.10.9) I f |A.| denotes the determinant of A and A i s the cofactor of l i s i n \k\, we have B A ' - J - A.. A , . ; . . . " . A IT i i i 'TV \ A A .. . A I A l i i i A i i .. 2k A (.<= i , ^ . - . , r ; . ' But f o r o< ,p> =1,2,.. .,r and i =r+1,r+2,... ,n by v i r t u e of (2.10.5) we have i k A = i t A + If* A A Hence we get (2.10.10) BA =. / a . a a. tyr+i a. a. T/n. - 49 - Substituting from (2.10.10) into (2.10.9) we f i n a l l y obtain a. a. CL i.-n. a. r-,n TTV. = TYV. or. Writing out the f u l l expressions f o r and M± the l a s t equations become These equations as established previously are the equations of motion free from Lagrange's m u l t i p l i e r s . 2.11. The Equations of Motion i n the Form of Determinants. (2.11.1) Starting from the equations of motion i _ j _ J _ _ J _9 5= a JA- _tL _____ _ we propose to rewrite them i n terms of determinants a l l of which can be obtained from a certain matrix according to a general scheme. For a l i n e a r nonholonomic system such a; 50 - problem was solved by I . 1.Metellcyn [21] If we put (2.11.2) (S.l,*,..,,1".) the equations (2.11.1) take the following form: (2.11.3) Pi _ a M = o (*s|>V-»r;U™,T+x,...JT0. Now l e t us consider the determinant (2.11.4) Zk H t H e I f we denote by A the cofactor of M (s=l,2, ...,n) i n |A°I , we f i n d , on expanding (2.11.4) i n terms of the cofactors of the top row, that (2.11.5) NA+^A ( - . l A - . T i U f l , ^ " . . * ) But we e a s i l y f i n d that (2.11.6) (2.11.7) A A- HU*,' = (-) OCX. 51 - where a ^ are given by (2.10.5) With the help of (2.11.6) and (2.11.7) the expansions (2.11.5) become or (2.11.8) Since (2.11.9) *(%%••••»%) icTf-tiAl,-- ) ,v)' the equations (2.11.8) by virtu e of (2.11.3) y i e l d (2.11.10) |AW| = 0 (i-f-n/r+l,...,*). As a consequence of (2.11.9) and the fact that n - r determinants of the type (2.11.4) vanish, i t follows that any determinant of order (r+1) obtained from the (r.+l)xn matrix (2.11.11) i i If* n i i i i nr Mr+1. i i i i . . -ii i k . i i lit-. - 52 - must also vanish. Hence the equations of motion (2.11.1) can be found by equating to zero any determinant of order r-f obtained from the matrix (2.11.11). - 53 CHAPTER 3 APPLICATIONS 3.1. Some General Considerations. There are not very many known examples of mech- anical systems moving with nonlinear non-holdnomic constraints.' In 1911 Appell gave an example of such constraints. However, nonlinear non-holonomic constraints can be r e a l i s e d i n problems concerning the regulation of the motion, or i n other problems of technical interest where the constraints between the moving parts are re a l i s e d by means of ele c t r o - magnetic devices. It i s expected that with technical development the use of nonlinear non-holonomic constraints w i l l also increase. The procedure f o r solving problems with nonlinear nonholonomic constraints i s quite straightforward. To obtain the equations of motion one has only to write down T, the k i n e t i c energy, and the external forces i n terms of the generalised coordinates, and substitute them i n one of the many forms of the equations established i n the previous chapter. Let us now consider some examples of thi s procedure: 1. A system of two wheels and th e i r axle moving on a horizontal plane. 2. A disc moving on a horizontal plane 0 3. A heavy b a l l moving on a horizontal plane. - 54 - Despite the fa c t that the equations of constraint i n a l l these examples are e s s e n t i a l l y l i n e a r nonholonomic, they can be a r t i f i c i a l l y thrown; into a nonlinear form. The purpose of doing so i s two-fold. F i r s t , i t provides us with examples of nonlinear non-holonomic constraints. Second- l y , i t serves to i l l u s t r a t e the general treatment of the theory developed i n the previous chapter* In view of the l i n e a r i t y of constraints the solut- ions of the above-mentioned examples are well-known, but the method depends on the use of the equations of motion i n terms of Lagrange's undetermined m u l t i p l i e r s . This, of course, requires the determination of these m u l t i p l i e r s p r i o r to the actual sol u t i o n of the problem. But i n the methods employed below we use the equations of motion estab- l i s h e d i n Chap, 2, which are free from such m u l t i p l i e r s . Consequently the calculations become simple. 3*2, Motion of a System of Two Wheels and Their Axle on a Horozontal Plane. Let the axle be a z, z homogeneous rod of length 2a and mass m-̂, and the wheels ,v be two homogeneous disc s , which are fix e d normally to the rod at the centres 0 and 0 each of radius a and mass m, and free to turn about i t . I Let ' 0, x, Tj z , be a - 55 - reference system fixed i n space and l e t the wheels move on the plane z-o (the wheel with centre o' having a contact without f r i c t i o n and that with centre 0 having a per f e c t l y rough contact). Suppose we introduce an intermediate trihedron Gu.vz at the centre, G, of the rod with Gu along the rod, GV horizontal and perpendicular to Gu. , and Gz v e r t i c a l . The parameters, characterizing the p o s i t i o n of the system, are the coordinates (x-^y-^) of the centre G, the angle Y which G u makes with O^x^ , and the angles of rotation 4> and X of the two discs with centres 0 and C-' respectively. The well-known theorem of Konig, when applied f i r s t to the entire system, then to each disc, immediately gives f o r the k i n e t i c energy, T, the following expression: (3.2 .1) » T « K+K-OI*,**.)' +(£ +-^)*V+^V+={_*\ Since the forces of gravity do no work we can assume that there are no externally applied forces. I f I denotes the instantaneous point of contact, we s h a l l express the kinemetical condition of the absence of s l i d i n g at I by means of nonlinear (with respect to the v e l o c i t i e s ) d i f f e r e n t i a l equations. The absence of s l i d i n g demands that the v e l o c i t y at the point I of the disc be zero. But this v e l o c i t y i s the resultant of the v e l o c i t y of G and of a (Y + <T»), p a r a l l e l to Gv , due to the rotations y and <j> o Hence we must have - 56 - (3.2.2) < ft. 3, These are the equations of constraint of which the f i r s t Is nonlinear i n v e l o c i t i e s . Solving the system (3.2.2) f o r x, and y, we get x | B a ( y + 4 ) s ^ T ' (3.2.3) Taking y , «j> and "JC as the independent v e l o c i t i e s , we have, i n the notation of Sec.2.3, and Hence we get f' s (m^m . )a(n • )* + (!2i + f* + +*+ X + containing the second derivatives, To = T 0 = o . By v i r t u e of the above expressions the function reduces to or, as f a r as the terms i n second derivatives are concerned, - 57 - Using the equations (2.3.11) we have f o r the equations of motion of the system considered: (3.2.4) X =0. The equations (3.2.4) can be integrated, y i e l d i n g the three f i r s t i n t e g r a l s : (3.2.5) < where 4>° , "X are the i n i t i a l values o f y , * , " * respectively. The equations (3.2.5) are equivalent to (3.2.6) y _ y " ' , <£=4>° ft^X-OC. Integrating the above equations we get (3.2.7) y_y°t , 4> _<j>°-fc , x_x°-fc. By virtue of (3.2.6) and (3.2.7) we get from the equations (3.2.3): =• «-(V+4>,)*I"'y** > Integrating the equations (3.2.8) and suitably (3.2.8) choosing the a r b i t r a r y constants, we get x = - i - 58 - The l a s t equations show that the trajectory of Y t 4> , described the centre G i s a c i r c l e , of radius a. with a uniform v e l o c i t y , 3.3. Motion of a Heavy Circular Disc on a Horizontal Plane, Let a c i r c u l a r disc, of unit mass and radius a, r o l l (without s l i d i n g ) along a fi x e d h o rizontal plane o, x, y(. Let the centre of I n e r t i a , G, of the disc be the centre of the figure and the central e l l i p s o i d of i n e r t i a be an e l l i p s o i d of revolution about Gz of the d i s c . The parameters characterising the position of the disc are the Eulerian angles 0,Y,4> and the coordinates x, , y ( of the point G, f o r which z, i s obviously equal to a sine I f Guvz i s an intermediate trihedron, the components p,q,r of the instantaneous rotation u> of the disc along the axes of Guvz are given by the expressions: (3.3.1) |>=d, <y=Y^e , r = 4» + Y£«jeJ whence we get (3.3.2) '6=\> Y ^ 6 ^ , y t o « e - ^ e * t e . I f i s the v e l o c i t y of the centre G, the v e l o c i t y of the point of contact I i s given by the expression However, the kinemetical condition of the absence of s l i d i n g demands that the v e l o c i t y of I be zero. Hence we have (3.3.3) ^ + U)X (kl a O . - 59 - Since the c o o r d i n a t e s of I r e f e r r e d to-i-VZ-are (o,-a,o), the components of U ) * _ j l a l o n g the axes of Guvz are -O.T , 0 , o|a. When p r o j e c t e d along the f i x e d axes G, x, and 0,7, they become Hence the r e l a t i o n ? (3.3.3) gi v e s the equations of c o n s t r a i n t i n the form: (3.3.4) I x i +11, = +f ^ V J the f i r s t of which i s n o n l i n e a r i n v e l o c i t i e s . S o l v i n g the equations (3.3.4) f o r x and y , we get (3.3.5) Hence (3.3.6k where, as f a r as terms i n the second d e r i v a t i v e s are concerned, we have (3.3.7) < - 60 - In the notation of Sec. 2.7 we have where A,C are the p r i n c i p a l moments of i n e r t i a of the disc with respect to Gu and G z respectively. Hence , (3.3.8) * T = (A -t-̂ ) A + CC + r , (3.3.9) ^ - x * + ̂ * + I f T,0 denotes the expression of T, f o r f i x e d values of x, and y, , then c l e a r l y we have Consequently (3.3.10) = TC * . * ^ ) + ^ X , ^ , - With the help of (3.3.7) and (3.3.8) we get (3.3.11) and H- = Acyy cose -(C+cL^yM^e , 3+ r 2 T / - = (A+*)!>. 30 (3.3.12) J = A ^ e + (C+a)rc*se, Again, with the help of (3.3.5), (3.3.6) and (3.3.9) we f i n d that (3.3.13) < - 61 - " _ _ a. Y - 6 + a- P coteA*n.e} m - o 2* and (3.3.14) ( 20.= -t*T«a, 3Y F i n a l l y , from (3.3.6), (3.3.7) and (3.3.10) we f i n d dB H i 3* The 9- equation, written with the help of (2.7.10) and s i m p l i f i e d with the help of (3.3.2) gives (3.3.15) (A+flf^-A^cote +(C +«.*)^ =-Cj< S i m i l a r l y , the -y- a n d 4* - equations are (3.3.16) -Ai/ta^e + (C+«*)rtwe]_i_^Aorf©)- /c^a^c*©-*) = 0, •<- " at - 62 - (3.3.17) + |^ -°- Simplifying (3.3.16) with the help of (3.3.17) we get (3.3.18) ^ +(»(A^Cot0_C^ _o. The equations (3.3.15), (3.3.17) and (3.3.18) are the well-known equations describing the motion of the d i s c . 5.4. Motion of a Heavy B a l l on a Fixed Horizontal Plane. Let a heavy non-homogeneous sphere, of centre 0 and radius a, r o l l and pivot without s l i d i n g on a horizon- t a l plane z-,~o of the fixed reference system O^x^y^z. . Let us also suppose that the centre 0 of the sphere i s ./ the centre of i n e r t i a and the central e l l i p s o i d i s an e l l i p s o i d of revolution about a diameter 0 Z of the sphere, where Oxyz. i s a trihedron r i g i d l y connected with the sphere. The parameters of the sphere are the coordinates ^ xl»^l» °? * n e point 0 and the three Eulerian angles _>,y, 4> . The condition that the sphere remains i n contact with the plane Z(=o gives ' Z - a- =0. t The condition of contact without s l i d i n g demands that ^ = 0 » Vj. being the v e l o c i t y of the point I of the sphere which i s i n contact with the plane. But where i s the v e l o c i t y of 0 and LO i s the instantaneous r o t a t i o n . The components of V ,10 and , along the axes - 63 - fi x e d i n space, are respectively U , , 7 - 1 , 0 ) , (p « - ) 1' X 1 and (o,o,-a), where (3.4.1) \ «̂ = Y + e ^ f , The requirement of the absence of s l i d i n g demands that (3.4.2) S i . <V, ' the f i r s t of which i s nonlinear i n v e l o c i t i e s . Taking into consideration the relations (3.4.1) the equations (3.4.2) are equivalent to D i f f e r e n t i a t i n g the equations (3.4.3) with respect to the time, we get (3.4.4) < The k i n e t i c energy, i n terms of the notations of Sec. 2.9, can be written as (3 .4.5) * T =y^(xtff{)^le\^W&yC(^^^ , where rrv i s the mass of the sphere and A and C are the moments of inert ia about 0 ll and Oz , Ou-VZ. being an inter- mediate trihedron. - 64 - Prom (3.4.5) we get on d i f f e r e n t i a t i o n with respect to the time - • *• jt» A % + 0(4. +yC _ J © ) ( $ + Y _ * e ,y9^v»)) T = + A ( e \ y ^ e + * T N ' « ^ e + i . 0 / ^ * 0 ) + +terms not containing the second derivatives. Using the notations of Sec.2.9 we have, with the help of (3.4.3), (3.4.4), (3.4.6), and (3.4.7), the following expressions: lUs. ^3^Jh^)& +(^~C)p^^B ^L^^C)^^^8 , Ye 3© K * % ___ - 3 A ( y ^ r 5 a +y e + GTy c^e +*<~<se_<f exsi-.e-yfl/^*6'), Hence the equations of motion are (3.4.8) (k^i)&J^-C)^^-L{^^C)^X^Z6 .0, (3.4.9) ^(Y/kv?0 +Y_>^^9)4 - C(y - « * 0 + * c ^ e-*a^ e - y 8 .-U-v^e)-O, (3.4.10) - w a J J Y - ^^e -^e^e^^e+C^+YCcde _Ye~^e ) - o . (3.4.6) (3.4.7) - 65 - Equation (3.4.10) can be written as dt^ 7 «ttv 7 or, (3.4.11) '»r^^'4**v®^(^<*^e)+^-^(<*,+Y^e)= -mft Y/iUve. Also the equation (3.4.9) can be put i n the following form: f o r which a f i r s t i n t e g r a l i s obviously (3.4.12) A ( Y ^ u e ^ + C ^ - i-Y ^ e ) c<rfe = constant. W© can ignore the equation (3.4.8) since i n the case under consideration we have • , . which, being a quadratic form i n 0,Y » leads to a f i r s t i n t e g r a l (3.4.13) T = constant. Hence the motion of the sphere i s completely determined by the equations (3.4.11), (3.4.12) and (3.4.13) - 66 - CHAPTER 4 QUASI-COORDINATES OR NON-HOLONOMIC COORDINATES 4 . 1 * Some General Considerations. Let P^qg* • • • D® t h e generalised coordinates defining the position of a mechanical system. Following Hamel [ l 8 J , l e t us introduce n parameters ^ 1 * w 2 » ' ' * » w n ' c a l l e d the k i n e t i c c h a r a c t e r i s t i c s , by means of the r e l a t i o n s : ( 4 . 1 . 1 ) V % % + * C t , S = ' > * " h where .a , â are functions of Q^,Q£» • * a n d * n general the P f f a f f i a n forms are non-integrable. Assuming that the determinant of the c o e f f i c i e n t s £-£S i s non-zero, we can express as l i n e a r functions of fe^. Let these functions be ( 4 . 1 . 2 ) V V S + A i c£>s=>,v-,~) where (4.1.3) V " ' V-S**« ^ ' A - ^ . |j 0.̂ 1 being the inverse of the matrix With each to^ we can associate a quantity d , defined by - 67 - (4.1.4) ^ - V ^ V V V 1 The quantities <L 7̂ are ca l l e d the d i f f e r e n t i a l s of the quasi-coordinates [28] or the d i f f e r e n t i a l s of non-holon- omic coordinates • I f the forms £s Ts + i . are exact d i f f e r e n t i a l s , 7 ^ exist and are the true coord- i n the usual sense; otherwise ^ do not e x i s t . The variations, representing the v i r t u a l displace ments, of q_ are given by (4.1 .5) Stfj ^ \ S % ' (A,s_i,*,».,~) . In case of a holonomic system, with T t degrees of freedom, a l l of S a r e Independent. However, I f the system Is subject to l i n e a r non-holonomic constraints of the type: A*$\ + \ t - ° ^S_i,*,...,Ti.;<tsi,ai...Jr<^') we can take so that the equations of constraint become U^-o (* =I)V-->T>). Corresponding to we have The remaining 8'fy ( I - r+l,r+2,...,n) are independent. In 1957 V.S.Novoselov [23] generalised the d e f i n i t i o n of non-holonomic coordinates. His d e f i n i t i o n includes as a special case the above given d e f i n i t i o n of nonholonomic coordinates f o r holonomic or l i n e a r - 68 - nonholonomic systems. I t i s well-suited i n case where the constraints being nonlinear nonholonomic are of the type of Cetaev. Following the point of view of Novoselov, l e t us define the k i n e t i c c h a r a c t e r i s t i c s by the r e l a t i o n s : (4.1.6) ^ = ^ O i V V (£,s=i,V-^), where «fy are not necessarily l i n e a r functions of q s . If the functional matrix m i s of rank r , we s h a l l have (4.1.7) \ = \ ( t j V % ) (4,S-<A-,*9< The v a r i a t i o n s , representing the v i r t u a l displace- ments, of q | are defined by (4.1.8) H 3 8 — > (l,s.!,*,-.*). The SA s i n (4.1.8) are c a l l e d the d i f f e r e n t i a l s of nonholonomic coordinates • For nonlinear non-holonomic systems subject to constraints of the type of Cetaev and expressed by the equations "t^'V'i^ = ° *̂i,*>...,r<7w;s»i,*,.»»«.) we take The remaining ( i = r + l , r + 2, • •.,n) are a r b i t r a r y funct- ions of the form (4.1.6). The n - r independent <5fî - 69 - s a t i s f y the relations k - ____ in. ( i _ l > v ^ i i - * + , » ' * ' * * » " - . n ) . In case i n the equations (4.1.6) are l i n e a r functions of q we c a l l the d i f f e r e n t i a l s of l i n e a r nonholonomic coordinates; otherwise they are c a l l e d non- l i n e a r nonholonomic coordinates. 4.2. Poisson 1s Theorem i n Linear Non-holonomic Coordinates. In 1944 the c l a s s i c a l theorem of Poisson, dealing with the properties of integrals of canonical equations of dynamics, was extended by V.V.Dobronravov [17) to the case of canonical equations expressed i n l i n e a r nonholonomic coordinates. This generalisation was achieved by assuming the so-called k i n e t i c c h a r a c t e r i s t i c s to be independent of the time. We propose to generalise his result by taking the k i n e t i c c h a r a c t e r i s t i c s to be time-dependent. Consider a holonomic system f o r which the k i n e t i c c h a r a c t e r i s t i c s and the corresponding d i f f e r e n t i a l s of l i n e a r non-holonomic coordinates cLfi^ are given respectively by the equations (4.1.1) and (4*1.4) For such systems G.Lamparlello \20\ i n 1942 established the equations of Volterra-Hamel i n the form ( 4 2 1 ) Y* _ _ r + - r __•. Q; r , * Here T i s the k i n e t i c energy of the mechanical system expressed as a function of the time t, the coordinates q , s and the k i n e t i c c h a r a c t e r i s t i c s u> ; Q, are the generalised - 70 - forces corresponding to the l i n e a r nonholonomic coordinates V In the above equations the operator ^— means the r e l a t i o n (4 2 2) — = kdr (>,s sgi,...,w>, and the Ys a r e defined by the relations where the a's and b's are given by the relations (4.1.1) and (4.1.3), respectively. I t i s known that (4.2.5) \ + T 4 s =0 • Let U be the po t e n t i a l function f o r the generalised forces Q and l e t s (4.2.6) V = -̂* ' cs.yt;...,*>. IS JIJJ I f the generalised Hamiltonian function i s defined by (4.2.7) Hlt^lj),^. (T +U) , Lampariello \2o\ showed that-the cononical equations of motion are (4.2.8) < Next l e t us introduce, following Dobronravov £L7̂ the generalised Poisson brackets denoted by double parentheses: where It was also shown that these generalised brackets possess the properties of the usual Poisson brackets, namely: (4.2 . 1 0 , + -0, (4.2 . 1 D * UU1) --*> In terms of the generalised Poisson brackets, we s h a l l investigate the condition that | ( t ; q s ; p s ) = constant be a f i r s t Integral of the canonical equations of motion (4.2.8). In f a c t , If we take the complete derivative o f ^ - 72 - with respect to t and substitute i n i t f o r p g and q g the expressions obtained from the equations (4.2.8), we have or, by virtue of (4.2.2) and (4.2.5), we have at ^ a f c ^ ?}>s " m s n s s U > f s ~ In the second term on the left-hand side we change the repeated s u f f i x t to s and make use of (4.2.9) to obtain the following: (4.2.12) £ + ( U , H ) ) =rs^ which i s the condition we were looking f o r * F i n a l l y , l e t us suppose that j!(t;q s;p a) - constant and "f^*»<ls>Ps^ =constant are two f i r s t integrals of the canonical equations (4.2.8). Then according to (4.2.12) we have 2i at (4.2.13) < Dt = r P lL - i 2i ./(f>H)) =Y;I> i k . i ^ i i , By vi r t u e of (4.2.13) the i d e n t i t y (4.2.11) y i e l d s - 73 - i s also a f i r s t i n t e g r a l of the canonical equations of motion (4.2.13). Thus the theorem of Poisson can be stated as follows: Let { ((tjq g;p s) _cons_t. and | ( t ; q g ; p g ) = const. be two integrals of the canonical equations of motion (4.2.13) . I f the expression on the left-hand side of (4.2.14) does not reduce to zero or a_ constant, and i f moreover i t i s not expressible i n terms of then the equation (4.2.14) constitutes a_ new i n t e g r a l of the - 74 - system (4.2.13). In p a r t i c u l a r , i f the k i n e t i c c h a r a c t e r i s t i c s do not depend on the time, i . e . i f Ys =o f o r s, the equation (4.2.14) reduces to UAD~- constant» y i e l d i n g a new i n t e g r a l of the system i - i ___ is - * i » a result which was proved by Dobronravov QL7}. In case of a l i n e a r non-holonomic system with T t - r degrees of freedom, the equations of constraint can be expressed by (4.2.15) ^ = 0 („*!,*,...,r). Hence the independent k i n e t i c c h a r a c t e r i s t i c s are ^ ( i = r + l , r + 2, . . . , _ ) . Consequently (4.2.16) [>_. =o ( . ( - i , * , . . . ^ ) and the remaining ^ ( i = r + l , r 2,... ,n) are Independent. Furthermore, . At or A* - constant ( <*.,1,2, ..., r ). That Is the l i n e a r non- holonomic systems, when referred to l i n e a r non-holonomic coordinates, assume a holonomic form. Taking into consider- ation the equations (4.2.15) and (4.2.16), the canonical equations of motion (4.2.8) s t i l l hold. As a consequence, - 75 - the theorem of Poisson i n i t s generalised form holds f o r li n e a r non-holonomic systems ^ provided that we take into account the relations (4.2.15) and (4.2.16). 4.3. Appell's Equations i n Nonlinear Non-holonomic Coordinates. Let us consider a holonomic mechanical system with generalised coordinates qi,q 0,...,q . Following Novoselov C233» l e t u s define the k i n e t i c c h a r a c t e r i s t i c s by the relations (4.3.1) ^ = U*fi'>%>'^ ( M = >>V^) which are nonlinear i n the q's. Assuming that the functional matrix 2% has the rank r, we have (4.3.2) \ t = \£>\> **) (l^,*.-,*). The variations of the q's are given by (4.3.3, ' V X ' * ' ^-A-.-) where S f l s are the variations of nonlinear nonholonomic coordinates 7TS . According to the r e l a t i o n (4.3.1) the variations of the cartesian coordinates, x^ , and the q's are related as follows H i '« - 76 - Or, by vi r t u e of ( 4 . 3 . 3 ) i . e . ( 4 . 3 . 4 ) Sx =_-_S7r ( V = i , * , . . . , 5A/ ;^5 . o* , . . . , -n . ) In ( 4 . 3 . 4 ) x represents x a f t e r substituting f o r q from ( 4 . 3 . 2 ) , i . e . ( 4 . 3 . 5 ) \ - t ^ V ^ ) ' Hence the p r i n c i p l e of d fAlembert-Lagrange, expressed by the equation takes the form ( 4 . 3 . 6 ) ( > n . x. _ X \ & X - O \ t »> v v/ ~ ° ' s Since the mechanical system i s holonomic, S ŝ are independent. Therefore the equation ( 4 . 3 . 6 ) leads to the system of equations ( 4 . 3 . 7 ) (•V) V _____ - X ( v = i , * ) . . . ^ ^ s = i . * , - ^ . Let us introduce the energy of acceleration, S, defined by S - J- T f t . x x and denote by s' the function S when q and q are changed 5 S - 77 - into to and w§ by means of the equations (4.3.2). Also, from (4.3.5) we have / so that (4.3.8) Now x = sLu> +terms not containing^ t or, by virtue of (4.3.8) (4.3.9) ~T-= xv " • Further, l e t us put (4.3.10) G'K'Z* ' t'-'.*.--"). As a consequence of (4.3.9) and (4.3.10) we can rewrite the equations (4.3.7) i n the following form: (4.3.11) ^ P - ^ ^ (S...A...,-) • which are the so-called equations of Appell© In case of a nonlinear nonholonomic system we can take the non-linear constraints of the type of Cetaev to be given by Hence the independent k i n e t i c c h a r a c t e r i s t i c s are ^ ( 1 = r - 78 - r-t-2,...,n). As a consequence the r e l a t i o n s (4.3.2) and (4.3,3) are replaced by (4.3.12) V V ^ V ^ ' and (4.3.13) H = &*i (_,S.»A...,tv;i_-r+^+»--.. respectively, where Sfi^are the variations of the TO- r independent 7\£ . Proceeding exactly as i n the holonomic case we derive the equations of motion f o r the nonlinear non- holonomic system i n the form (4.3.14) — = V ; (i-r+i/f**,,...^) where s' i s a function of t , q g , U)± and ŵ , and Q' - X 2f_i The equations (4.3.14) are Appell's equations of motion f o r nonlinear non-holonomic syst ems o - 79 - B I B L I O G R A P H Y I Appell,P. Sur une forme generale des equations de la dynamique. CR. Acad. S c i . Pa r i s . 129(1899), 423-427. ^ Sur u n e forme nouvelle des equations de l a dynamique. CR. Acad. S c i . P a r i s . 129(1899), 459-460. 3 ; Sur les exprimees par des relations non l inelaires entre les v i t e s s e s . CR. Acad. S c i . Paris 152(1911), 1197-1199. • 4 Exemple de mouvement d'un point assu jetjti a une l i a i s o n exprimeV par une r e l a t i o n non li n e a l r e . entre les composantes de la v i t e s s e . Rend. C i r c . Mat. Palermao. 32(1911), 48-50. 5 Sur une forme generale des equations de l a dynamique. Mem. des sciences math. Pasc.l (1925), 1-48. 6 : Traits' de mecanique r a t i o n n e l l e . T.II, 6th ed., G a u t h i e r - V i l l a r s , Paris (1953). 7 Cenoy, I. Quelques formes nouvelle des e'quations generales du mouvement des systemes materiels. CR. Acad. Bulgare S c i . Math. Nat. 2(1949), no. 1, 13-16. 8 On a new form of the equations of analytic dynamics. Doklady Akad. Nauk SSSR(N.S.) 89 (1953), 21-24. (Russian) 9 On some transformations of the equations of motion and on geodesic t r a j e c t o r i e s of mechanical systems. Doklady Akad. Nauk SSSR(N.S.) 89 (1953), 225-228. (Russian) 10 On Gauss's p r i n c i p l e of least constraint. Doklady Akad. Nauk SSSR(N.S.) 89(1953), 415-418. (Russian) II Cetaev, N.G. On Gauss's p r i n c i p l e . lzv.Kazan.Fiz.- Mat.Obsc. 6(1933), no.3, 68-71. (Russian) - 80 - 12 Delassus, P. Sur la r e a l i s a t i o n materielle des l i a i s o n s . C.R.Acad.Sci.Paris. 152(1911), 1739-1743. 13 Sur les l i a i s o n s non l i n e a i r e s . C.R.Acad.Sci. Pa r i s . 153(1911), 626-628. 14 Sur ^les l i a i s o n s non l i n e a i r e s et les mouvement etudies par M.Appell. C.R.Acad.Sci. Paris. 153(1911), 707-710. 15 ^ Sur les l i a i s o n s d'ordre queleonque des systemes materiels. C.R.Acad.Sci.Paris. 154 (1912), 964-967. 16 Sur les l i a i s o n s et mouvements. Ann. S c i . Ecole. Norm. Sup. (3) 29(1912), 305. 17 Dobronravov, V.V. Poisson's theorem i n non-holonomic coordinates. C.R.(Doklady) Akad.Sci. URSS(N.S.) 44 (1944), 231-234. 18 Hamel, G. Die Lagrange-Eulerschen Gleichungen der Mechanik. Zeltschr. f . Math. u. Phys. 50 (1904), 1-57. 19 Johnson, L i e f . Dynamique ge'ne'rale des systemes non holonomes. Skr.Norske.Vide.Akad. Oslo. I. no 4 (1941), 1-75. 20 Lampariello, Giovanni. Generalizzazione del metodo di Hamilton - Jacob! a l i a dinamica dei sistemi analonomi. A t t i Accad. I t a l i a . Rend. CI. S c i . F i s . Mat. Nat. (7) 4 (1948), 20-28. 21 Metellcyn, I . I . Reduction of equations of motion of a non-holonomic system to the form free from undetermined m u l t i p l i e r s . Moskov. Gos. Univ. Ucenye Zap. (1934), no. 2, 127-130. (Russian) 22 Novoselov, V.S. Reduction of the problem of non- holonomic mechanics to a conditional problem of mechanics of holonomic systems. Leningrad, Gos. Univ. Ucenye Zap. 217. Ser. Mat. Nauk 31 (1957), 28-49. 23 Application of non-linear non- holonomic coordinates i n a n a l y t i c a l mechanics. Leningrad. Gos. Univ. Ucenye Zap. 217. Ser. Mat. Nauk 31 (1957), 50-83. (Russian) - 81 - 2 4 Extended equations of motion of non-linear non-holonomic systems. Leningrad. Gos. Univ. Ucenye Zap. 217. Ser. Mat. Nauk 31 (1957) 84-89. 25 Peres, J . Mecanique generale. Masson et C^e, Paris (1953). 26 Pogosov, G.S. Equations of motion f o r a system with non-holonomic non-linear constraints. Vestnik Moskov. Univ. (1948), no.10, 93-97. (Russian) 27 Volterra, V. Sopra una classe di equazioni dinamiche. A t t i Reale Accad. S c i . Torino. 33 (1898), 451-475. 28 Whittaker, E.T. Treatise on the a n a l y t i c a l dynamics of p a r t i c l e s and r i g i d bodies. 4th ed., Cambridge University Press, (1952). 29 Woronetz, P. Uber die Bewegung eines starren Korpers, der ohne Gleitung auf einer beliebigen Fischer r o l l t . Math. Ann. 70 (1911), 410^453.
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