Документ взят из кэша поисковой машины. Адрес оригинального документа : http://ics.org.ru/upload/iblock/970/119-some-comments-to-the-paper-by-a-m-perelomov-quot-a-note-on-geodesics-on-ellipsoid-quot-rcd-2000-5-1-_ru.pdf Дата изменения: Wed Oct 28 19:30:51 2015 Дата индексирования: Sun Apr 10 00:32:35 2016 Кодировка:

;

A. M. PERELOMOV

SOME COMMENTS TO THE PAPER
A. V. BORISOV I. S. MAMAEV
Department of Theoretical Mechanics MoscowStateUniversity,Vorob'ievy gory 119899 Moscow, Russia E-mail: borisov@uni.udm.ru

Laboratory of Dynamical Chaos and Nonlinearity Udmurt State University Universitetskaya 1, 426034 Izhevsk, Russia E-mail: mamaev@uni.udm.ru

between the geodesic ow on two-dimensional ellipsoid and a Hamiltonian system on algebra e(3) in the Clebsch case was rst indicated in 3]. A generalization for three-dimensional ellipsoid with the use of quaternions contains in 2], where the connection of this problem with the integrable cases of rigid body dynamics in superposition of homogeneous potential elds is also mentioned. It is also noted there, that a natural generalization of the construction given in 3] for the case of arbitrary dimension is possible. In this case a system on the singular orbit di eomorphic to T S n;1 in algebra e(n) is considered. Let us give this construction in a complete form in the notations of A. M. Perelomovwork. 2. The geodesic ow on ellipsoid (1)2 preserves the total energy (2), which de nes a sphere in the space of velocities xi = yi i =1 ::: n. At the same time the ow appears on this sphere. We _ shall show that after the time transformation (10) these ows go into each other with the help of a linear substitution. Indeed, after the substitution (10) the equations of motion (3) take the form

1. Basing on the results of 1, 2]1, let us make some comments to this work. The isomorphism

d x = ;A;1x A;1 x y d y = ;y A;1 y A;1x (A:1) d d where x =(x1 ::: xn ) y =(y1 : : : yn ) A =diag(a1 : : : an ), ( ) is the scalar product in Rn .
It is obvious that the substitution

A

;1=2 y = x0

A;

1

=2 x

= y0

yields the same system as that of (A.1), but with x0 y0 exchanged their places. Further it will be more convenient to carry out this linear transformation. 3. Let us use the Lagrangian description of the geodesic ow on ellipsoid (1) the Lagrangian n 1X_ function is of the form L = 2 x2 in this case. Let us make the substitution i
i=1

A
Q0 (q)=

;1=2 x = q

(A:2) (A:3) (A:4)

it maps the ellipsoid (1) into the sphere

n X i
=1

qi2 =(q q)= 1

and the Lagrangian function takes the form n 1X __ L = 2 ai q_i2 =(q Aq) : i=1
1 2

The references for the addendum are below. Here and below references on the work of A. M. Perelomov are given.

92

REGULAR AND CHAOTIC DYNAMICS, V. 5,

1, 2000


A NOTE ON GEODESICS ON ELLIPSOID

;

Let us proceed to the Hamiltonian description with the constraint (A.3) 4]. For this purpose we shall introduce momenta, which are canonically conjugated with the variables q by the formula
p=

@L + q = Aq + q _ _ @q
; p =; q

where the unde ned multiplier can be found from the equation (A.3)

A;1 q : A;1 q _ With the help of Legendre transformation H = pq ; L q q!q p we obtain the Hamiltonian function _ ; ; ; p A;1 p q A;1 q ; p A;1 q 2 H= ; : (A:5) q A;1 q
Let us introduce moments of momentum by analogy with (9)

lij = piqj ; pj qi

yi = qi

i j =1 : :: n:

(A:6)

Note, that in this case y have a slightly di erent meaning. The Poisson brackets of new variables coincide with (5, 6), and de ne the algebra e(n) with the generators lij yi i =1 : : : n. It is possible to show that the formulas (A.6) together with (A.3) give the mapping Rn S n;1 onto a singular orbit of algebra e(n), which is di eomorphic to (co)tangent foliation to the sphere TS n;1 T S n;1. For this purpose it is enough to observe that (A.6) glues together all the points of the form p + y with an arbitrary and xed p q . In new variables the Hamiltonian takes the form

H

n2 X lij 1 = = (y A;1 y) ij =1 ai aj

;

; Tr LA;1 LA;1 ; y

A

;1 y

(A:7)

where L = jjlij jj is a matrix of moments of momenta. 4. Following 2], we shall prove Proposition 1. The Hamiltonian system (A.5) on the manifold of constant energy H = 1 is trajectory equivalent to the system with the Hamiltonian
; ; H 0 = ; Tr LA;1LA;1 ; y A;1y

(A:8)

on the zero energy level (The system (A.8) Proof. The equations of tions (15, 16)

coincides with (15).)

H 0 =0.

motion on e(n) may be written with the help of the commutative rela_ L = L @H + @L
y

@H @y

; @H @y

y

_ y=

; @H y : @L

Here, ( )T denotes the transposition. Let us present the Hamiltonians of the systems (A.7) and (A.8) in the form ) H = F (L) B (y

H 0 = F (L) ; B (y) :
1, 2000 93

REGULAR AND CHAOTIC DYNAMICS, V. 5,


;
F _1 L = B L @H ; B 2 @L _ y = ; 1 @H y B @L
y

A. M. PERELOMOV

The equations of motion of both systems have the form

@H @y

T

; @H @y

y

T

Making the time substitution d = B;1 dt and taking into consideration the fact that the relation F (L)= B (y) is ful lled on the energy level H =1, we obtain the second system from the rst one. The system with the Hamiltonian (A.8) satis es the integrability conditions of the Clebsch case. A generalization of this case for arbitrary dimensions is obtained by A. M. Perelomov 5]. Let us note that the second identity (12) means that in the Clebsch system the zero energy level (in our case H 0 =0) is xed. It is shown in 6] that if the geodesic ow has a partial integral f (x p) at some energy value, it also has a complete integral, which is equal to
p

_ L = L @H ; @L _ y = ; @H y : @L

y

@H @y

T

; @H @y

y

T

p F (x p)= f x jpj

where jpj = gij pi pj , and the Hamiltonian H = gij (x)pi pj . 5. In 7] the isomorphism between the geodesic ow on the ellipsoid and the Neumann problem on the zero level of one of the integrals which has the form of (A.5) is indicated. This connection is given by the Gauss pro jection q = Bx : Later, in 8], this isomorphism was generalized for the problem of a particle on ellipsoid with quadratic potential and the Neumann problem on a xed level of the integral (A.5). Both these analogies were established with the help of the equations of motion, their Hamiltonian description is too tedious.

jBxj

References
1] A. V. Borisov, I. S. Mamaev. Poisson structures and Lie algebras in Hamiltonian mechanics. Izhevsk, 1999. (in Russian). 2] A. V. Borisov, I. S. Mamaev. Non-linear Poisson brackets and isomorphisms in dynamics. Regular and Chaotic Dynamics, V. 2, 1997, 3/4, P. 72{89. 3] V. V. Kozlov. Twointegrable problems of classical mechanics. Vestnik MGU, Ser. mat. mekh., 1981, 4, P.80{83. 4] V. I. Arnold, V. V. Kozlov, A. I. Neishtadt. Mathematical aspects of classical and celestial mechanics. Sovr. probl. math., V. 3, VINITI, 1985. 5] A. M. Perelomov. Some remarks about integrabilityof equations of motion of a rigid body in a uid. Func. anal. and app., 1981, V. 15, 2, P. 83{85. 6] A. V. Bolsinov, V. V. Kozlov, A. T. Fomenko. Maupertius principle and geodesic ows, originated from the integrable cases of rigid body dynamics. Usp. mat. nauk., 1995, V. 50, 3, P. 3{32. 7] H. Knorrer. Geodesics on quadrics and mechanical problem of C. Neumann. J. Reine Angew. Math., 1982, V. 334, P. 69{78. 8] A. P. Veselov. Two remarks about the connection of Jacobi and Neumann integrable systems. Math. Z. 1994, V. 216, P. 337{345.

94

REGULAR AND CHAOTIC DYNAMICS, V. 5,

1, 2000