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Russian Math . Surveys 70:6 11671181

Uspekhi Mat . Nauk 70:6 203212

COMMUNICATIONS OF THE MOSCOW MATHEMATICAL SOCIETY

Equations of motion of non-holonomic systems
A. V. Borisov and I. S. Mamaev We consider a non-holonomic mechanical system whose configuration space is some (smooth) manifold N . Its phase space (space of states) is a submanifold M T N which in local coordinates is given by constraint equations linear in the velocities: M = {(q, q) | aµi (q)q i = 0, µ = 1, . . . , k}, where q = (q 1 , . . . , q n ) and q = (q 1 , . . . , q n ) (here and below, summation over repeated indices is assumed). Systems with inhomogeneous constraints are not considered in this note (see [5] for details). In addition, the system is characterized by the Lagrangian function L(q, q), which is calculated without the constraints (that is, it is defined everywhere in T N ) and satisfies the non-degeneracy condition det 2 L/ q i q j = 0. The equations of motion of the system (that is, the vector field on M ) which describe its dynamics are defined by the D'AlembertLagrange principle [1], according to which the work of the reaction forces for arbitrary virtual displacements q = ( q 1 , . . . , q n ) satisfying the constraint equations is zero: L qi
·

-

L - Qi q i = 0, qi

aµi (q) q i = 0,

µ = 1, . . . , k ,

(1)

where Q1 (q, q), . . . , Qn (q, q) are the generalized non-potential external forces. In order to explicitly write the vector field on M defined by (1), we choose in each tangent space to T Nq a new basis of vector fields (q), = 1, . . . , n - k, nµ (q), µ = 1, . . . , k, such that the are tangent to the distribution of constraints and the nµ are i transverse to them: aµi (q) (q) 0, aµi ni = 0. The generalized velocities in the new basis are represented as q = (q) + wµ nµ , (2) where = ( 1 , . . . , n-k ) and w = (w1 , . . . , wk ) are new local coordinates which parameterize the tangent spaces T Nq and are called quasi-velocities in mechanics. Equation (1) and the constraints are now represented as L qi
·

-

L i - Q i = 0 , qi

w µ = 0,

= 1, . . . , n - k,

µ = 1, . . . , k .

(3)

It can be shown that for any vector field u(q) = (u1 (q), . . . , un (q)) on N the following natural relation holds: d dt L u qi
i

- u(L) =

L qi

·

-

L i u, qi

(4)

where u is the lift of the vector field u to the tangent bundle T N .
This work is supported by the Russian Science Foundation under grant 14-50-00005. AMS 2010 Mathematics Subject Classification . Primary 37J60, 37C10. DOI 10.1070/RM2015v070n06ABEH004976.

c 2015 Russian Academy of Sciences (DoM), London Mathematical So ciety, Turpion Ltd.

1168

Communications of the Moscow Mathematical So ciety

The lift of the vector fields to the tangent bundle T N in the coordinate basis is i F i F given by (F (q, q)) = i + q k i , and in the basis (2) it has the form q qk q (F (q, , w)) = F F F F F + c + cµ + d wµ + d wµ , µ µ qi wµ w [ , ] = c + cµ nµ , [ , nµ ] = d + d n , µ µ
i

where [ · , · ] is the Lie bracket of vector fields. By (4), the dynamical equation (3) can be rewritten in the invariant form (L)
i = Q , where L(q, , w) = L(q, q) i q( ,w)

d L - dt and where for brevity we use the

special notation f (q, , w) w=0 = f (q, ) for the operation of restricting an arbitrary function to the constraints. Finally, we obtain the equations of motion on M in the form L
·

- (L ) = -c

L L - cµ wµ

i + Q , i i q i = .

(5)

We will show that if the external forces are potential forces (that is, Qi = 0, i = 1, . . . , n), then the equations of motion are represented in the pseudo-Hamiltonian form. To this L end, we apply the Legendre transformation M = , H (q, M) = ( M - L ) M , = 1, . . . , n - k, where M = (M1 , . . . , Mn-k ). In terms of the variables (q, M) the system (5) becomes H i H M = - i - c M -c q M
µ pµ

H , M

qi =

i

H , M

(6)

where pµ (q, M) = ( L/ wµ ) M . The same equations were obtained in [6] in a different way. Denote the set of new variables (q, M) by x = (q 1 , . . . , q n , M1 , . . . , Mn-k ). Then the equations (6) can be written as xi = Jij (x) H , xj i = 1, . . . , 2n - k, (7)

where the Jij (x) are the components of the skew-symmetric matrix of the form J= 0 -T TT , - (q, M) = c

(q)M + c

µ

(q)pµ (q, M),

i T(q) = (q).

The matrix J satisfies the Jacobi identity if and only if the constraints are holonomic. Therefore, although non-holonomic systems admit the pseudo-Hamiltonian representation (7), their dynamics differs significantly from Hamiltonian systems; examples of this can be found in [2][4]. Bibliography [1] . . , . . , . . , , , . 2002; English transl., V. I. Arnol'd, V. V. Kozlov, and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, Encyclopaedia Math. Sci., vol. 3, Springer-Verlag, Berlin 2006, xiv+518 pp.

Communications of the Moscow Mathematical Society

1169

[2] I. A. Bizyaev, A. V. Borisov, and I. S. Mamaev, Regul . Chaotic Dyn . 19 198213. [3] A. V. Borisov, A. A. Kilin, and I. S. Mamaev, Regul . Chaotic Dyn . 18:6 832859. [4] A. V. Borisov, I. S. Mamaev, and I. A. Bizyaev, Regul . Chaotic Dyn . 18 277328. [5] A. V. Borisov, I. S. Mamaev, and I. A. Bizyaev, Regul . Chaotic Dyn . 20 383400. [6] A. J. van der Schaft and B. M. Maschke, Rep . Math . Phys . 34:2 (1994),
Alexey V. Borisov Steklov Mathematical Institute of Russian Academy of Sciences E-mail : borisov@rcd.ru Ivan S. Mamaev Steklov Mathematical Institute of Russian Academy of Sciences E-mail : mamaev@rcd.ru

:2 (2014), (2013), :3 (2013), :3 (2015), 225233.

Presented by D. V. Treschev Accepted 13/OCT/15 Translated by THE AUTHORS