Topological Features of Integrable Problems of Non-holonomic Dynamics
by
Galina Okouneva
Canada Center for Remote Sensing

We consider the motion of a rigid body under a non-holonomic constraint which was first considered by G. K. Suslov (in 1946): a component of the vector of instantaneous velocity is equal to zero. Earlier the author proved the integrability of this problem in the force field with a linear-quadratic potential. As the Euler-Poisson equations of motion of this system are non-Hamiltonian the well-known Liouville theorem is not applicable here and the description of the topological structure of the phase space re quires a special technique. The problem is also integrable in the force field with the potentials:

U\alpha = 1/(\alpha+ 1 - \gamma12 - \gamma22) + B\gamma12 + A\gamma22,

U = V(B\gamma12 + A\gamma22),

where V(x) is an arbitrary smooth function, \gamma = (\gamma₁, \gamma₂, \gamma₃) is the unit vector for the vertical line; A, B and C are the principle inertia momenta of the rigid body.

The topological structure of the common level surfaces of two integrals is quite complicated. The non-degenerate invariant manifolds can be both compact and non-compact two-dimensional surfaces, in particular, a torus with four holes, a sphere with four holes and a sphere with a number of handles and a number of holes.

Date received: June 29, 1997

Copyright © 1997 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-55.