Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

Abstract

We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and twisting of an $n$-dimensional balanced ball over a fixed sphere is considered. For a special inertia operator (depending on $n$ parameters) we prove complete integrability when the radius of the ball is twice the radius of the sphere. In the case of $SO(l)\times SO(n-l)$ symmetry, noncommutative integrability for any ratio of the radii is established.

18 pages, 2 figures, final version, to appear in Nonlinearity

References in corpus (9)

• Nonholonomic LR systems as Generalized Chaplygin systems with an Invariant Measure and Geodesic Flows on Homogeneous Spaces
• Chaplygin ball over a fixed sphere: explicit integration
• Rubber rolling over a sphere
• Moving energies as first integrals of nonholonomic systems with affine constraints
• Integrable discretization and deformation of the nonholonomic Chaplygin ball
• Noether symmetries and integrability in time-dependent Hamiltonian mechanics
• Rolling balls over spheres in R^n
• Integrability and dynamics of the n-dimensional symmetric Veselova top
• Symmetries of line bundles and Noether theorem for time-dependent nonholonomic systems

Cited by in corpus (3)

• Gyroscopic Chaplygin systems and integrable magnetic flows on spheres
• Generalisation of Chaplygin's Reducing Multiplier Theorem with an application to multi-dimensional nonholonomic dynamics
• Integrability and dynamics of the n-dimensional symmetric Veselova top