Conservation of `moving' energy in nonholonomic systems with affine constraints and integrability of spheres on rotating surfaces

Abstract

Energy is in general not conserved for mechanical nonholonomic systems with affine constraints. In this article we point out that, nevertheless, in certain cases, there is a modification of the energy that is conserved. Such a function coincides with the energy of the system relative to a different reference frame, in which the constraint is linear. After giving sufficient conditions for this to happen, we point out the role of symmetry in this mechanism. Lastly, we apply these ideas to prove that the motions of a heavy homogeneous solid sphere that rolls inside a convex surface of revolution in uniform rotation about its vertical figure axis, are (at least for certain parameter values and in open regions of the phase space) quasi-periodic on tori of dimension up to three.

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Cited by in corpus (10)

• Moving energies as first integrals of nonholonomic systems with affine constraints
• Hamiltonization and separation of variables for Chaplygin ball on a rotating plane
• Hamiltonization of solids of revolution through reduction
• Rolling balls over spheres in R^n
• Dynamics of the Chaplygin ball on a rotating plane
• Integrability and dynamics of the n-dimensional symmetric Veselova top
• A Geometric Characterization of Certain First Integrals for Nonholonomic Systems with Symmetries
• On the dynamics of a heavy symmetric ball that rolls without sliding on a uniformly rotating surface of revolution
• Radial kinetic nonholonomic trajectories are Riemannian geodesics!
• On some aspects of the dynamics of a ball in a rotating surface of revolution and of the kasamawashi art