Moving energies as first integrals of nonholonomic systems with affine constraints

TL;DR

This paper investigates the properties and conservation conditions of moving energies, a class of modified energy functions, in nonholonomic systems with affine constraints, demonstrating their relevance through various classical and generalized mechanical systems.

Contribution

It enlarges the class of moving energies and shows their role as first integrals in several nonholonomic systems, including Lie group systems and rolling bodies.

Findings

Moving energies can be conserved under specific conditions.

Certain known first integrals are instances of moving energies.

Conserved moving energies are identified for various affine nonholonomic systems.

Abstract

In nonholonomic mechanical systems with constraints that are affine (linear nonhomogeneous) functions of the velocities, the energy is typically not a first integral. It was shown in [Fass\`o and Sansonetto, JNLS, 26, (2016)] that, nevertheless, there exist modifications of the energy, called there moving energies, which under suitable conditions are first integrals. The first goal of this paper is to study the properties of these functions and the conditions that lead to their conservation. In particular, we enlarge the class of moving energies considered in [Fass\`o and Sansonetto, JNLS, 26, (2016)]. The second goal of the paper is to demonstrate the relevance of moving energies in nonholonomic mechanics. We show that certain first integrals of some well known systems (the affine Veselova and LR systems), which had been detected on a case-by-case way, are instances of moving energies. Moreover, we determine conserved moving energies for a class of affine systems on Lie groups that include the LR systems, for a heavy convex rigid body that rolls without slipping on a uniformly rotating plane, and for an $n$-dimensional generalization of the Chaplygin sphere problem to a uniformly rotating hyperplane.