Classical relativistic nonholonomic mechanics and time-dependent $G$-Chaplygin systems with affine constraints

TL;DR

This paper develops a relativistic, time-dependent nonholonomic mechanics framework using moving frames, introduces invariant $G$-Chaplygin systems with affine constraints, and explores Hamiltonization for systems with time-dependent constraints.

Contribution

It presents a novel relativistic formulation of nonholonomic mechanics with moving frames and introduces time-dependent $G$-Chaplygin systems with affine constraints, addressing the Hamiltonization problem.

Findings

Modified Chaplygin multiplier method for time-dependent systems.

Reduced system becomes a standard Lagrangian system with a new time.

Application to a rolling disc over a variable radius circle.

Abstract

We study the relativistic formulation of a classical time-dependent nonholonomic Lagrangian mechanics from the perspective of moving frames. We also introduce time-dependent $G$-Chaplygin systems with affine constraints, which are natural objects for the invariant formulation of nonholonomic systems with symmetries. As far as the author is aware, the Hamiltonization problem for time-dependent constraints has not yet been studied. As a first step in this direction, we consider a rolling without sliding of a balanced disc of radius $r$ over a vertical circle of variable radius $R(t)$. We modify the Chaplygin multiplier method and prove that the reduced system becomes the usual Lagrangian system with respect to the new time.