A new perspective on nonholonomic brackets and Hamilton-Jacobi theory

TL;DR

This paper unifies three different nonholonomic brackets and leverages this to advance Hamilton-Jacobi theory and quantization methods for nonholonomic systems.

Contribution

It proves the equivalence of three nonholonomic brackets and introduces new insights into Hamilton-Jacobi theory and quantization for these systems.

Findings

The three nonholonomic brackets are shown to coincide.

New developments in Hamilton-Jacobi theory for nonholonomic systems.

Progress in the quantization of nonholonomic systems.

Abstract

The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in this case, of the canonical bracket on the cotangent bundle of the configuration manifold. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids. Recently, reviewing two older papers by R. J. Eden, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket \`a la Eden has allowed us to make important advances in the study of Hamilton-Jacobi theory and the quantization of nonholonomic systems.