A unified framework for mechanics. Hamilton-Jacobi equation and applications

TL;DR

This paper develops a unified Hamilton-Jacobi framework for diverse mechanical systems, including nonholonomic, dissipative, and time-dependent systems, using skew-symmetric algebroids with a 1-cocycle, and illustrates it with examples.

Contribution

It introduces a comprehensive framework that unifies Hamilton-Jacobi equations across various mechanical systems via skew-symmetric algebroids with a 1-cocycle.

Findings

Unified Hamilton-Jacobi equations for multiple mechanical systems

Application of skew-symmetric algebroids with a 1-cocycle

Illustrative examples demonstrating the theory

Abstract

In this paper, we construct Hamilton-Jacobi equations for a great variety of mechanical systems (nonholonomic systems subjected to linear or affine constraints, dissipative systems subjected to external forces, time-dependent mechanical systems...). We recover all these, in principle, different cases using a unified framework based on skew-symmetric algebroids with a distinguished 1-cocycle. Several examples illustrate the theory.