Geometric Hamilton-Jacobi theory for systems with external forces

TL;DR

This paper extends Hamilton-Jacobi theory to forced Hamiltonian and Lagrangian systems, providing methods for reduction, reconstruction, and applications to symmetric and Chaplygin systems.

Contribution

It introduces a Hamilton-Jacobi framework for forced systems, including reduction techniques and specific solutions for Rayleigh and symmetric systems.

Findings

Developed Hamilton-Jacobi theory for forced systems

Presented reduction and reconstruction methods for symmetric systems

Provided examples illustrating the theory's application

Abstract

In this paper, we develop a Hamilton-Jacobi theory for forced Hamiltonian and Lagrangian systems. We study the complete solutions, particularize for Rayleigh systems and present some examples. Additionally, we present a method for the reduction and reconstruction of the Hamilton-Jacobi problem for forced Hamiltonian systems with symmetry. Furthermore, we consider the reduction of the Hamilton-Jacobi problem for a \v{C}aplygin system to the Hamilton-Jacobi problem for a forced Lagrangian system.