# Generalized geometric Hamilton-Jacobi theorem on Lie algebroids

## TL;DR

This paper develops generalized geometric Hamilton-Jacobi theorems for Hamiltonian systems on Lie algebroids, extending classical results to a broader geometric framework with time-dependent versions.

## Contribution

It introduces new geometric formulations and proofs of Hamilton-Jacobi theorems on Lie algebroids, including time-dependent cases, expanding the theoretical understanding of these systems.

## Key findings

- Two geometric versions of Hamilton-Jacobi theorem on Lie algebroids proved.
- Generalization to time-dependent Hamilton-Jacobi theorems achieved.
- Framework applicable to a broad class of Hamiltonian systems on Lie algebroids.

## Abstract

In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system on Lie algebroids are given. Here we use the general properties of Lie algebroids to express and prove two geometric version of the Hamilton-Jacobi theorem for Hamiltonian system on Lie algebroids. Then this results are generalized and two types of time-dependent Hamilton-Jacobi theorem of Hamiltonian system on Lie algebroids are obtained.