# Symmetries and conservation laws of Hamiltonian systems

## TL;DR

This paper explores the symmetries and conservation laws in Hamiltonian systems, establishing how these invariants relate to the geometric structures on the cotangent bundle and providing an example from optimal control.

## Contribution

It introduces a geometric framework for identifying symmetries and conservation laws in Hamiltonian systems using covariant derivatives and Jacobi endomorphisms, linking them to nonlinear connections.

## Key findings

- Invariant equations for symmetries derived
- Canonical nonlinear connection characterized by symmetries
- Application demonstrated in optimal control example

## Abstract

In this paper we study the infinitesimal symmetries, Newtonoid vector fields, infinitesimal Noether symmetries and conservation laws of Hamiltonian systems. Using the dynamical covariant derivative and Jacobi endomorphism on the cotangent bundle we find the invariant equations of infinitesimal symmetries and Newtonoid vector fields and prove that the canonical nonlinear connection induced by a regular Hamiltonian can be determined by these symmetries. Finally, an example from optimal control theory is given.