Non-Noether symmetries in singular dynamical systems

TL;DR

This paper explores the relationship between non-Noether symmetries and conserved quantities across various types of dynamical systems, extending the classical understanding of symmetries and conservation laws.

Contribution

It establishes the correspondence between non-Noether symmetries and conserved quantities in systems on symplectic, presymplectic, and Poisson manifolds.

Findings

Non-Noether symmetries generate conserved quantities in diverse dynamical systems.

The paper generalizes the concept of conservation laws beyond Hamiltonian symmetries.

It provides a unified framework linking symmetries and conserved quantities in different geometric settings.

Abstract

It's well known that Noether symmetries lead to the conservation laws. Conserved quantities are constructed out of generator of the symmetry - invariant Hamiltonian vector field. Considering more general class of vector fields - non-Hamiltonian ones leads to the notion of non-Noether symmetry and conservation laws (Lutzky's integrals of motion) with interesting properties. In the present paper correspondence between non-Noether symmetries and conserved quantities in different types of dynamical systems (DS on symplectic, presymplectic and Poisson manifolds) is considered.