# Quasi-Noether systems and quasi-Lagrangians

## TL;DR

This paper explores quasi-Noether systems and quasi-Lagrangians, extending Noether's theorem to systems where symmetries are sub-symmetries, and analyzes their properties and conservation laws.

## Contribution

It introduces a generalized framework for quasi-Noether systems and quasi-Lagrangians, extending the classical Noether theorem to broader classes of differential equations.

## Key findings

- Quasi-Noether systems share conservation laws with Lagrangian systems.
- A generalized notion of quasi-Lagrangians extends Noether's theorem.
- Examples demonstrate the compatibility of the critical point condition with conserved integrals.

## Abstract

We study differential systems for which it is possible to establish a correspondence between symmetries and conservation laws based on Noether identity: quasi-Noether systems. We analyze Noether identity and show that it leads to the same conservation laws as Lagrange (Green-Lagrange) identity. We discuss quasi-Noether systems, and some of their properties, and generate classes of quasi-Noether differential equations of the second order. We next introduce a more general version of quasi-Lagrangians which allows us to extend Noether theorem. Here, variational symmetries are only sub-symmetries, not true symmetries. We finally introduce the critical point condition for evolution equations with a conserved integral, demonstrate examples of its compatibility, and compare the invariant submanifolds of quasi-Lagrangian systems with those of Hamiltonian systems.