Lie algebras of conservation laws of variational ordinary differential equations

TL;DR

This paper presents a new version of Noether's theorem linking conservation laws of variational ODEs to specific geometric symmetries, enhancing understanding of their algebraic structure.

Contribution

It introduces a novel geometric formulation of Noether's theorem connecting conservation laws with vector fields preserving certain distributions.

Findings

Establishes a one-to-one correspondence between conservation laws and geometric vector fields.

Provides a new geometric perspective on symmetries and conservation laws in variational ODEs.

Enhances the theoretical framework for analyzing symmetries in differential equations.

Abstract

We establish a new version of the first Noether Theorem, according to which the (equivalence classes of) first integrals of given Euler-Lagrange equations in one independent variable are in exact one-to-one correspondence with the (equivalence classes of) vector fields satisfying two simple geometric conditions, namely they simultaneously preserve the holonomy distribution of the jets space and the action from which the Euler-Lagrange equations are derived.