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Approximate Noether symmetries of perturbed Lagrangians and approximate conservation laws

M. Gorgone, F. Oliveri

TL;DR

The paper addresses extending Noether's theorem to approximate symmetries for variational problems with small perturbations. It develops a consistent perturbative framework in which both the dependent variables and the symmetry generators are expanded in the small parameter $\varepsilon$, and the action is required to be invariant up to $O(\varepsilon^{p+1})$. An explicit approximate Noether theorem is formulated and applied to construct first-order approximate conservation laws, demonstrated on three ODE problems: the perturbed harmonic oscillator, a second-order system, and the planar restricted three-body problem. The results yield rich families of approximate invariants and are computed with the ReLie package, underscoring the method's practicality and consistency with perturbation theory. The work lays groundwork for extending approximate Noether theory to higher-order Lagrangians and field theories.

Abstract

In this paper, within the framework of the consistent approach recently introduced for approximate Lie symmetries of differential equations, we consider approximate Noether symmetries of variational problems involving small terms. Then, we state an approximate Noether theorem leading to the construction of approximate conservation laws. Some illustrative applications are presented.

Approximate Noether symmetries of perturbed Lagrangians and approximate conservation laws

TL;DR

The paper addresses extending Noether's theorem to approximate symmetries for variational problems with small perturbations. It develops a consistent perturbative framework in which both the dependent variables and the symmetry generators are expanded in the small parameter , and the action is required to be invariant up to . An explicit approximate Noether theorem is formulated and applied to construct first-order approximate conservation laws, demonstrated on three ODE problems: the perturbed harmonic oscillator, a second-order system, and the planar restricted three-body problem. The results yield rich families of approximate invariants and are computed with the ReLie package, underscoring the method's practicality and consistency with perturbation theory. The work lays groundwork for extending approximate Noether theory to higher-order Lagrangians and field theories.

Abstract

In this paper, within the framework of the consistent approach recently introduced for approximate Lie symmetries of differential equations, we consider approximate Noether symmetries of variational problems involving small terms. Then, we state an approximate Noether theorem leading to the construction of approximate conservation laws. Some illustrative applications are presented.

Paper Structure

This paper contains 8 sections, 1 theorem, 47 equations.

Table of Contents

  1. Introduction
  2. The Consistent Approach to Approximate Lie Symmetries
  3. Approximate Noether Theorem
  4. Applications
  5. Perturbed Harmonic Oscillator
  6. A Second Order System of ODEs
  7. The Three-Body Problem
  8. Concluding Remarks

Key Result

Theorem 3.2

Let us consider a variational system of differential equations arising from a first order Lagrangian function. Let expression (gen-approx) be the generator of an approximate Lie symmetry of the approximate Lagrangian action (action), say where $\phi^i_{(k)}\left(\mathbf{x},\mathbf{u}_{(0)},\ldots,\mathbf{u}_{(k)}\right)$ ($i=1,\ldots,n$) are functions of their arguments to be suitably determined.

Theorems & Definitions (7)

  • Definition 3.1
  • Theorem 3.2: Approximate Noether theorem
  • proof
  • Remark 3.3
  • Remark 3.4
  • Remark 4.1
  • Remark 4.2