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Noether symmetry groups, locally conserved integrals, and dynamical symmetries in classical mechanics

Stephen C. Anco

Abstract

Several aspects of the connection between conserved integrals (invariants) and symmetries are illustrated within a hybrid Lagrangian-Hamiltonian framework for dynamical systems. Three examples are considered: a nonlinear oscillator with time-dependent frequency (one degree of freedom); geodesics of a spheroid (two degrees of freedom); Calogero-Moser-Sutherland system of interacting particles (three degrees of freedom). For each system, a local generalization of Liouville integrability is shown. Specifically, the variational point symmetries in a Lagrangian setting lead to corresponding locally conserved integrals which are found to commute in the Poisson bracket imported from the equivalent Hamiltonian setting. Action-angle variables are then introduced in the Lagrangian setting, which leads to explicit integration of the Euler-Lagrange equations of motion locally in time.

Noether symmetry groups, locally conserved integrals, and dynamical symmetries in classical mechanics

Abstract

Several aspects of the connection between conserved integrals (invariants) and symmetries are illustrated within a hybrid Lagrangian-Hamiltonian framework for dynamical systems. Three examples are considered: a nonlinear oscillator with time-dependent frequency (one degree of freedom); geodesics of a spheroid (two degrees of freedom); Calogero-Moser-Sutherland system of interacting particles (three degrees of freedom). For each system, a local generalization of Liouville integrability is shown. Specifically, the variational point symmetries in a Lagrangian setting lead to corresponding locally conserved integrals which are found to commute in the Poisson bracket imported from the equivalent Hamiltonian setting. Action-angle variables are then introduced in the Lagrangian setting, which leads to explicit integration of the Euler-Lagrange equations of motion locally in time.

Paper Structure

This paper contains 24 sections, 5 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Summary of framework
  3. Variational symmetries and conserved integrals
  4. Poisson bracket and its uses
  5. Symmetry transformation groups
  6. Locally Liouville integrable systems
  7. Nonlinear oscillator with time-dependent frequency
  8. Point symmetries and conserved integrals
  9. Liouville integrability
  10. Dynamical symmetries and Poisson brackets
  11. Noether symmetry group
  12. Geodesics of spheroids
  13. Point symmetries
  14. Conserved integrals and Liouville integrability
  15. Variational symmetries
  16. ...and 9 more sections

Key Result

Theorem 1

For a dynamical system EL.eqn.gen possessing a variational principle, a function $C(t,q,\dot{q}^i)$ is a locally conserved integral $\dot{C}|_{\mathcal{E}} =0$ if and only if the vector field ${\bf X} =P^i(t,q,\dot{q}) \partial_{q^i}$ is an infinitesimal variational symmetry inv.cond, where $C$ and and Here $\mathcal{C}$ denotes any curve in the coordinate space $(t,q,\dot{q}^i)$, starting at an

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Theorem 4