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A hybrid Lagrangian-Hamiltonian framework and its application to conserved integrals and symmetry groups

Stephen C. Anco

Abstract

A hybrid framework is developed that highlights and unifies the most important aspects of the Noether correspondence between symmetries and conserved integrals in Lagrangian and Hamiltonian mechanics. Several main results are shown: (1) a modern form of Noether's theorem is presented that uses only the equations of motion, with no knowledge required of an explicit Lagrangian; (2) the Poisson bracket is formulated with Lagrangian variables and used to express the action of symmetries on conserved integrals; (3) features of point symmetries versus dynamical symmetries are clarified and explained; (4) both autonomous and non-autonomous systems are treated on an equal footing. These results are applied to dynamical systems that are locally Liouville integrable. In particular, they allow finding the complete Noether symmetry group of such systems.

A hybrid Lagrangian-Hamiltonian framework and its application to conserved integrals and symmetry groups

Abstract

A hybrid framework is developed that highlights and unifies the most important aspects of the Noether correspondence between symmetries and conserved integrals in Lagrangian and Hamiltonian mechanics. Several main results are shown: (1) a modern form of Noether's theorem is presented that uses only the equations of motion, with no knowledge required of an explicit Lagrangian; (2) the Poisson bracket is formulated with Lagrangian variables and used to express the action of symmetries on conserved integrals; (3) features of point symmetries versus dynamical symmetries are clarified and explained; (4) both autonomous and non-autonomous systems are treated on an equal footing. These results are applied to dynamical systems that are locally Liouville integrable. In particular, they allow finding the complete Noether symmetry group of such systems.
Paper Structure (17 sections, 17 theorems, 138 equations, 1 figure)

This paper contains 17 sections, 17 theorems, 138 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conserved integrals of dynamical systems
  3. Noether's theorem in modern form
  4. Variational symmetries
  5. Multipliers
  6. Noether correspondence between variational symmetries and conserved integrals
  7. Dynamical symmetry groups
  8. Gauge freedom in symmetry generators
  9. Poisson brackets and action of symmetries on conserved quantities
  10. Poisson bracket
  11. Action of variational symmetries on conserved integrals
  12. Lie algebras of variational symmetries and conserved integrals
  13. Liouville integrable systems
  14. Conserved integrals from action-angle variables
  15. Variational symmetries
  16. ...and 2 more sections

Key Result

Proposition 1

A vertical vector field X.gen is an infinitesimal variational symmetry if and only if it satisfies

Figures (1)

  • Figure 1: Point transformation of curves

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 1
  • Lemma 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • proof
  • Definition 4
  • ...and 17 more