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arXiv · 2603.26624

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

Stephen C. Anco
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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.

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Stephen C. Anco. 2026-03-27. Noether symmetry groups, locally conserved integrals, and dynamical symmetries in classical mechanics. https://arxiv.org/abs/2603.26624

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