Symmetries of Hamiltonian Equations and Λ-Constants of Motion

DOI

10.1142/S1402925109000315How to use a DOI?

Keywords

Hamiltonian equations of motion; Lie point symmetries; Λ-symmetries; Λ-constants of motion; Λ-invariant Lagrangians; reduction procedures

Abstract

We consider symmetries and perturbed symmetries of canonical Hamiltonian equations of motion. Specifically we consider the case in which the Hamiltonian equations exhibit a Λ-symmetry under some Lie point vector field. After a brief survey of the relationships between standard symmetries and the existence of first integrals, we recall the definition and the properties of Λ-symmetries. We show that in the presence of a Λ-symmetry for the Hamiltonian equations, one can introduce the notion of “Λ-constant of motion”. The presence of a Λ-symmetry leads also to a nice and useful reduction of the form of the equations. We then consider the case in which the Hamiltonian problem is deduced from a Λ-invariant Lagrangian. We illustrate how the Lagrangian Λ-invariance is transferred into the Hamiltonian context and show that the Hamiltonian equations are Λ-symmetric. We also compare the “partial” (Lagrangian) reduction of the Euler–Lagrange equations with the reduction which can be obtained for the Hamiltonian equations. Several examples illustrate and clarify the various situations.

Copyright

© 2009 The Authors. Published by Atlantis Press and Taylor & Francis

Open Access

This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - Giampaolo Cicogna
PY  - 2021
DA  - 2021/01/07
TI  - Symmetries of Hamiltonian Equations and Λ-Constants of Motion
JO  - Journal of Nonlinear Mathematical Physics
SP  - 43
EP  - 60
VL  - 16
IS  - Supplement 1
SN  - 1776-0852
UR  - https://doi.org/10.1142/S1402925109000315
DO  - 10.1142/S1402925109000315
ID  - Cicogna2021
ER  -