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- Title
A Geometric Approach to the Sundman Transformation and Its Applications to Integrability.
- Authors
Cariñena, José F.
- Abstract
A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed.
- Subjects
GEOMETRIC approach; JACOBI forms; TENSOR fields; DIFFERENTIAL equations; AUTONOMOUS differential equations; NOETHER'S theorem; HAMILTON-Jacobi equations
- Publication
Symmetry (20738994), 2024, Vol 16, Issue 5, p568
- ISSN
2073-8994
- Publication type
Article
- DOI
10.3390/sym16050568