Unimodularity and invariant volume forms for Hamiltonian dynamics on Poisson-Lie groups
| Autor: | I Gutierrez-Sagredo, D Iglesias Ponte, J C Marrero, E Padrón, Z Ravanpak |
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| Rok vydání: | 2022 |
| Předmět: |
Statistics and Probability
Mathematics - Differential Geometry Differential Geometry (math.DG) Modeling and Simulation FOS: Mathematics General Physics and Astronomy FOS: Physical sciences Statistical and Nonlinear Physics Mathematical Physics (math-ph) Dynamical Systems (math.DS) Mathematics - Dynamical Systems Mathematical Physics 37C40 37J39 53D17 70G45 70H05 |
| DOI: | 10.48550/arxiv.2207.05511 |
| Popis: | In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and the unimodularity of the Poisson-Lie structure. In particular, we prove that Hamiltonian vector fields on a Lie group endowed with a unimodular Poisson-Lie structure preserve a multiple of any left-invariant volume on the group. Conversely, we also prove that if there exists a Hamiltonian function such that the identity element of the Lie group is a nondegenerate singularity and the associated Hamiltonian vector field preserves a volume form, then the Poisson-Lie structure is necessarily unimodular. Furthermore, we illustrate our theory with different interesting examples, both on semisimple and unimodular Poisson-Lie groups. Comment: 17 pages |
| Databáze: | OpenAIRE |
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