Completely integrable system with Jacobi multipliers and its KAM stability
Year: 2024
Type: article
Abstract: In the present paper, we develop a general theory for the reduction of systems with Jacobi multiplier and volume-preserving symmetries on $ n $-manifolds ($ n\geq2 $). If the system has $ r $ commuting volume-preserving symmetries, we can reduce it to another system with a Jacobi multiplier, where the solutions of this reduced system are determined by the solutions of its first $ n-r $ equations. Specifically, if the system has $ n-2 $ such symmetries, then the solutions of the transformed equations are determined by the solutions of its first two equations, and these first two equations constitute a one-degree-of-freedom Hamiltonian system. We also use geometric methods to provide a local way to find the complete first integrals of the system. Additionally, we provide examples illustrating that using our method to find first integrals is equivalent to solving certain partial differential equations. Finally, we used a KAM-type theorem to investigate the stability of the solutions of this system.
DOI: 10.3934/dcds.2024150
Author: Xuefeng Zhao
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4404613063