Jacobi-Lie systems: Fundamentals and low-dimensional classification

Abstract

A Lie system is a system of differential equations describing the integral curves of a $t$-dependent vector field taking values in a finite-dimensional real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. We define and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called Jacobi-Lie systems. We classify Jacobi-Lie systems on $\mathbb{R}$ and ${\mathbb{R}}^2$. Our results shall be illustrated through examples of physical and mathematical interest.

15 pages. Examples, references and comments added. Based on the contribution presented at "The 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications", July 07-11, 2014, Madrid, Spain. To appear in the Proceedings of the 10th AIMS Conference

References in corpus (2)

• k-symplectic Lie systems: theory and applications
• Lie symmetries for Lie systems: applications to systems of ODEs and PDEs

Cited by in corpus (3)

• Cosymplectic geometry, reductions, and energy-momentum methods with applications
• Contact Lie systems on Riemannian and Lorentzian spaces: from scaling symmetries to curvature-dependent reductions
• Mixed superposition rules for Lie systems, compatible geometric structures, and applications