k-symplectic Lie systems: theory and applications

Abstract

A Lie system is a system of first-order ordinary 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 so-called Vessiot-Guldberg Lie algebra. We suggest the definition of a particular class of Lie systems, the $k$-symplectic Lie systems, admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields with respect to the presymplectic forms of a $k$-symplectic structure. We devise new $k$-symplectic geometric methods to study their superposition rules, time independent constants of motion and general properties. Our results are illustrated by examples of physical and mathematical interest. As a byproduct, we find a new interesting setting of application of the $k$-symplectic geometry: systems of first-order ordinary differential equations.

arXiv comment29 pages. An example and several minor details were corrected

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• Geometry of Riccati equations over normed division algebras
• Jacobi structures on real two- and three-dimensional Lie groups and their Jacobi-Lie systems
• Polysymplectic Reduction and the Moduli Space of Flat Connections
• Quasi-Lie schemes for PDEs
• A new application of k-symplectic Lie systems
• Stochastic K-symplectic integrators for stochastic non-canonical Hamiltonian systems and applications to the Lotka--Volterra model
• Lie systems and Schrödinger equations
• Reduction and reconstruction of multisymplectic Lie systems
• Stratified Lie systems: Theory and applications
• An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold