Reduction and reconstruction of multisymplectic Lie systems

Abstract

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot--Guldberg Lie algebra. In this work, multisymplectic structures are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples occurring in physics, mathematics, and control theory.

arXiv comment29 pages, no figures. A new multisymplectic reconstruction theorem and several applications thereof added. Several parts have been simplified and the presentation has been improved

References in corpus (24)

• Superposition rules, Lie theorem and partial differential equations
• Lie systems: theory, generalisations, and applications
• A contact geometry framework for field theories with dissipation
• Lie-Hamilton systems on the plane: properties, classification and applications
• Multivector Fields and Connections. Setting Lagrangian Equations in Field Theories
• From constants of motion to superposition rules for Lie-Hamilton systems
• Lie--Hamilton systems: theory and applications
• Dirac--Lie systems and Schwarzian equations
• Applications of Lie systems in dissipative Milne--Pinney equations
• Superposition rules for higher-order systems and their applications
• A Poisson Bracket on Multisymplectic Phase Space
• Lie-Hamilton systems on the plane: Applications and superposition rules
• k-symplectic Lie systems: theory and applications
• Periodic first integrals for Hamiltonian systems of Lie type
• Lie-Hamilton systems on curved spaces: A geometrical approach
• Reduction of multisymplectic manifolds
• Lie symmetries for Lie systems: applications to systems of ODEs and PDEs
• Invariant Forms and Automorphisms of Locally Homogeneous Multisymplectic Manifolds
• Mixed superposition rules and the Riccati hierarchy
• Poisson-Hopf algebra deformations of Lie-Hamilton systems
• Remarks on multisymplectic reduction
• Multisymplectic structures and invariant tensors for Lie systems
• A time-dependent energy-momentum method
• Geometric features of Vessiot--Guldberg Lie algebras of conformal and Killing vector fields on ${\mathbb{R}}^2$