Contact Lie systems: theory and applications
Autor:
Lucas, Javier de
; Rivas, Xavier
Fecha:
2023Palabra clave:
Revista / editorial:
Journal of Physics A: Mathematical and TheoreticalTipo de Ítem:
Articulo Revista IndexadaDirección web:
https://iopscience.iop.org/article/10.1088/1751-8121/ace0e7
Resumen:
A Lie system is a time-dependent system of differential equations describing the integral curves of a time-dependent vector field that can be considered as a curve in a finite-dimensional Lie algebra of vector fields V. We call V a Vessiot-Guldberg Lie algebra. We define and analyse contact Lie systems, namely Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields relative to a contact manifold. We also study contact Lie systems of Liouville type, which are invariant relative to the flow of a Reeb vector field. Liouville theorems, contact Marsden-Weinstein reductions, and Gromov non-squeezing theorems are developed and applied to contact Lie systems. Contact Lie systems on three-dimensional Lie groups with Vessiot-Guldberg Lie algebras of right-invariant vector fields and associated with left-invariant contact forms are classified. Our results are illustrated with examples having relevant physical and mathematical applications, e.g. Schwarz equations, Brockett systems, quantum mechanical systems, etc. Finally, a Poisson coalgebra method to derive superposition rules for contact Lie systems of Liouville type is developed.
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