A representation-theoretical approach to higher-dimensional Lie-Hamilton systems: The symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$

Abstract

A new procedure for the construction of higher-dimensional Lie-Hamilton systems is proposed. This method is based on techniques belonging to the representation theory of Lie algebras and their realization by vector fields. The notion of intrinsic Lie-Hamilton system is defined, and a sufficiency criterion for this property given. Novel four-dimensional Lie-Hamilton systems arising from the fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(4,\mathbb{R})$ are obtained and proved to be intrinsic. Two distinguished subalgebras, the two-photon Lie algebra ${\mathfrak{h}}_6$ and the Lorentz Lie algebra $\mathfrak{so}(1,3)$, are also considered in detail. As applications, coupled time-dependent systems which generalize the Bateman oscillator and the one-dimensional Caldirola-Kanai models are constructed, as well as systems depending on a time-dependent electromagnetic field and generalized coupled oscillators. A superposition rule for these systems, exhibiting interesting symmetry properties, is obtained using the coalgebra method.

44 pages. Some typos and misprints have been corrected. Several comments and an appendix have been added

References in corpus (4)

• Superposition rules, Lie theorem and partial differential equations
• Contact Lie systems
• Exact solutions and superposition rules for Hamiltonian systems generalizing time-dependent SIS epidemic models with stochastic fluctuations
• Poisson-Hopf deformations of Lie-Hamilton systems revisited: deformed superposition rules and applications to the oscillator algebra

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