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

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 sp(4, R) are obtained and proved to be intrinsic. Two distinguished subalgebras, the two-photon Lie algebra h₆ and the Lorentz Lie algebra 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.