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 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.
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