Lie-Hamilton systems associated with the symplectic Lie algebra $\mathfrak{sp}(6, \mathbb{R})$

TL;DR

This work extends the LH-system framework to the rank-3 symplectic algebra $\mathfrak{sp}(6,\mathbb{R})$ by using its 6-dimensional defining representation to build three-dimensional LH dynamics on $T^*\mathbb{R}^3$. The authors construct a full $\mathfrak{sp}(6,\mathbb{R})$ LH structure, derive $t$-independent constants of motion via a coalgebra approach, and obtain a nonlinear superposition rule, with the first nontrivial constant $F^{(2)}$ and higher prolongations enabling a 6-solution rule. They apply the construction to a time-dependent electromagnetic field and to time-dependent coupled oscillators, and explore a restricted $\mathfrak{su}(3)$ subalgebra embedding, showing how branching rules yield coupled systems that can be interpreted on Minkowski and 2D spaces. The results generalize previous sp(4,R) LH-system work to higher rank and suggest avenues for extending to $\mathfrak{sp}(2n,\mathbb{R})$ and other semisimple subalgebras, with future work focusing on explicit solvability conditions for physically meaningful cases.

Abstract

New classes of Lie-Hamilton systems are obtained from the six-dimensional fundamental representation of the symplectic Lie algebra $\mathfrak{sp}(6,\mathbb{R})$. The ansatz is based on a recently proposed procedure for constructing higher-dimensional Lie-Hamilton systems through the representation theory of Lie algebras. As applications of the procedure, we study a time-dependent electromagnetic field and several types of coupled oscillators. The irreducible embedding of the special unitary Lie algebra $\mathfrak{su}(3)$ into $\mathfrak{sp}(6, \mathbb{R})$ is also considered, yielding Lie-Hamilton systems arising from the sum of the quark and antiquark three-dimensional representations of $\mathfrak{su}(3)$, which are applied in the construction of t-dependent coupled systems. In addition, t-independent constants of the motion are obtained explicitly for all these Lie-Hamilton systems, which allows the derivation of a nonlinear superposition rule