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

TL;DR

This paper introduces new classes of Lie-Hamilton systems derived from the symplectic Lie algebra $rak{sp}(6,b R)$ using representation theory, with applications to electromagnetic fields and coupled oscillators, and explicitly finds constants of motion.

Contribution

It develops a novel procedure for constructing higher-dimensional Lie-Hamilton systems from Lie algebra representations, including $rak{sp}(6,b R)$ and $rak{su}(3)$, with explicit constants of motion and superposition rules.

Findings

Constructed new Lie-Hamilton systems from $rak{sp}(6,b R)$ representations.

Applied the method to electromagnetic fields and coupled oscillators.

Derived explicit constants of motion and nonlinear superposition rules.

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