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

The paper constructs an explicit realization of sp(4,R) on T*R^2 via the fundamental representation, exhibiting 10 quadratic Hamiltonians h1,…,h10 that close under the Poisson bracket and generate a Lie–Hamilton (LH) system; it then proves Theorem 6.1: every subalgebra g ⊂ sp(4,R) yields a LH system on T*R^2 with Vessiot–Guldberg algebra isomorphic to g (see the explicit vector fields (6.2), system (6.3), Hamiltonians (6.4) and Theorem 6.1 in Section 6 of the paper ). The candidate solution proves the same statement by the standard isomorphism A ↦ h_A(z)=1/2 z^T(-JA)z sending sp(4,R) to quadratic Hamiltonians with X_{h_A}(z)=A z and {h_A,h_B}=h_{[A,B]}, then taking time-dependent linear combinations to obtain a LH system. This matches the paper’s construction in content and method (linear Hamiltonian vector fields on (T*R^2,ω)), differing mainly in presentation (coordinate-free matrix map versus an explicit basis).