Linear Hamiltonian Vector Fields on Lie Groups

Linear vector fields on Lie groups constitute a fundamental class of dynamical systems, as their flows are one-parameter subgroups of automorphisms and their infinitesimal behavior is entirely determined by derivations of the Lie algebra. When a Lie group is endowed with a Hamiltonian-type geometric structure, a natural problem is to determine whether such linear dynamics admit a global variational realization, and how such realizations can be interpreted in terms of reduced models of fluid motion. In the even-dimensional case, where the Lie group carries a symplectic structure, we establish a complete global criterion for the existence of Hamiltonians generating linear symplectic vector fields. The problem reduces to a single global obstruction: the de Rham cohomology class of the 1-form ιXω. Thus, every linear symplectic vector field on a simply connected Lie group is globally Hamiltonian, and when the obstruction vanishes, we provide an explicit constructive procedure to recover the Hamiltonian. On the affine group Aff+(1), this yields a fully explicit, finite-dimensional Hamiltonian model of a 1D ideal fluid with affine symmetries. We then treat odd-dimensional Lie groups, where symplectic geometry is unavailable. Using contact geometry as the canonical replacement, we prove a Hamiltonian lifting theorem ensuring the existence and uniqueness of the associated dynamics. The Reeb vector field appears as a distinguished vertical direction resolving the ambiguities of degenerate Hamiltonian systems. On the Heisenberg group H3, this gives a fully explicit contact Hamiltonian model of an effective non-conservative fluid mode. Finally, we interpret symplectic and contact theories within a unified geometric framework and discuss their relevance to geometric formulations of ideal (symplectic) and effective (contact) fluid equations on Lie groups.