General characterisation of Hamiltonians generating velocity-independent forces

Abstract: Dynamics generated from Hamiltonians enjoy potential pathways to quantisation, but standard Hamiltonians are only capable of generating conservative forces. Classes of Hamiltonians have been proposed in Berry et al. capable of generating non-conservative velocity-independent forces. Such Hamiltonians have been classified in the past, under the strict assumption that they are polynomial in momentum. This assumption is relaxed here to analyticity. In doing so, broader classes of Hamiltonians are discovered. By considering the Hamiltonian as a function of state space without introducing the Lagrangian and constructing a metric-like tensor, we develop strong general constraints on Hamiltonians generating velocity-independent forces and exhibit a surprising dichotomy between classes of such Hamiltonians. These results are applicable to any spatial domain of any dimension admitting well-defined Hamiltonian dynamics. As an example application, we apply these constraints to classify all Hamiltonian velocity-independent forces in two spatial dimensions, as well as all such Hamiltonians which do not generate an isotropic simple harmonic motion. The case of one spatial dimension is also discussed for the sake of completeness.