  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.
