  We consider how to describe Hamiltonian mechanics in generalised
probabilistic theories with the states represented as quasi-probability
distributions. We give general operational definitions of energy-related
concepts. We define generalised energy eigenstates as the purest stationary
states. Planck's constant plays two different roles in the framework: the phase
space volume taken up by a pure state and a dynamical factor. The Hamiltonian
is a linear combination of generalised energy eigenstates. This allows for a
generalised Liouville time-evolution equation that applies to quantum and
classical Hamiltonian mechanics and more. The approach enables a unification of
quantum and classical energy concepts and a route to discussing energy in a
wider set of theories.
