  Hamiltonian mechanics describes the evolution of a system through its
Hamiltonian. The Hamiltonian typically also represents the energy observable, a
Noether-conserved quantity associated with the time-invariance of the law of
evolution. In both quantum and classical mechanics, Hamiltonian mechanics
demands a precise relationship between time evolution and observable energy,
albeit using slightly different terminology. We distil basic conditions
satisfied in both quantum and classical mechanics, including canonical
coordinate symmetries and inner product invariance. We express these conditions
in the framework of generalised probabilistic theories, which includes
generalizing the definition of energy eigenstates in terms of time-invariant
properties of the Hamiltonian system. By postulating these conditions to hold,
we derive a unified Hamiltonian system model. This unified framework describes
quantum and classical mechanics in a consistent language, facilitating their
comparison. We moreover discuss alternative theories: an equation of motion
given by a mixture of commutation relations, an information-restricted version
of quantum theory, and Spekken's toy theory. The findings give a deeper
understanding of the Hamiltonian in quantum and classical theories and point to
several potential research topics.
