Hamiltonian

A Hamiltonian is a function on the phase space. Consider when the phase space is $T^{\ast }M$. Differentiating the Hamiltonian to get a section of $T^{\ast }{T}^{\ast }M$, we can then map it to a vector field on $T^{\ast }M$ (namely, a section of $T{T}^{\ast }M$) by the musical isomorphism $\mathrm{\#}$, which is called Hamilton vector field and can be integrate to the Hamilton phase flow.
advantage: Converse of Noether's theorem; $on-shell$ (legal to plug in conservation condition)