A priori properties of classical Hamiltonian (unitarity in Classical Mechanics)

Question

A canonical equation of motion has form: \begin{equation} \dot{p}_i = -\frac{\partial H}{\partial q_i} = \left\{ p,H\right\}, \quad \dot{q}_i = \frac{\partial H}{\partial p_i} = \left\{q,H\right\}.\tag{1} \end{equation}

For a classical particle in external potential field the view of Hamiltonian $H(q,p) = \frac{p^2}{2m} + V(q)$ is a postulate. But what properties of Hamiltonian we know before we postulate a particular form?

In Quantum Mechanics, for example, we know from unitarity of evolution of state $\hat{U}\hat{U}^{\dagger} = \hat{\mathbb{I}}$, for the infitesimal time shift $\Delta t$ the $\hat{U} = \hat{\mathbb{I}} - i\Delta t \hat{H}$, so, the Hamiltonian is a Hermitian operator $\hat{H} = \hat{H}^{\dagger}$ and is a generator of group $U(1)$. So, in Quantum Mechanics from postulate of unitarity of evolution we have determined that the evolution of the state is governing by $\hat{H}$ (Schrödinger equation $i\hbar \dot{\left|\Psi\right\rangle} = \hat{H} \left|\Psi\right\rangle$ ), but without particular form of it. Can we get something similar in classical mechanics? I.e., I want to understand if there is a principle in Classical Mechanics, similar to the unitarity of Quantum Mechanics, which will give us the opportunity to get the Hamiltonian form of equations of motion?

Answer 1

For what it's worth, in the classical setting, the Hamiltonian function $H(q,p,t)$ should for starters be a (i) differentiable real function, (ii) typically bounded from below, and (iii) its Hamiltonian vector field should generate time evolution.