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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?

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    $\begingroup$ Not sure $H$ is necessarily of the form above. True for natural systems but there are many effective Hamiltonians for which there is no clear potential (v.g. rotating wire or rotating bowls or problems of this type.) $\endgroup$ – ZeroTheHero Jul 14 at 23:31
  • $\begingroup$ Clearly, $H$ should be a real-valued, differentiable function of many variables! $\endgroup$ – SRS Jul 15 at 4:41
  • $\begingroup$ @ZeroTheHero Of course, $H$ is not necessarily $T + V$. Form of $H$ for particular physicsl system is an postulate. $T + V$ - as I mention, for classical particle in external potential field. $\endgroup$ – Sergio Jul 15 at 6:25
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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.

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  • $\begingroup$ Thank you, the statement "Hamiltonian vector field should generate time evolution" maybe is the answer, I try to understand. $\endgroup$ – Sergio Jul 15 at 8:01

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