Classical Hamiltonian system

definiendum	$\langle \mathcal M, H\rangle \in \mathrm{it}$
postulate	$\mathcal M$ … smooth manifold
range	$\Gamma_{\mathcal M}\equiv \mathcal M\times T^*\mathcal M$
postulate	$H:\Gamma_{\mathcal M} \times \mathbb R \to \mathbb R$
postulate	$H$ … differentiable in $\Gamma_{\mathcal M}$

The Hamiltonian function is related to the Lagrangian function via Legendre transformation.