Classical Hamiltonian system

Set

 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 $\Gamma_{\mathcal M}$

Discussion

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

Reference

Wikipedia: Hamiltonian mechanics,