| 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.
Wikipedia: Hamiltonian mechanics,