===== Classical Hamiltonian system =====
==== Set ====

| @#FFBB00: definiendum | @#FFBB00: $\langle \mathcal M, H\rangle \in \mathrm{it} $ |
| @#55EE55: postulate   | @#55EE55: $ \mathcal M$ ... smooth manifold |
| @#DDDDDD: range       | @#DDDDDD: $ \Gamma_{\mathcal M}\equiv \mathcal M\times T^*\mathcal M $ |
| @#55EE55: postulate   | @#55EE55: $ H:\Gamma_{\mathcal M} \times \mathbb R \to \mathbb R $ |
| @#55EE55: postulate   | @#55EE55: $ H $ ... differentiable in $\Gamma_{\mathcal M}$ |

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

=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Hamiltonian_mechanics|Hamiltonian mechanics]], 

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=== Parents ===
=== Context ===
[[ODE system]]
=== Requirements ===
[[Smooth manifold]]
=== Related ===
[[Euler-Lagrange equations]]