===== Hamiltonian equations =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ \langle \mathcal M, H\rangle $ ... Classical Hamiltonian system |

| @#FFBB00: definiendum | @#FFBB00: $ \pi \in \mathrm{it} $ |

| @#55EE55: postulate   | @#55EE55: $ \pi:C(\mathbb R,\Gamma_{\mathcal M}) $  |

| @#55EE55: postulate   | @#55EE55: $ \pi'(t) = X_H(\pi(t),t) $  |

>todo: Hamiltonian vector field

==== Discussion ====
== Equivalent definitions ==
| @#55CCEE: context     | @#55CCEE: $ \langle \mathcal M, H\rangle ... Hamiltonian system $ 

| @#DDDDDD: range       | @#DDDDDD: $ {\bf q} \in \mathcal M $ |
| @#DDDDDD: range       | @#DDDDDD: $ {\bf p} \in T^*\mathcal M $ |
| @#DDDDDD: range       | @#DDDDDD: $ H:: H({\bf q},{\bf p},t)$ | 

| @#FFBB00: definiendum | @#FFBB00: $ \langle q,p \rangle \in \mathrm{it} $ |

| @#55EE55: postulate   | @#55EE55: $ q:C(\mathbb R,\mathcal M) $  |
| @#55EE55: postulate   | @#55EE55: $ p:C(\mathbb R,T^*\mathcal M) $  |

| $i\in\mathrm{range}(\mathrm{dim}(\mathcal M))$ |

| @#55EE55: postulate   | @#55EE55: $ \frac{\partial}{\partial t}q^i(t) = \frac{\partial}{\partial {\bf p}_i}  H(q(t),p(t),t) $ |
| @#55EE55: postulate   | @#55EE55: $ \frac{\partial}{\partial t}p_i(t) = -\frac{\partial}{\partial {\bf q}^i} H(q(t),p(t),t) $ |

i.e.

$ \pi:: t\mapsto\langle q^1(t),\dots,q^s(t),p_1(t),\dots,p_s(t) \rangle $

$ \langle q(t),p(t) \rangle\equiv \langle q^1(t),\dots,q^s(t),p_1(t),\dots,p_s(t) \rangle $
=== Reference ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Hamiltonian_mechanics|Hamiltonian mechanics]]

==== Parents ====
=== Subset of ===
[[ODE system]]
=== Related ===
[[Classical Hamiltonian system]]