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hamiltonian_equations [2013/11/04 19:09] nikolaj |
hamiltonian_equations [2014/03/21 11:11] |
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- | ===== Hamiltonian equations ===== | ||
- | ==== Definition ==== | ||
- | | @#88DDEE: $ \langle \mathcal M, H\rangle $ ... Classical Hamiltonian system | | ||
- | | @#FFBB00: $ \pi \in \mathrm{it} $ | | ||
- | |||
- | | @#DDDDDD: $ \Gamma_{\mathcal M}\equiv \mathcal M\times T^*\mathcal M $ | | ||
- | | @#55EE55: $ \pi:C(\mathbb R,\Gamma_{\mathcal M}) $ | | ||
- | |||
- | | @#55EE55: $ \pi'(t) = X_H(\pi(t)) $ | | ||
- | |||
- | >todo: Hamiltonian vector field | ||
- | |||
- | ==== Discussion ==== | ||
- | == Equivalent definitions == | ||
- | | @#88DDEE: $ \langle \mathcal M, H\rangle ... Hamiltonian system $ | ||
- | |||
- | | @#DDDDDD: $ {\bf q} \in \mathcal M $ | | ||
- | | @#DDDDDD: $ {\bf p} \in T^*\mathcal M $ | | ||
- | | @#DDDDDD: $ H:: H({\bf q},{\bf p},t)$ | | ||
- | |||
- | | @#FFBB00: $ \langle q,p \rangle \in \mathrm{it} $ | | ||
- | |||
- | | @#55EE55: $ q:C(\mathbb R,\mathcal M) $ | | ||
- | | @#55EE55: $ p:C(\mathbb R,T^*\mathcal M) $ | | ||
- | |||
- | | $i\in\mathrm{range}(\mathrm{dim}(\mathcal M))$ | | ||
- | |||
- | | @#55EE55: $ \frac{\partial}{\partial t}q^i(t) = \frac{\partial}{\partial {\bf p}_i} H(q(t),p(t),t) $ | | ||
- | | @#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]] | ||
- | === Context === | ||
- | [[Classical Hamiltonian system]] |