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hamiltonian_equations [2013/11/04 19:08] nikolaj |
hamiltonian_equations [2013/11/05 09:43] nikolaj |
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| ===== Hamiltonian equations ===== | ===== Hamiltonian equations ===== | ||
| ==== Definition ==== | ==== Definition ==== | ||
| - | | @#88DDEE: $ \langle \mathcal M, H\rangle $ ... Hamiltonian system | | + | | @#88DDEE: $ \langle \mathcal M, H\rangle $ ... Classical Hamiltonian system | |
| | @#FFBB00: $ \pi \in \mathrm{it} $ | | | @#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:C(\mathbb R,\Gamma_{\mathcal M}) $ | | ||
| - | | @#55EE55: $ \pi'(t) = X_H(\pi(t)) $ | | + | | @#55EE55: $ \pi'(t) = X_H(\pi(t),t) $ | |
| >todo: Hamiltonian vector field | >todo: Hamiltonian vector field | ||
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| | $i\in\mathrm{range}(\mathrm{dim}(\mathcal M))$ | | | $i\in\mathrm{range}(\mathrm{dim}(\mathcal M))$ | | ||
| - | | @#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}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) $ | | | @#55EE55: $ \frac{\partial}{\partial t}p_i(t) = -\frac{\partial}{\partial {\bf q}^i} H(q(t),p(t),t) $ | | ||
