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