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 hamiltonian_equations [2013/11/04 19:24]nikolaj hamiltonian_equations [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/11/05 09:43 nikolaj 2013/11/04 19:24 nikolaj 2013/11/04 19:09 nikolaj 2013/11/04 19:09 nikolaj 2013/11/04 19:08 nikolaj 2013/11/04 19:08 nikolaj old revision restored (2013/11/04 18:41) Next revision Previous revision 2013/11/05 09:43 nikolaj 2013/11/04 19:24 nikolaj 2013/11/04 19:09 nikolaj 2013/11/04 19:09 nikolaj 2013/11/04 19:08 nikolaj 2013/11/04 19:08 nikolaj old revision restored (2013/11/04 18:41) Line 1: Line 1: ===== 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: $\pi \in \mathrm{it}$ | - | @#55EE55: $\pi:​C(\mathbb R,​\Gamma_{\mathcal M})$  | + | @#55EE55: postulate ​  | @#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 Line 13: Line 13: ==== 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. Line 41: Line 41: === Subset of === === Subset of === [[ODE system]] [[ODE system]] - === Context ​=== + === Related ​=== [[Classical Hamiltonian system]] [[Classical Hamiltonian system]]