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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 $ | +| @#55EE55postulate ​  | @#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]]
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