Differences

This shows you the differences between two versions of the page.

Link to this comparison view

hamiltonian_equations [2013/11/04 19:24]
nikolaj
hamiltonian_equations [2014/03/21 11:11]
Line 1: Line 1:
-===== Hamiltonian equations ===== 
-==== Definition ==== 
-| @#88DDEE: $ \langle \mathcal M, H\rangle $ ... Classical Hamiltonian system | 
  
-| @#FFBB00: $ \pi \in \mathrm{it} $ | 
- 
-| @#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]] 
Link to graph
Log In
Improvements of the human condition