Hamiltonian equations
Set
| context | $ \langle \mathcal M, H\rangle $ … Classical Hamiltonian system |
| definiendum | $ \pi \in \mathrm{it} $ |
| postulate | $ \pi:C(\mathbb R,\Gamma_{\mathcal M}) $ |
| postulate | $ \pi'(t) = X_H(\pi(t),t) $ |
todo: Hamiltonian vector field
Discussion
Equivalent definitions
| range | $ {\bf q} \in \mathcal M $ |
| range | $ {\bf p} \in T^*\mathcal M $ |
| range | $ H:: H({\bf q},{\bf p},t)$ |
| definiendum | $ \langle q,p \rangle \in \mathrm{it} $ |
| postulate | $ q:C(\mathbb R,\mathcal M) $ |
| postulate | $ p:C(\mathbb R,T^*\mathcal M) $ |
| $i\in\mathrm{range}(\mathrm{dim}(\mathcal M))$ |
| postulate | $ \frac{\partial}{\partial t}q^i(t) = \frac{\partial}{\partial {\bf p}_i} H(q(t),p(t),t) $ |
| postulate | $ \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
Parents
Subset of
