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Hamiltonian equations

Definition

$ \langle \mathcal M, H\rangle $ … Classical Hamiltonian system
$ \pi \in \mathrm{it} $
$ \pi:C(\mathbb R,\Gamma_{\mathcal M}) $
$ \pi'(t) = X_H(\pi(t)) $
todo: Hamiltonian vector field

Discussion

Equivalent definitions
$ {\bf q} \in \mathcal M $
$ {\bf p} \in T^*\mathcal M $
$ H:: H({\bf q},{\bf p},t)$
$ \langle q,p \rangle \in \mathrm{it} $
$ q:C(\mathbb R,\mathcal M) $
$ p:C(\mathbb R,T^*\mathcal M) $
$i\in\mathrm{range}(\mathrm{dim}(\mathcal M))$
$ \frac{\partial}{\partial t}q^i(t) = \frac{\partial}{\partial {\bf p}_i} H(q(t),p(t),t) $
$ \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

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Context

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