Persamaan Hamilton

[cotrans]

Hosana in excelcis.

\section{Persamaan Hamilton}

Andaikan ada sebuah Lagrangian $L \mapsto (q, \dot{q}, t)$, di mana $q$ adalah satu-satunya koordinat umum, $t$ adalah waktu, dan $\dot{q} := dq/dt$, serta $q \mapsto t$ dan $\dot{q} \mapsto t$.  Andaikan ada sebuah momentum umum $p$ yang didefinisikan sebagai
$$p := \left(\frac{\partial L}{\partial\dot{q}}\right)_{q,t}.$$
Tentu saja, $p \mapsto (q, \dot{q}, t)$, sehingga tentu saja $\dot{q} \mapsto (q, p, t)$.

Karena $L$ memenuhi persamaan Euler-Lagrange, yaitu
$$\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}}\right)_{q,t} = \left(\frac{\partial L}{\partial q}\right)_{\dot{q},t},$$
maka
$$\dot{p} = \left(\frac{\partial L}{\partial q}\right)_{\dot{q},t}.$$
Tentu saja, $\dot{p} \mapsto (q, \dot{q}, t)$.

Andaikan ada sebuah Hamiltonian $H \mapsto (q, p, t)$, yang didefinisikan sebagai $H := \dot{q}p - L$.

Karena $L = L_{q, \dot{q}, t}(q, \dot{q}_{q,p,t}(q, p, t), t)$, maka
$$\left(\frac{\partial H}{\partial q}\right)_{p, t} = \left(\frac{\partial\dot{q}}{\partial q}\right)_{p,t}p - \left(\frac{\partial L}{\partial q}\right)_{p,t}.$$
Karena
$$\left(\frac{\partial L}{\partial q}\right)_{p,t} = \left(\frac{\partial L}{\partial q}\right)_{\dot{q},t} + \left(\frac{\partial L}{\partial\dot{q}}\right)_{q,t}\left(\frac{\partial\dot{q}}{\partial q}\right)_{p,t},$$
maka
$$\left(\frac{\partial H}{\partial q}\right)_{p, t} = \left(\frac{\partial\dot{q}}{\partial q}\right)_{p,t}\left(p - \left(\frac{\partial L}{\partial\dot{q}}\right)_{q,t}\right) - \left(\frac{\partial L}{\partial q}\right)_{\dot{q},t}$$
sehingga
$$\left(\frac{\partial H}{\partial q}\right)_{p, t} = -\dot{p}.$$
Demikian pula,
$$\left(\frac{\partial H}{\partial p}\right)_{q,t} = \left(\frac{\partial\dot{q}}{\partial p}\right)_{q,t} + \dot{q} - \left(\frac{\partial L}{\partial p}\right)_{q,t}.$$
Karena
$$\left(\frac{\partial L}{\partial p}\right)_{q,t} = \left(\frac{\partial L}{\partial\dot{q}}\right)_{q,t}\left(\frac{\partial\dot{q}}{\partial p}\right)_{q,t},$$
maka
$$\left(\frac{\partial H}{\partial p}\right)_{q,t} = \left(\frac{\partial\dot{q}}{\partial p}\right)_{q,t}\left(p - \left(\frac{\partial L}{\partial\dot{q}}\right)_{q,t}\right) + \dot{q}$$
sehingga
$$\left(\frac{\partial H}{\partial p}\right)_{q,t} = \dot{q}.$$

Benedictus qui venit in nomine Domini.