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
\[ \left(\frac{\partial H}{\partial p}\right)_{q,t} = \dot{q}. \]