## Lagrange's equation A variation of the [[Hamilton's principle]] gives the Lagrange's equation, which has the same meaning as Newton's 2nd law, the form is $\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial \mathcal{L}}{\partial \dot{q_i}} - \frac{\partial \mathcal{L}}{\partial q_i} = 0$ where $i = 1, \ldots, n$. >[!note] >The above equation describes the actual evolution, i.e., $\delta I = 0$ > >In this expression, the $\frac{\partial \mathcal{L}}{\partial \dot{q_i}}$ is (generalized) momentum $p_i$. (Notice its partial $\dot{q_i}$, the velocity) $\mathcal{L}$ is Lagrangian, $t$ is time, $q_i$ is generalized coordinates, and it can be shown that if the potential $V$ is time independent, then the total energy $H = T+V$ is conserved, i.e., energy conservation. >[!info] >Read more on https://en.wikipedia.org/wiki/Lagrangian_mechanics