Euler-Lagrange Equation

Statement

For an action functional

$$S[q] = \int_{t_1}^{t_2} L(q,\dot q,t)\,dt,$$

stationarity under variations with fixed endpoints gives

$$\frac{d}{dt}\frac{\partial L}{\partial \dot q_i} - \frac{\partial L}{\partial q_i}=0.$$

Meaning

The equation is Newton’s law written in coordinates adapted to constraints. It is the local form of the principle of stationary action.