Euler-Lagrange equation

The necessary condition for $y$ to extremize $\int L(x,y,y')\,dx$ : $\partial L/\partial y - d/dx(\partial L/\partial y')=0$ .

The Euler-Lagrange equation $\partial L/\partial y - \frac{d}{dx}(\partial L/\partial y')=0$ is the necessary condition for $y$ to extremize the functional $J[y]=\int_a^b L(x,y,y'')\,dx$ , derived from requiring the first variation to vanish for every admissible perturbation.

Examples

For $L=(y')^2$ , the Euler-Lagrange equation reduces to $y''=0$ -- straight lines extremize arc length in flat space.