Mathcenter Forum - Euler–Lagrange equation

In the calculus of variations, the Euler equation[1]
is a second-order partial differential equation whose solutions are the functions
for which a given functional is stationary.

It was developed by Swiss mathematician Leonhard Euler and
Italian mathematician Joseph-Louis Lagrange in the 1750s.

Because a differentiable functional is stationary at its local extrema,
the Euler–Lagrange equation is useful for solving optimization problems in which,
given some functional, one seeks the function minimizing or maximizing it.

This is analogous to Fermat's theorem in calculus,
stating that at any point where a differentiable function attains a local extremum
its derivative is zero.