Question

How is the euler lagrange eqution in general related to the directional derivative?

Answer 1

euler lagrange equation is a second order partial diffrential equation whose solutions are the functions for which a given functional is stationary.

in some cases the equation can be solved directly in closed form. for other cases one uses numerical techniques for gradient descent which gives rise to a partial differential equation. in effect calculus of variation extends vector calculus to enable us to evaluate derivatives of functional, from vector calculus then the directional derivative in a direction where the product expands