In learning Hamilton and Lagranian Mechanics, calculus of variation is needed.

The essence is to solve min/max problem not on a value but on a function parameterized by the value.

Application of it not only is Lagranian mechanics, but also the geodesic computation on both a flat plan and a smooth curved plane. In explaining geodesic Eigenchris and faculty of khan both did a good job. in Euler-Lagrange equation, its goal is to find the stationary state where kinetic energy minus the potential energy maintain same, i.e. the differential of both is zero. When variable x is fixed, it’s the Beltrami Identity, which is related The Brachistochrone Problem and Solution.

When there are multiple variables and function of these variables together with the first derivatives of these variables, and boundary constraints in the starting and ending point, it’s the general version of Euler-Lagrange problem.