THE EULER-LAGRANGE EQUATION VIA STRETCHED SPRINGS
8.1 Some Background on the Euler-Lagrange Equation
This short chapter contains a purely mechanical interpretation of the Euler-Lagrange functional as the potential energy of an imaginary spring. This interpretation makes for an almost immediate derivation of the Euler-Lagrange equations and gives a very transparent mechanical explanation of the conservation of energy. Moreover, each individual term in the Euler-Lagrange equation acquires a concrete mechanical meaning.
Here is some motivation for the reader not familiar with the Euler-Lagrange equations.
A basic problem of the calculus of variations is to find a function x(t) which minimizes an integral involving x and its derivative :