## 8

**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 :

where ...