O'Reilly logo

The Mathematical Mechanic by Mark Levi

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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 ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required