The Euler equations for the arc length in two dimensions yields a straight line:

A simple pendulum has the Lagrangian :

The solution to the pendulum equation can be expressed using the function JacobiAmplitude:

The Lagrangian of a point particle in two dimensions has two dependent variables, and yields Newton's equations:

The Lagrangian of a point particle in two dimensions with a central potential:

Second- and higher-order derivatives may be included in the integrand. A Lagrangian for motion on a spring using higher-order terms:

The integrand has several independent variables:

The Euler equations yield Laplace's equation:

The Euler equations for the integrand :

The "textbook" answer:

Check:

The brachistochrone problem asks for the curve of quickest descent. The time taken for a particle to fall an arc length is . If measures the decrease in height from an initial point of release, then the velocity satisfies:

The equation for a curve joining two points, where a particle starting at rest from the higher point takes the least amount of time to reach the lower point:

It is well known that the solution to the brachistochrone problem is a cycloid:

The Lagrangian for a vibrating string yields the classical wave equation: