returns the Euler–Lagrange differential equation obeyed by u[x] derived from the functional f, where f depends on the function u[x] and its derivatives, as well as the independent variable x.
returns the Euler–Lagrange differential equation obeyed by u[x,y,…].
returns a list of Euler–Lagrange differential equations obeyed by u[x,y,…],v[x,y,…],….
Details and Options
- To use EulerEquations, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].
Examplesopen allclose all
Basic Examples (2)
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 for the integrand :
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: