Variational Methods Package Symbol

EulerEquations

EulerEquations[f, u[x], x]
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.

EulerEquations[f, u[x, y, ...], {x, y, ...}]
returns the Euler-Lagrange differential equation obeyed by u[x, y, ...].

EulerEquations[f, {u[x, y, ...], v[x, y, ...], ...}, {x, y, ...}]
returns a list of Euler-Lagrange differential equations obeyed by u[x, y, ...], v[x, y, ...], ....

MORE INFORMATION

To use EulerEquations, you first need to load the Variational Methods Package using Needs["VariationalMethods`"].

EXAMPLES

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Basic Examples (2)

The Euler equations for the arc length

in 2 dimensions yields a straight line:

A simple pendulum has the Lagrangian

:

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

Needs["VariationalMethods`"]

The Euler equations for the arc length

in 2 dimensions yields a straight line:

Needs["VariationalMethods`"]

A simple pendulum has the Lagrangian

:

In[2]:=

Out[2]=

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

In[3]:=

Out[3]=

Scope (4)

The Lagrangian of a point particle in 2 dimensions has 2 dependent variables, and yields Newton's equations:

The Lagrangian of a point particle in 2 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:

Applications (3)

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 ds is ds/v. If y measures the decrease in height from an initial point of release, then the velocity v 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: