This is documentation for Mathematica 6, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)

Variational Methods

The basic problem of the calculus of variations is to determine the function u (x) that extremizes a functional . In general, there can be more than one independent variable and the integrand f can depend on several functions and their higher derivatives.
The extremal functions are solutions of the Euler(-Lagrange) equations that are obtained by setting the first variational derivatives of the functional F with respect to each function equal to zero. Since many ordinary and partial differential equations that occur in physics and engineering can be derived as the Euler equations for appropriate functionals, variational methods are of general utility.
VariationalD[f,u[x],x],VariationalD[f,u[x,y,...],{x,y,...}]give the first variational derivative of the functional F defined by the integrand f, where f depends on one function u and one independent variable x or several independent variables x, y, ...
VariationalD[f,{u[x,y,...],v[x,y,...],...},{x,y,...}]give a list of the first variational derivatives of the functional F defined by the integrand f, where f depends on several functions u, v, ... and several independent variables x, y, ...
EulerEquations[f,u[x],x],EulerEquations[f,u[x,y,...],{x,y,...}]give the Euler equation for the integrand f, where f depends on one function u and one independent variable x or several independent variables x, y, ...
EulerEquations[f,{u[x,y,...],v[x,y,...],...},{x,y,...}]give a list of the Euler equations for the integrand f, where f depends on several functions u, v, ... and several independent variables x, y, ...

First variational derivatives and Euler equations.

VariationalD gives the first variational derivatives of a functional F defined by the integrand f. f may depend on several functions u, v, w, ...; their derivatives of arbitrary order; and variables x, y, z, ... . EulerEquations returns the Euler(-Lagrange) equations given the integrand f. Again f may depend on several functions u, v, w, ...; their derivatives of arbitrary order; and variables x, y, z... .
This loads the package.
In[1]:=
Click for copyable input
This is the first variational derivative of .
In[2]:=
Click for copyable input
Out[2]=
Here f is the Lagrangian for the simple pendulum and EulerEquations gives the pendulum equation.
In[3]:=
Click for copyable input
Out[3]=
This package defines several coordinates systems as well as the Grad function.
In[4]:=
Click for copyable input
The default coordinate system is set to Cartesian and the coordinates are set to x, y, and z.
In[5]:=
Click for copyable input
This generates Laplace's equation.
In[6]:=
Click for copyable input
Out[6]=
FirstIntegrals[f,u[x],x],FirstIntegrals[f,{u[x],v[x],...},x]give first integrals when the integrand f is independent of one or more of {u[x], v[x], ...}, or independent of x
FirstIntegral[u]first integral associated with the variable u (appears in the output of FirstIntegrals)

First integrals.

When there is only one independent variable x, FirstIntegrals gives conserved quantities in the following cases: (1) if f does not depend on a coordinate u explicitly, it is referred to as an ignorable coordinate and the corresponding Euler equation possesses an obvious first integral (a conserved generalized momentum), and (2) if f depends on u, v, ... and their first derivatives only and has no explicit x dependence, FirstIntegrals also returns the first integral corresponding to the Hamiltonian.
The Lagrangian for central force motion has an ignorable coordinate (angular momentum conservation) and is independent of time t (energy conservation). FirstIntegrals yields both the first integral corresponding to coordinate and the first integral corresponding to the Hamiltonian.
In[7]:=
Click for copyable input
Out[7]=
The Ritz variational principle affords a powerful technique for the approximate solution of (1) eigenvalue problems Au=wu where A is an operator and w (x, y, ...) is a weight function and (2) problems of the form Bu (x, y, ...)=h (x, y, ...) where B is a positive definite operator and h is given. A judicious choice for the trial function ut (x, y, ...) that satisfies boundary conditions and depends on variational parameters {a, b, ...} must be given in both cases. For (1) VariationalBound[{f,g}, u[x,y,... ], {{x,xmin,xmax}, {y,ymin,ymax},... }, ut, {a,amin,amax}, {b,bmin,bmax}, ... ] extremizes where f=uAu and g=uwu. The result is an upper bound on the corresponding eigenvalue and optimal values for the parameters. For (2) VariationalBound[f, u[x, y, ... ], {{x, xmin, xmax}, {y, ymin, ymax}, ... }, ut, {a, amin, amax}, {b, bmin, bmax}, ... ] extremizes the functional with f=uBu -2uh and yields the value of the functional and the optimal parameters. VariationalBound can also be used to extremize general functionals given appropriate trial functions. NVariationalBound performs the same functions as VariationalBound numerically. It uses the internal function FindMinimum and has the same options and input format for parameters.
VariationalBound[{f,g},u[x,y,...],{{x,xmin,xmax},{y,ymin,ymax},...},ut,
{a,amin,amax},{b,bmin,bmax},...]give an upper bound for the eigenvalue and the optimal values of a, b, ... in the range {{amin, amax}, {bmin, bmax}, ... }
VariationalBound[f,u[x,y,...],{{x,xmin,xmax},{y,ymin,ymax},...},ut,
{a,amin,amax},{b,bmin,bmax},...]give the value of the functional and optimal values of a, b, ...
NVariationalBound[{f,g},u[x,y,...],{{x,xmin,xmax},{y,ymin,ymax},...},ut,
{a,a0,amin,amax},{b,b0,bmin,bmax},...]evaluate numerically an upper bound for the eigenvalue and the optimal values of a, b, ... in the range {{amin, amax}, {bmin, bmax}, ... } given initial values a0, b0, ...
NVariationalBound[f,u[x,y,...],{{x,xmin,xmax},{y,ymin,ymax},...},ut,
{a,a0,amin,amax},{b,b0,bmin,bmax},...]evaluate numerically the value of the functional and optimal values of a, b, ... given initial values a0, b0, ...

Ritz variational bounds.

A trial (wave) function for the 2s state of the hydrogen atom with one node at a yields the exact energy in units of Rydberg. Note that the volume element r2 is included in functional parameters f and g, and the default range for the parameters is (-, ).
In[8]:=
Click for copyable input
Out[8]=
The problem of the torsion of a rod of square cross section involves solving 2u=-1 where u vanishes on the boundary. VariationalBound gives optimal values of parameters for the approximate solution.
In[9]:=
Click for copyable input
Out[9]=
The ground state energy of the one-dimensional quantum anharmonic oscillator is determined for the given trial (wave) function by NVariationalBound. Note that the default range for the parameters is (-, ) and the initial values are specified.
In[10]:=
Click for copyable input
Out[10]=