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JetCalculus[EulerLagrange] - calculate the Euler-Lagrange equations for a Lagrangian

Calling Sequences

     EulerLagrange(L)

     EulerLagrange( )

     EulerLagrange()

Parameters

     L         - a function on a jet space defining the Lagrange function for a variational problem (single or multiple integral)

              - a differential bi-form on a jet space defining the Lagrangian form for a variational problem (single or multiple integral)

              - a differential bi-form of vertical degree > 0

 

Description

Examples

Description

• 

Let be a fiber bundle, with base dimension  and fiber dimension  and let  be the -th jet bundle. Introduce local coordinates , ..., where, as usual, if  is a section andis the -jet of then

    and dim.

A function  on  defines the action integral or fundamental integral,

 ,

for a -th order multiple integral problem in the calculus of variations. The Euler-Lagrange equations are the system of , order partial differential equations for the extremals  of the action integral . The general formula for the components of the Euler-Lagrange operator are

,

where is the total derivative with respect to . In the special case of a single integral variational problem, this formula can be written as

 

while for a double integral problem, we have

 

.

See Gelfand and Fomin for an excellent introduction to the calculus of variations.

 

• 

For the first calling sequence, EulerLagrange(L) returns the list of functions on .

• 

The differential forms on the jet spaces  can be bi-graded by their horizontal and vertical/contact degree. A differential form of horizontal degree and vertical degree 0 is called a Lagrangian form or Lagrangian bi-form. In terms of local coordinates on , a Lagrangian bi-form  can be expressed as

, ..., .

The associated Euler-Lagrange form  is a differential bi-form of horizontal degree  and vertical degree . It is defined in terms of the usual Euler-Lagrange expressions by

  where

 For geometrical aspects of the calculus of variations, the representation of the Euler-Lagrange equations as the components of a differential bi-form is very useful.

• 

The third calling sequence EulerLagrange() returns a list of  differential bi-forms of vertical degree 1 less than the vertical degree of . Here the partial derivatives with respect to the jets of dependent variables  in the usual formula for the Euler-Lagrange operator acting on functions are replaced by interior products of the corresponding vector fields, that is,

    where  denotes the interior product with the vector field

• 

 The command EulerLagrange is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form EulerLagrange(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-EulerLagrange(...).

Examples

 

Example 1.

Create a space of 1 independent variable and 3 dependent variables.

 

Define the standard Lagrangian from mechanics as the difference between the kinetic and potential energy.

E > 

(2.1)

 

Calculate the Euler-Lagrange equations for .

E > 

(2.2)

 

The convert/DGdiff command will change this output from jet space notation to standard differential equations notation.

E > 

(2.3)

 

Here are the same calculations done with differential forms.

E > 

(2.4)
E > 

(2.5)

 

Example 2.

Create a space of 1 independent variable and 1 dependent variable.

E > 

 

Calculate the Euler-Lagrange equations for an arbitrary second order Lagrangian.

E > 

E > 

E > 

(2.6)

 

Compare with the usual formula for the Euler-Lagrange expression in terms of the total derivatives (calculated using TotalDiff) of the partial derivative of L with respect to the jet coordinates .

E > 

(2.7)
E > 

(2.8)
E > 

(2.9)

 

Here are the same calculations again using an alternative jet space notation. See Preferences for details.

E > 

(2.10)
E > 

 

Calculate the Euler-Lagrange equations for an arbitrary second order Lagrangian.

E > 

E > 

E > 

(2.11)
E > 

(2.12)
E > 

Example 3.

Create a space of 3 independent variables and 1 dependent variable. Derive the Laplace's equation from its variational principle.

E > 

E > 

(2.13)
E > 

(2.14)
E > 

(2.15)

 

Repeat this computation using differential forms.

E > 

(2.16)
E > 

(2.17)

 

Example 4.

Create a space of 3 independent variables and 3 dependent variables. Derive 3-dimensional Maxwell equations from the variational principle.

E > 

 

Define the Lagrangian.

M > 

(2.18)

 

Compute the Euler-Lagrange equations.

M > 

(2.19)

 

Change notation to improve readability.

M > 

M > 

(2.20)

 

Example 5.

In this example we apply the Euler-Lagrange operator to some contact forms. We start with the case of 1 independent variable and 1 dependent variable.

M > 

 

First we try a form  of vertical degree 1.

E > 

(2.21)
E > 

(2.22)

 

Try a form  of vertical degree 2.

E > 

(2.23)
E > 

(2.24)

 

Here is the explicit formula for computing EulerLagrange(omega2).

E > 

(2.25)
E > 

(2.26)
E > 

(2.27)

 

Now we compute some simple examples in the case of 2 independent variables and 2 dependent variables.

E > 

 

Try a form of vertical degree 1.

E > 

(2.28)
E > 

(2.29)

 

Try a form of vertical degree 2.

E > 

(2.30)
E > 

(2.31)

 

Try a form  of vertical degree 3.

E > 

(2.32)
E > 

(2.33)

 

The Euler-Lagrange operator of the horizontal exterior derivative of any form vanishes, for example:

E > 

(2.34)
E > 

(2.35)

See Also

DifferentialGeometry

JetCalculus

Prolong

Transformation

Pullback

DifferentialEquationData

 


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