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.
Calculate the Euler-Lagrange equations for .
The convert/DGdiff command will change this output from jet space notation to standard differential equations notation.
Here are the same calculations done with differential forms.
Example 2.
Create a space of 1 independent variable and 1 dependent variable.
Calculate the Euler-Lagrange equations for an arbitrary second order Lagrangian.
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 .
Here are the same calculations again using an alternative jet space notation. See Preferences for details.
Calculate the Euler-Lagrange equations for an arbitrary second order Lagrangian.
Example 3.
Create a space of 3 independent variables and 1 dependent variable. Derive the Laplace's equation from its variational principle.
Repeat this computation using differential forms.
Example 4.
Create a space of 3 independent variables and 3 dependent variables. Derive 3-dimensional Maxwell equations from the variational principle.
Define the Lagrangian.
Compute the Euler-Lagrange equations.
Change notation to improve readability.
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.
First we try a form of vertical degree 1.
Try a form of vertical degree 2.
Here is the explicit formula for computing EulerLagrange(omega2).
Now we compute some simple examples in the case of 2 independent variables and 2 dependent variables.
Try a form of vertical degree 1.
Try a form of vertical degree 2.
Try a form of vertical degree 3.
The Euler-Lagrange operator of the horizontal exterior derivative of any form vanishes, for example: