We define a manifold with coordinates.
Example 1.
Find all vector fields which commute with the vector field .
Find all vector fields whose coefficients depend only on which commute with the vector field .
Example 2.
Find the infinitesimal symmetries for the metric .
Show the defining differential equations for these symmetries. Here we explicitly define the general form of the symmetry vector and specify the unknowns.
We can use the auxilaryequations option to find the symmetries X of the metric g for which
Example 3.
Find the joint infinitesimal symmetries for the 0 connection and the volume form .
Example 4.
Here is a famous calculation due to E. Cartan. See Fulton and Harris Representation Theory page 357. We find the linear infinitesimal symmetries of the 3-form defined on the 7-manifold N with coordinates .
It is a simple matter to use the package LieAlgebras to check that this Lie algebra is indecomposable and simple and is a realization of the exceptional Lie algebra .
Example 5.
Find the point symmetries of the Lagrangian for the (2 +1) wave equation. The result is a 8-dimensional Lie algebra.
Example 6.
Find the infinitesimal conformal symmetries of the metric . These are the vector fields such that or span.
Note that the first argument is now a list of a list.
The conformal symmetries of define a 10-dimensional Lie algebra.
Example 7.
Find the infinitesimal symmetries of a distribution of vector fields . These are the vector fields such that (Y) for each .
Example 8.
Find the symmetries of a metric which depend upon 2 parameters , where .
Example 9.
The command InfinitesimalSymmetriesOfGeometricObjectFields can also be used to calculate the symmetries of a tensor defined on a Lie algebra.