LieBracket(X, Y)

Define a 2-dimensional manifold M..

Example 1.

Define a pair of vector fields X1 and Y1.

Calculate the Lie bracket of X1 and Y1.

Let's check this result against the commutator definition of the Lie bracket acting on functions.  To apply a vector field to a function we use the LieDerivative command.

Example 2.  

Here is the general coordinate formula for the Lie bracket of two vector fields defined on a 2-dimensional manifold.

Example 3.  

Two vector fields are said to commute if their Lie bracket is 0.  For example:

Example 4.  

The Lie bracket satisfies the Jacobi identity [[X, Y], Z] + [[Z, X], Y] + [[Y, Z], X] = 0.  For example:

Example 5.  

Use LieAlgebraData and DGsetup to initialize a Lie algebra.  

Calculate the Lie bracket of 2 vectors in this Lie algebra.

DifferentialGeometry, ExteriorDerivative, LieDerivative