InfinitesimalTransformation(phi, par, initialpoint)

Example 1.

The group of Euclidean motions in the plane, consisting of translations in the coordinate directions and rotations about the origin. We initialize the coordinates on the plane and define a 3-parameter transformation consisting of all the Euclidean motions.

To calculate the structure equations for this Lie algebra of vector fields, use the LieAlgebraData command from the LieAlgebras package.  Here [e1, e2, e3] denote the vectors in Gamma and only the non-trivial brackets are displayed.

Example 2.

The group of fractional linear transformations on the line.

The identity transformation is given by a = 1, b = 0, c = 0, d = 1.  Only the non-zero parameter values need to be specified.

Note that these vectors fields are not linearly independent over the real numbers (Gamma2[1] = - Gamma2[4]).  This is because the parameter values [a = t, b = 0, c = 0, d = 1] and [a = 1, b = 0, c = 0, d = 1/t] generate the same 1-parameter group of transformations, that is, the action is not effective.

We can remove the linearly dependent elements of Gamma2 a with the DGbasis command.

Alternatively, we can make the action effective by normalizing the parameters to a*b - c*d = 1.  (Now the group is SL2, the set of all 2 x 2 matrices with a determinant of 1.)

Example 3.

The group of fractional linear transformations in the plane.

Again we have to remove linearly dependent vectors:

DifferentialGeometry, LieAlgebras, DGbasis, Flow, GetComponents, LieAlgebraData, Transformation