SymmetryTransformation(S, DepVars, NewVars, 'options'='value')

Consider a PDE problem with two independent variables and one dependent variable, , and consider the list of infinitesimals of a symmetry group

In the input above you can also enter the symmetry  without infinitesimals' labels, as in . The corresponding infinitesimal generator is

A  is invariant under the symmetry transformation generated by  in that , where, in this formula,  represents the prolongation necessary to act on  (see InfinitesimalGenerator).

The actual form of this finite, one-parameter, symmetry transformation relating the original variables  to new variables, , that leaves invariant any PDE system admitting the symmetry represented by  above is obtained via

where  is a (Lie group) transformation parameter. To express this transformation using jetnotation use

That this transformation leaves invariant any PDE system invariant under  above is visible in the fact that it also leaves invariant the infinitesimals ; to verify this you can use ChangeSymmetry

which is the same as  (but written in terms of  instead of ). So to this list of infinitesimals corresponds, written in terms of , this infinitesimal generator

which is also equal to , only written in terms of .

If the new variables, ,  are not indicated, variables prefixed by the underscore _ to represent the new variables are introduced

An example where the Lie group parameter  appears only through the subexpression

A symmetry transformation with the parameter redefined

CanonicalCoordinates, ChangeSymmetry, InfinitesimalGenerator, Invariants, InvariantSolutions, InvariantTransformation, PDEtools, SimilarityTransformation, SymmetrySolutions