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 labels for the infinitesimals, as in , or use the corresponding infinitesimal generator
The equation invariant under the symmetry transformation underlying the infinitesimal generator is (you can equally use G instead of S)
This is the explicit form of the PDE system as explained in the Description, in terms of two arbitrary functions and
The invariance of can be verified in various ways, the simplest of which is perhaps to use SymmetryTest, which verifies that is a symmetry of .
A less abstract verification would be to explicitly construct the symmetry transformation related to , for example, in terms of new variables .
Now, change the variables in using , and recall that the Lie group parameter is real:
The equation above is identical to , that is the meaning of invariant in the context of symmetries. Alternatively, the following is the prolongation to order 1 of the infinitesimal generator (5.2) (that is, ready to act on functions depending on and partial derivatives of of order 1 at most).
If you apply this differential operator to , you obtain zero because of the invariance of the under the transformation related to . To apply , an operator in jet notation, you need to rewrite in the same notation
Underlying this zero, there is the way is constructed: it is an arbitrary function of the Invariants associated to , that is, the solutions of the differential operator (you can equally pass instead of )
InvariantEquation also handles dynamical symmetries, that is, symmetries that depend on the partial derivatives of the dependent variables of the problem. In these cases it is however of no use to directly compute the invariants, as in the example above, because they will all be of higher order. The approach used is then different. Consider for instance these infinitesimals depending on
As mentioned, the differential invariants, for instance, of order 1, automatically come depending on higher derivatives
In fact, for order = n, and besides the invariant of order 0, all the other ones will always be of order . To see the actual invariants of order 1 you would need to eliminate the higher derivatives using the related invariant equation, thus defeating the use of these invariants with the purposes of constructing invariant equations of order . The approach used by InvariantEquation results instead in
Note the depend on first and second derivatives of , as it should be. In explicit form (use declare to have a more compact and readable output with derivatives displayed in jet notation)
where the PDE system returned contains as many differential equations as partial derivatives of order 2, that is, 3 equations, and the same number of arbitrary functions .