computes the finite form of the (symmetry) transformation leaving invariant any PDE system admitting a given symmetry
SymmetryTransformation(S, DepVars, NewVars, 'options'='value')
a list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator
a function or a list of functions indicating the dependent variables of the problem
optional - a function or a list of functions representing the new dependent variables
jetnotation = ...
(optional) can be true (same as jetvariables), false (default), jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available
simplifier = ...
optional - indicates the simplifier to be used instead of the default simplify/size
optional - to simplify a subexpression that involves the Lie group parameter replacing it by another group parameter
Given a list with the infinitesimals of a symmetry symmetry transformation.
When there is only one dependent variable, DepVars and NewVars can be a function. Otherwise they must be a list of functions representing dependent variables. If NewVars are not given, SymmetryTransformation will generate a list of globals to represent them.
You can optionally specify a simplifier, to be used instead of the default which is simplify/size, as well as requesting the output to be in jet notation by respectively using the optional arguments simplifier = ... and jetnotation. Note that the option simplifier = ... can be used not just to "simplify" the output but also to post-process this output in the way you want, for instance using a procedure that you have written to discard, change or do what you find necessary with the transformation.
In some cases, the Lie group parameter introduced by SymmetryTransformation appears embedded into a subexpression, for example as in ⅇ_ε, and only appears through functions of that subexpression. To have these cases returned with _ε instead of - say - ⅇ_ε, use the option redefinegroupparameter.
To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.
Consider a PDE problem with two independent variables and one dependent variable, u⁡x,t, and consider the list of infinitesimals of a symmetry group
In the input above you can also enter the symmetry S without infinitesimals' labels, as in x,1,u. The corresponding infinitesimal generator is
A PDESYS is invariant under the symmetry transformation generated by G in that G⁡PDESYS=0, where, in this formula, G represents the prolongation necessary to act on PDESYS (see InfinitesimalGenerator).
The actual form of this finite, one-parameter, symmetry transformation relating the original variables t,x,u⁡x,t to new variables, r,s,v⁡r,s, that leaves invariant any PDE system admitting the symmetry represented by G above is obtained via
where _ε is a (Lie group) transformation parameter. To express this transformation using jetnotation use
That this transformation leaves invariant any PDE system invariant under G above is visible in the fact that it also leaves invariant the infinitesimals S; to verify this you can use ChangeSymmetry
which is the same as S (but written in terms of v⁡r,s instead of u⁡x,t). So to this list of infinitesimals corresponds, written in terms of v⁡r,s, this infinitesimal generator
which is also equal to G, only written in terms of v⁡r,s.
If the new variables, v⁡r,s, are not indicated, variables prefixed by the underscore _ to represent the new variables are introduced
An example where the Lie group parameter _ε appears only through the subexpression ⅇ_ε
A symmetry transformation with the parameter redefined
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