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 generator, or the corresponding infinitesimal generator differential operator, SymmetryTransformation computes the actual finite form of that 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
S ≔ _ξx=x,_ξt=1,_ηu=u
In the input above you can also enter the symmetry S without infinitesimals' labels, as in x,1,u. The corresponding infinitesimal generator is
G ≔ InfinitesimalGenerator⁡S,u⁡x,t
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
TR,NewVars ≔ solve⁡,x,t,u⁡x,t,map⁡lhs,
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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