return the commutator of two infinitesimals of a symmetry transformation
SymmetryCommutator(S1, S2, DepVars, options=value)
two lists with the infinitesimals of a symmetry transformation, either as lists or procedures (infinitesimal generators)
function or a list of them indicating the dependent variables of the problem
jetnotation = ...
(optional) true (default, the notation found in S1), false, jetnumbers, jetvariables, jetvariableswithbrackets or jetODE; to respectively return or not using the different jet notations available
output = ...
(optional) list or operator; specifies whether the output should be a list of infinitesimal components or its corresponding infinitesimal generator differential operator
prolongation = ...
(optional) positive integer indicating the desired prolongation order of the commutator; default is the prolongation order found in the given S1.
Given a pair of infinitesimals of a symmetry transformation, S1 and S2, either as lists or differential operators (see infinitesimal generator), the SymmetryCommutator command returns the commutator of these symmetries, S1,S2=S1@S2−S2@S1.
If S1 and S2 are infinitesimal generator differential operators, the result is also a differential operator. If S1 and S2 are given as lists, then the coefficients in that differential operator, so conforming the infinitesimal, are returned within a list. When S1 and S2 are not of the same kind (operator or list), the output is in the format of S1 unless indicated otherwise using the option output = ...
The prolongation order of the commutator S1,S2 returned is the one found in S1 unless indicated otherwise using the option prolongation = n where n is a non-negative integer.
The jet notation used in the output is the one of S1 unless indicated otherwise using the option jetnotation = ... where the right-hand side is any of jetnumbers' (default), jetODE, jetvariables or jetvariableswithbrackets; for details about the available jet notations see ToJet.
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 two lists of infinitesimals corresponding to a symmetry transformation where there are two independent variables and one dependent variable, u⁡x,t.
S1,S2 ≔ _ξx=x,_ξt=1,_ηu=t,_ξx=1,_ξt=1t,_ηu=x2
The corresponding infinitesimal generators in operator format are
G1 ≔ InfinitesimalGenerator⁡S1,u⁡x,t
G2 ≔ InfinitesimalGenerator⁡S2,u⁡x,t
The symmetry commutator is S1,S2=S1@S2−S2@S1; when S1 is a operator, the output is then a differential operator
The output can be requested as an ordered list of infinitesimal components
The input can also be given in mixed formats, in which case the output is returned in the format of the first infinitesimal
The prolongation order of the commutator is by default the one of the given infinitesimals, but can also be specified using the optional argument prolongation = n, where n is a positive integer.
To request the output in a different notation, for instance jetnumbers (see ToJet), use the optional argument jetnotation = ....
Note that in the output above the infinitesimals (right-hand-sides) and also their labels (left-hand-sides) are written in jetnumbers notation. You can also specify the output format to be an operator
The PDEtools[SymmetryCommutator] command was introduced in Maple 15.
For more information on Maple 15 changes, see Updates in Maple 15.
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