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InfinitesimalGenerator

  

receives a list with the infinitesimals of a symmetry generator and returns the corresponding infinitesimal generator as a differential operator procedure

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

InfinitesimalGenerator(S, DepVars, options=value)

Parameters

S

-

list with the infinitesimals of a symmetry generator or the corresponding infinitesimal generator operator

DepVars

-

function or list of functions indicating the dependent variables of the problem

expanded = ...

-

(optional) can be true or false (default); to expand or not the prolongation of the infinitesimals as opposed to returning just the table procedure that computes it

jetnotation = ...

-

(optional) can be true (default, the notation found in S), false, jetvariables, jetvariableswithbrackets, jetnumbers or jetODE; to respectively return or not using the different jet notations available

output = ...

-

(optional) can be list, expression or operator; indicating the output to be a list of infinitesimal components or its corresponding infinitesimal generator differential operator

prolongation = ...

-

(optional) positive integer indicating the desired prolongation order; default is 0

Description

• 

Given a list of infinitesimals of a symmetry generator, the InfinitesimalGenerator command returns a differential operator representing the infinitesimal generator. Note the distinction here made between infinitesimals and infinitesimal generator: the latter is an operator constructed as a linear combination of differential operators, while the infinitesimals are represented by the list you can construct with the coefficients of each differential operator entering the infinitesimal generator. InfinitesimalGenerator also works with anticommutative variables set using the Physics package.

• 

You can also pass the symmetry S directly as an infinitesimal generator operator, for instance with the purpose of having it rewritten in different jetnotation or prolonged - see related options below.

• 

By default the infinitesimal generator returned is not prolonged more than S - to request any other prolongation pass the optional argument prolongation=n where n is a nonnegative integer. As a handy shortcut notation, instead of the optional argument prolongation=n you can also just specify n, a nonnegative integer, as the third argument passed to InfinitesimalGenerator.

• 

By default, the infinitesimal generator is also returned not expanded. To request it otherwise, pass the optional argument expanded or expanded = true. To understand this option, recall that given a prolongation order - say k - there exist as many prolonged infinitesimals as partial derivatives of order 0<k exist in the jet space where the symmetry generator acts, that is, mn+kk1, where n and m are respectively the numbers of independent and dependent variables. In addition, the infinitesimal generator to be returned includes a sum over the independent and another one over the dependent variables, where each of the summands has for coefficient the corresponding prolongation of the infinitesimal η (see Eta_k) associated to each dependent variable. All these sums return expanded when the optional argument expanded is given. As an indicator of what this means, in a problem with 2 independent and 2 dependent variables, the length of the infinitesimal generator prolonged to order 2, when the option expanded is given, is more than 40 times larger than when it is not given.

• 

By default, the infinitesimal generator is returned as an operator, that is a mapping to be applied to a function (see the Examples section). Equivalent representations would be: an expression, i.e., the result of applying the operator to some generic function, or a list with the infinitesimal components. To compute these representations use the option output = ... where the right-hand side is respectively expression or list. When output = expression, differentiation is represented by the inert Diff command and the function to which the mapping is applied is represented generically by a label, typically _&Phi; (provided that _&Phi; is not assigned already, or found in the infinitesimals S).

• 

The jetnotation used in the output is the one of S unless indicated otherwise using the option jetnotation = ... where the right-hand side is any of jetvariables (default), jetvariableswithtbrackets, jetnumbers' or jetODE; 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.

Examples

withPDEtools&comma;InfinitesimalGenerator&comma;declare

InfinitesimalGenerator&comma;declare

(1)

Consider the generic form of a list of infinitesimals of a PDE problem in - say - two independent and two dependent variables ux&comma;t,vx&comma;t: there are then two ξ infinitesimals associated to each of the independent variables and two η infinitesimals associated to each of the dependent variables, as in

DepVarsu&comma;vx&comma;t

DepVarsux&comma;t&comma;vx&comma;t

(2)

Sseq&xi;jx&comma;t&comma;u&comma;v&comma;j&equals;x&comma;t&comma;seq&eta;jx&comma;t&comma;u&comma;v&comma;j&equals;u&comma;v

Sξxx&comma;t&comma;u&comma;v&comma;ξtx&comma;t&comma;u&comma;v&comma;ηux&comma;t&comma;u&comma;v&comma;ηvx&comma;t&comma;u&comma;v

(3)

The operator returned by InfinitesimalGenerator, say the first prolongation, is constructed basically without consuming any computational resources or time

G1InfinitesimalGeneratorS&comma;DepVars&comma;prolongation&equals;1

G1f&rarr;add&xi;xjxjf&comma;j&equals;1..2&plus;add&eta;umumf&plus;&eta;um&comma;xumxf&plus;&eta;um&comma;tumtf&comma;m&equals;1..2

(4)

The shortcut InfinitesimalGenerator(S, DepVars, 1) for indicating the prolongation returns works as well.

Consider now the same infinitesimal generator but expanded:

G2InfinitesimalGeneratorS&comma;DepVars&comma;prolongation&equals;1&comma;expanded

G2f&rarr;&xi;xx&comma;t&comma;u&comma;vxf&plus;&xi;tx&comma;t&comma;u&comma;vtf&plus;&eta;ux&comma;t&comma;u&comma;vuf&plus;&eta;vx&comma;t&comma;u&comma;vvf&plus;u&xi;xx&comma;t&comma;u&comma;vux2v&xi;xx&comma;t&comma;u&comma;vuxvxu&xi;tx&comma;t&comma;u&comma;vuxutv&xi;tx&comma;t&comma;u&comma;vutvx&plus;uxu&eta;ux&comma;t&comma;u&comma;v&plus;vxv&eta;ux&comma;t&comma;u&comma;vx&xi;xx&comma;t&comma;u&comma;vuxx&xi;tx&comma;t&comma;u&comma;vut&plus;x&eta;ux&comma;t&comma;u&comma;vuxf&plus;u&xi;xx&comma;t&comma;u&comma;vuxvxv&xi;xx&comma;t&comma;u&comma;vvx2u&xi;tx&comma;t&comma;u&comma;vuxvtv&xi;tx&comma;t&comma;u&comma;vvxvt&plus;uxu&eta;vx&comma;t&comma;u&comma;v&plus;vxv&eta;vx&comma;t&comma;u&comma;vx&xi;xx&comma;t&comma;u&comma;vvxx&xi;tx&comma;t&comma;u&comma;vvt&plus;x&eta;vx&comma;t&comma;u&comma;vvxf&plus;u&xi;xx&comma;t&comma;u&comma;vuxutv&xi;xx&comma;t&comma;u&comma;vuxvtu&xi;tx&comma;t&comma;u&comma;vut2v&xi;tx&comma;t&comma;u&comma;vutvtt&xi;xx&comma;t&comma;u&comma;vuxt&xi;tx&comma;t&comma;u&comma;vut&plus;utu&eta;ux&comma;t&comma;u&comma;v&plus;vtv&eta;ux&comma;t&comma;u&comma;v&plus;t&eta;ux&comma;t&comma;u&comma;vutf&plus;u&xi;xx&comma;t&comma;u&comma;vutvxv&xi;xx&comma;t&comma;u&comma;vvxvtu&xi;tx&comma;t&comma;u&comma;vutvtv&xi;tx&comma;t&comma;u&comma;vvt2t&xi;xx&comma;t&comma;u&comma;vvxt&xi;tx&comma;t&comma;u&comma;vvt&plus;utu&eta;vx&comma;t&comma;u&comma;v&plus;vtv&eta;vx&comma;t&comma;u&comma;v&plus;t&eta;vx&comma;t&comma;u&comma;vvtf

(5)

Both G1 and G2 produce the same result when applied to any function - say Λ - of x,t,u,v and their derivatives written in jet notation:

L&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&colon;

To avoid redundant display cluttering the presentation use the declare schema for compact mathematical display (derivatives are also displayed indexed)

declare&xi;&comma;&eta;x&comma;t&comma;u&comma;v&comma;L

ξx&comma;t&comma;u&comma;vwill now be displayed asξ

ηx&comma;t&comma;u&comma;vwill now be displayed asη

Λx&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux,x&comma;ux,t&comma;ut,t&comma;vx,x&comma;vx,t&comma;vt,twill now be displayed asΛ

(6)

This results from the application of the non-expanded G1 to Λ

G1L

&xi;xx&comma;t&comma;u&comma;vdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;x&plus;&xi;tx&comma;t&comma;u&comma;vdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;t&plus;&eta;ux&comma;t&comma;u&comma;vdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;u&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uux2diff&xi;xx&comma;t&comma;u&comma;v&comma;vuxvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;uuxutdiff&xi;tx&comma;t&comma;u&comma;v&comma;vutvx&plus;uxdiff&eta;ux&comma;t&comma;u&comma;v&comma;u&plus;vxdiff&eta;ux&comma;t&comma;u&comma;v&comma;vdiff&xi;xx&comma;t&comma;u&comma;v&comma;xuxdiff&xi;tx&comma;t&comma;u&comma;v&comma;xut&plus;diff&eta;ux&comma;t&comma;u&comma;v&comma;xdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;ux&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uuxutdiff&xi;xx&comma;t&comma;u&comma;v&comma;vuxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;uut2diff&xi;tx&comma;t&comma;u&comma;v&comma;vutvtdiff&xi;xx&comma;t&comma;u&comma;v&comma;tuxdiff&xi;tx&comma;t&comma;u&comma;v&comma;tut&plus;utdiff&eta;ux&comma;t&comma;u&comma;v&comma;u&plus;vtdiff&eta;ux&comma;t&comma;u&comma;v&comma;v&plus;diff&eta;ux&comma;t&comma;u&comma;v&comma;tdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;ut&plus;&eta;vx&comma;t&comma;u&comma;vdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;v&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uuxvxdiff&xi;xx&comma;t&comma;u&comma;v&comma;vvx2diff&xi;tx&comma;t&comma;u&comma;v&comma;uuxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;vvxvt&plus;uxdiff&eta;vx&comma;t&comma;u&comma;v&comma;u&plus;vxdiff&eta;vx&comma;t&comma;u&comma;v&comma;vdiff&xi;xx&comma;t&comma;u&comma;v&comma;xvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;xvt&plus;diff&eta;vx&comma;t&comma;u&comma;v&comma;xdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;vx&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uutvxdiff&xi;xx&comma;t&comma;u&comma;v&comma;vvxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;uutvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;vvt2diff&xi;xx&comma;t&comma;u&comma;v&comma;tvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;tvt&plus;utdiff&eta;vx&comma;t&comma;u&comma;v&comma;u&plus;vtdiff&eta;vx&comma;t&comma;u&comma;v&comma;v&plus;diff&eta;vx&comma;t&comma;u&comma;v&comma;tdiff&Lambda;x&comma;t&comma;u&comma;v&comma;ux&comma;ut&comma;vx&comma;vt&comma;ux&comma;x&comma;ux&comma;t&comma;ut&comma;t&comma;vx&comma;x&comma;vx&comma;t&comma;vt&comma;t&comma;vt

(7)

Let's verify that the non-expanded G1 and the expanded G2 produce the same output

G1LG2L

0

(8)

Apart from the default output, an operator, you can optionally request this output to be an expression (the operator applied) or a list with the components of the infinitesimal generator

InfinitesimalGeneratorS&comma;DepVars&comma;prolongation&equals;1&comma;expanded&comma;output&equals;expression

&xi;xx&comma;t&comma;u&comma;vDiff_&Phi;&comma;x&plus;&xi;tx&comma;t&comma;u&comma;vDiff_&Phi;&comma;t&plus;&eta;ux&comma;t&comma;u&comma;vDiff_&Phi;&comma;u&plus;&eta;vx&comma;t&comma;u&comma;vDiff_&Phi;&comma;v&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uux2diff&xi;xx&comma;t&comma;u&comma;v&comma;vuxvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;uuxutdiff&xi;tx&comma;t&comma;u&comma;v&comma;vutvx&plus;uxdiff&eta;ux&comma;t&comma;u&comma;v&comma;u&plus;vxdiff&eta;ux&comma;t&comma;u&comma;v&comma;vdiff&xi;xx&comma;t&comma;u&comma;v&comma;xuxdiff&xi;tx&comma;t&comma;u&comma;v&comma;xut&plus;diff&eta;ux&comma;t&comma;u&comma;v&comma;xDiff_&Phi;&comma;ux&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uuxvxdiff&xi;xx&comma;t&comma;u&comma;v&comma;vvx2diff&xi;tx&comma;t&comma;u&comma;v&comma;uuxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;vvxvt&plus;uxdiff&eta;vx&comma;t&comma;u&comma;v&comma;u&plus;vxdiff&eta;vx&comma;t&comma;u&comma;v&comma;vdiff&xi;xx&comma;t&comma;u&comma;v&comma;xvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;xvt&plus;diff&eta;vx&comma;t&comma;u&comma;v&comma;xDiff_&Phi;&comma;vx&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uuxutdiff&xi;xx&comma;t&comma;u&comma;v&comma;vuxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;uut2diff&xi;tx&comma;t&comma;u&comma;v&comma;vutvtdiff&xi;xx&comma;t&comma;u&comma;v&comma;tuxdiff&xi;tx&comma;t&comma;u&comma;v&comma;tut&plus;utdiff&eta;ux&comma;t&comma;u&comma;v&comma;u&plus;vtdiff&eta;ux&comma;t&comma;u&comma;v&comma;v&plus;diff&eta;ux&comma;t&comma;u&comma;v&comma;tDiff_&Phi;&comma;ut&plus;diff&xi;xx&comma;t&comma;u&comma;v&comma;uutvxdiff&xi;xx&comma;t&comma;u&comma;v&comma;vvxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;uutvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;vvt2diff&xi;xx&comma;t&comma;u&comma;v&comma;tvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;tvt&plus;utdiff&eta;vx&comma;t&comma;u&comma;v&comma;u&plus;vtdiff&eta;vx&comma;t&comma;u&comma;v&comma;v&plus;diff&eta;vx&comma;t&comma;u&comma;v&comma;tDiff_&Phi;&comma;vt

(9)

In order to use the result above you need to replace the label _&Phi; by any function of the jet variables, in this example x,t,u and v, and do that before activating the inert derivatives using value. For example

valuesubs_&Phi;&equals;fx&comma;t&comma;u&comma;v&comma;

&xi;xx&comma;t&comma;u&comma;vdifffx&comma;t&comma;u&comma;v&comma;x&plus;&xi;tx&comma;t&comma;u&comma;vdifffx&comma;t&comma;u&comma;v&comma;t&plus;&eta;ux&comma;t&comma;u&comma;vdifffx&comma;t&comma;u&comma;v&comma;u&plus;&eta;vx&comma;t&comma;u&comma;vdifffx&comma;t&comma;u&comma;v&comma;v

(10)

The other possible representation is a list

InfinitesimalGeneratorS&comma;DepVars&comma;prolongation&equals;1&comma;expanded&comma;output&equals;list

&xi;xx&comma;t&comma;u&comma;v&comma;&xi;tx&comma;t&comma;u&comma;v&comma;&eta;ux&comma;t&comma;u&comma;v&comma;&eta;vx&comma;t&comma;u&comma;v&comma;diff&xi;xx&comma;t&comma;u&comma;v&comma;uux2diff&xi;xx&comma;t&comma;u&comma;v&comma;vuxvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;uuxutdiff&xi;tx&comma;t&comma;u&comma;v&comma;vutvx&plus;uxdiff&eta;ux&comma;t&comma;u&comma;v&comma;u&plus;vxdiff&eta;ux&comma;t&comma;u&comma;v&comma;vdiff&xi;xx&comma;t&comma;u&comma;v&comma;xuxdiff&xi;tx&comma;t&comma;u&comma;v&comma;xut&plus;diff&eta;ux&comma;t&comma;u&comma;v&comma;x&comma;diff&xi;xx&comma;t&comma;u&comma;v&comma;uuxvxdiff&xi;xx&comma;t&comma;u&comma;v&comma;vvx2diff&xi;tx&comma;t&comma;u&comma;v&comma;uuxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;vvxvt&plus;uxdiff&eta;vx&comma;t&comma;u&comma;v&comma;u&plus;vxdiff&eta;vx&comma;t&comma;u&comma;v&comma;vdiff&xi;xx&comma;t&comma;u&comma;v&comma;xvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;xvt&plus;diff&eta;vx&comma;t&comma;u&comma;v&comma;x&comma;diff&xi;xx&comma;t&comma;u&comma;v&comma;uuxutdiff&xi;xx&comma;t&comma;u&comma;v&comma;vuxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;uut2diff&xi;tx&comma;t&comma;u&comma;v&comma;vutvtdiff&xi;xx&comma;t&comma;u&comma;v&comma;tuxdiff&xi;tx&comma;t&comma;u&comma;v&comma;tut&plus;utdiff&eta;ux&comma;t&comma;u&comma;v&comma;u&plus;vtdiff&eta;ux&comma;t&comma;u&comma;v&comma;v&plus;diff&eta;ux&comma;t&comma;u&comma;v&comma;t&comma;diff&xi;xx&comma;t&comma;u&comma;v&comma;uutvxdiff&xi;xx&comma;t&comma;u&comma;v&comma;vvxvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;uutvtdiff&xi;tx&comma;t&comma;u&comma;v&comma;vvt2diff&xi;xx&comma;t&comma;u&comma;v&comma;tvxdiff&xi;tx&comma;t&comma;u&comma;v&comma;tvt&plus;utdiff&eta;vx&comma;t&comma;u&comma;v&comma;u&plus;vtdiff&eta;vx&comma;t&comma;u&comma;v&comma;v&plus;diff&eta;vx&comma;t&comma;u&comma;v&comma;t

(11)

Note the compact display in the output above, due to the use of declare in previous examples. To see the contents behind this compact display use show

show

ξxx&comma;t&comma;u&comma;v&comma;ξtx&comma;t&comma;u&comma;v&comma;ηux&comma;t&comma;u&comma;v&comma;ηvx&comma;t&comma;u&comma;v&comma;uξxx&comma;t&comma;u&comma;vux2vξxx&comma;t&comma;u&comma;vuxvxuξtx&comma;t&comma;u&comma;vuxutvξtx&comma;t&comma;u&comma;vutvx+uxuηux&comma;t&comma;u&comma;v+vxvηux&comma;t&comma;u&comma;vxξxx&comma;t&comma;u&comma;vuxxξtx&comma;t&comma;u&comma;vut+xηux&comma;t&comma;u&comma;v&comma;uξxx&comma;t&comma;u&comma;vuxvxvξxx&comma;t&comma;u&comma;vvx2uξtx&comma;t&comma;u&comma;vuxvtvξtx&comma;t&comma;u&comma;vvxvt+uxuηvx&comma;t&comma;u&comma;v+vxvηvx&comma;t&comma;u&comma;vxξxx&comma;t&comma;u&comma;vvxxξtx&comma;t&comma;u&comma;vvt+xηvx&comma;t&comma;u&comma;v&comma;uξxx&comma;t&comma;u&comma;vuxutvξxx&comma;t&comma;u&comma;vuxvtuξtx&comma;t&comma;u&comma;vut2vξtx&comma;t&comma;u&comma;vutvttξxx&comma;t&comma;u&comma;vuxtξtx&comma;t&comma;u&comma;vut+utuηux&comma;t&comma;u&comma;v+vtvηux&comma;t&comma;u&comma;v+tηux&comma;t&comma;u&comma;v&comma;uξxx&comma;t&comma;u&comma;vutvxvξxx&comma;t&comma;u&comma;vvxvtuξtx&comma;t&comma;u&comma;vutvtvξtx&comma;t&comma;u&comma;vvt2tξxx&comma;t&comma;u&comma;vvxtξtx&comma;t&comma;u&comma;vvt+utuηvx&comma;t&comma;u&comma;v+vtvηvx&comma;t&comma;u&comma;v+tηvx&comma;t&comma;u&comma;v

(12)

You can also use InfinitesimalGenerator to prolong a given infinitesimal generator or to rewrite the operator in different jet notation. This is G2 rewritten using jetnumbers notation (compare with (5))

InfinitesimalGeneratorG2&comma;DepVars&comma;expanded&comma;notation&equals;jetnumbers

f&rarr;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;xf&plus;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;tf&plus;&eta;ux&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u&lsqb;&rsqb;f&plus;&eta;vx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v&lsqb;&rsqb;f&plus;u&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u12v&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1v1u&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1u2v&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u2v1&plus;u1u&lsqb;&rsqb;&eta;ux&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;&plus;v1v&lsqb;&rsqb;&eta;ux&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;x&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1x&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u2&plus;x&eta;ux&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1f&plus;u&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1v1v&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v12u&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1v2v&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v1v2&plus;u1u&lsqb;&rsqb;&eta;vx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;&plus;v1v&lsqb;&rsqb;&eta;vx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;x&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v1x&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v2&plus;x&eta;vx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v1f&plus;u&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1u2v&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1v2u&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u22v&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u2v2t&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u1t&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u2&plus;u2u&lsqb;&rsqb;&eta;ux&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;&plus;v2v&lsqb;&rsqb;&eta;ux&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;&plus;t&eta;ux&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u2f&plus;u&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u2v1v&lsqb;&rsqb;&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v1v2u&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;u2v2v&lsqb;&rsqb;&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v22t&xi;xx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v1t&xi;tx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v2&plus;u2u&lsqb;&rsqb;&eta;vx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;&plus;v2v&lsqb;&rsqb;&eta;vx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;&plus;t&eta;vx&comma;t&comma;u&lsqb;&rsqb;&comma;v&lsqb;&rsqb;v2f

(13)

By default, when reprocessing an infinitesimal generator as in the input/output above, the prolongation of the returned  generator is the same as that of the given one, in this example G2, unless the given generator is not expanded (for example, G1), in which case the returned generator has prolongation = 0. To overcome this limitation in the reprocessing of not expanded infinitesimal generators you can indicate the desired prolongation using the option prolongation = ...

The InfinitesimalGenerator command also works with anticommutative variables, natively, without using the approach explained in PerformOnAnticommutativeSystem.

withPhysics

`*`&comma;`.`&comma;Annihilation&comma;AntiCommutator&comma;Antisymmetrize&comma;Assume&comma;Bra&comma;Bracket&comma;Check&comma;Christoffel&comma;Coefficients&comma;Commutator&comma;CompactDisplay&comma;Coordinates&comma;Creation&comma;D_&comma;Dagger&comma;Decompose&comma;Define&comma;D&gamma;&comma;Einstein&comma;EnergyMomentum&comma;Expand&comma;ExteriorDerivative&comma;Factor&comma;FeynmanDiagrams&comma;FeynmanIntegral&comma;Fundiff&comma;Geodesics&comma;GrassmannParity&comma;Gtaylor&comma;Intc&comma;Inverse&comma;Ket&comma;KillingVectors&comma;KroneckerDelta&comma;LagrangeEquations&comma;LeviCivita&comma;Library&comma;LieBracket&comma;LieDerivative&comma;Normal&comma;NumericalRelativity&comma;Parameters&comma;PerformOnAnticommutativeSystem&comma;Projector&comma;Psigma&comma;Redefine&comma;Ricci&comma;Riemann&comma;Setup&comma;Simplify&comma;SortProducts&comma;SpaceTimeVector&comma;StandardModel&comma;Substitute&comma;SubstituteTensor&comma;SubstituteTensorIndices&comma;SumOverRepeatedIndices&comma;Symmetrize&comma;TensorArray&comma;Tetrads&comma;ThreePlusOne&comma;ToContravariant&comma;ToCovariant&comma;ToFieldComponents&comma;ToSuperfields&comma;Trace&comma;TransformCoordinates&comma;Vectors&comma;Weyl&comma;`^`&comma;dAlembertian&comma;d_&comma;diff&comma;g_&comma;gamma_

(14)

Set first θ and Q as suffixes for variables of type/anticommutative (see Setup)

Setupanticommutativepre&equals;&theta;&comma;Q

* Partial match of 'anticommutativepre' against keyword 'anticommutativeprefix'

_______________________________________________________

anticommutativeprefix=Q&comma;θ

(15)

Set now the generic form of the infinitesimals for a PDE system like this one formed by pde[1] and pde[2]. For this purpose, we need anticommutative infinitesimals for the dependent variable Q and two of the independent variables, θ1 and θ2; we use here the capital Greek letters Ξ and Η for the anticommutative infinitesimal symmetry generators and the corresponding lowercase Greek letters for commutative ones

Setupanticommutativepre&equals;&Xi;&comma;&Rho;&comma;additionally

* Partial match of 'anticommutativepre' against keyword 'anticommutativeprefix'

_______________________________________________________

anticommutativeprefix=Q&comma;Ρ&comma;Ξ&comma;θ

(16)

H&xi;1&comma;&xi;2&comma;&Xi;1&comma;&Xi;2&comma;&Eta;x&comma;y&comma;&theta;1&comma;&theta;2

Hξ1x&comma;y&comma;θ1&comma;θ2&comma;ξ2x&comma;y&comma;θ1&comma;θ2&comma;Ξ1x&comma;y&comma;θ1&comma;θ2&comma;Ξ2x&comma;y&comma;θ1&comma;θ2&comma;Ηx&comma;y&comma;θ1&comma;θ2

(17)

PDEtools:-declareH

Ηx&comma;y&comma;θ1&comma;θ2will now be displayed asΗ

Ξx&comma;y&comma;θ1&comma;θ2will now be displayed asΞ

ξx&comma;y&comma;θ1&comma;θ2will now be displayed asξ

(18)

The corresponding InfinitesimalGenerator

InfinitesimalGeneratorH&comma;Qx&comma;y&comma;&theta;1&comma;&theta;2

fξ1&DifferentialD;f&DifferentialD;x+ξ2&DifferentialD;f&DifferentialD;y+Ξ1&DifferentialD;f&DifferentialD;θ1+Ξ2&DifferentialD;f&DifferentialD;θ2+Η&DifferentialD;f&DifferentialD;Q

(19)

The same infinitesimal but prolonged to 1st and 2nd order

InfinitesimalGeneratorH&comma;Qx&comma;y&comma;&theta;1&comma;&theta;2&comma;prolongation&equals;1&comma;expanded

fξ1&DifferentialD;f&DifferentialD;x+ξ2&DifferentialD;f&DifferentialD;y+Ξ1&DifferentialD;f&DifferentialD;θ1+Ξ2&DifferentialD;f&DifferentialD;θ2+Η&DifferentialD;f&DifferentialD;Q+Ηxx&xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qxx&xi;2x&comma;y&comma;&theta;1&comma;&theta;2QyQθ1x&Xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qθ2x&Xi;2x&comma;y&comma;&theta;1&comma;&theta;2&DifferentialD;f&DifferentialD;Qx+Ηyy&xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qxy&xi;2x&comma;y&comma;&theta;1&comma;&theta;2QyQθ1y&Xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qθ2y&Xi;2x&comma;y&comma;&theta;1&comma;&theta;2&DifferentialD;f&DifferentialD;Qy+Ηθ1+θ1&xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qx+θ1&xi;2x&comma;y&comma;&theta;1&comma;&theta;2QyQθ1θ1&Xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qθ2θ1&Xi;2x&comma;y&comma;&theta;1&comma;&theta;2&DifferentialD;f&DifferentialD;Qθ1+Ηθ2+θ2&xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qx+θ2&xi;2x&comma;y&comma;&theta;1&comma;&theta;2QyQθ1θ2&Xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qθ2θ2&Xi;2x&comma;y&comma;&theta;1&comma;&theta;2&DifferentialD;f&DifferentialD;Qθ2

(20)

InfinitesimalGeneratorH&comma;Qx&comma;y&comma;&theta;1&comma;&theta;2&comma;prolongation&equals;2&comma;expanded

fξ1&DifferentialD;f&DifferentialD;x+ξ2&DifferentialD;f&DifferentialD;y+Ξ1&DifferentialD;f&DifferentialD;θ1+Ξ2&DifferentialD;f&DifferentialD;θ2+Η&DifferentialD;f&DifferentialD;Q+Ηxx&xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qxx&xi;2x&comma;y&comma;&theta;1&comma;&theta;2QyQθ1x&Xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qθ2x&Xi;2x&comma;y&comma;&theta;1&comma;&theta;2&DifferentialD;f&DifferentialD;Qx+Ηyy&xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qxy&xi;2x&comma;y&comma;&theta;1&comma;&theta;2QyQθ1y&Xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qθ2y&Xi;2x&comma;y&comma;&theta;1&comma;&theta;2&DifferentialD;f&DifferentialD;Qy+Ηθ1+θ1&xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qx+θ1&xi;2x&comma;y&comma;&theta;1&comma;&theta;2QyQθ1θ1&Xi;1x&comma;y&comma;&theta;1&comma;&theta;2Qθ2θ1&Xi;2x&comma;y&comma;&theta;1&comma;&theta;2&DifferentialD;f&DifferentialD;Qθ1+Ηθ2+θ2&xi;1x