DEtools - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Differential Equations : Lie Symmetry Method : Commands for ODEs : DEtools/Xcommutator

DEtools

  

Xcommutator

  

calculate the commutator of two generators of one-parameter Lie groups

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

Xcommutator(X1, X2, k, y(x), ODE)

Parameters

X1, X2

-

lists of the coefficients of the symmetry generators (infinitesimals) as in [xi, eta]

y(x)

-

'dependent variable'; it can be any indeterminate function of one variable

k

-

(optional) extension of the generators entering the commutator

ODE

-

(optional) right hand side is used to replace the highest derivative in the result; required if dynamical symmetries are given

Description

• 

The Xcommutator command receives two generators of one-parameter Lie groups, either in the form of a pair of infinitesimals [xi, eta] or in the form of differential operators, and the dependent variable y(x), and returns the commutator of these generators.

• 

If k is given, the k extension of the generators X1 and X2 is calculated at first, and the command returns the commutator of these extended generators (that is, another extended generator; see eta_k and infgen ).

• 

If the given generators are in the form of a list containing the infinitesimals, the result is returned as a list; otherwise, if X1 and X2 are given as differential operators (mappings) then the result is returned as a mapping (see examples).

• 

This function is part of the DEtools package, and so it can be used in the form Xcommutator(..) only after executing the command with(DEtools). However, it can always be accessed through the long form of the command by using DEtools[Xcommutator](..).

Examples

withDEtools:

X1x,y

X1:=x,y

(1)

X2ax,cy+y2

X2:=ax,cy+y2

(2)

The commutator of X1 with X2

XcommutatorX1,X2,yx

_ξ=0,_η=y2

(3)

The commutator of the third extensions of X1 and X2 involves four elements.

XcommutatorX1,X2,yx,3

_ξ=0,_η=y2,_η1=2_y1y,_η2=2_y122_y2y,_η3=6_y1_y22_y3y

(4)

The generators G1 and G2 associated to the lists X1 and X2

X1

x,y

(5)

G1infgenX1,yx

G1:=_F1→xx_F1yy_F1

(6)

X2

ax,cy+y2

(7)

G2infgenX2,yx

G2:=_F1→axx_F1+cy+y2y_F1

(8)

If G1 and G2 are differential operators (mappings), the commutator of G1 and G2 is returned as a mapping as well:

XcommutatorG1,G2,yx

_F2→y_F2y2

(9)

The commutator of the third extensions of G1 and G2

XcommutatorG1,G2,yx,3

_F2→2_y1_F2_y1y+2_y122_y2y_y2_F2+6_y1_y22_y3y_y3_F2y_F2y2

(10)

See Also

DEtools

dsolve,Lie

eta_k

infgen

PDEtools

Xchange

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam