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DEtools

  

canoni

  

look for a pair of canonical coordinates for a given Lie group of symmetries

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

canoni([xi(x, y), eta(x, y)], y(x), s(r))

Parameters

[xi(x, y), eta(x, y)]

-

list of the coefficients of the infinitesimal symmetry generator (infinitesimals)

y(x)

-

original dependent variable

s(r)

-

canonical dependent variable

ode

-

first order ODE

Description

• 

The canoni command tries to find a set of transformations from the original coordinates to the canonical coordinates by knowing the coefficients of the symmetry generator (infinitesimals) of a one-parameter Lie group.

• 

If there is more than one derivative in the ODE, canoni requires an extra argument (anywhere in the calling sequence), say y(x), specifying the dependent variable.

• 

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

Examples

withDEtools,symgen,canoni

symgen,canoni

(1)

withPDEtools,dchange

dchange

(2)

An ODE with an arbitrary function F

ODEⅆⅆxyx=2ax2yx+2Fxyx24axa

ODE:=ⅆⅆxyx=2ax2yx+2Fxyx24axa

(3)

A pair of infinitesimals for it

infinitesimalssymgenODE

infinitesimals:=_ξ=yx2,_η=2a

(4)

The canonical coordinates

trcanoniinfinitesimals,yx,sr

tr:=r=xyx2+4ax,sr=12yxa

(5)

The inverse transformation

itrop1,solvetr,x,yx

itr:=x=4a4sr2a2r,yx=2sra

(6)

The change of variables, using the canonical coordinates, reducing eq to a quadrature (see dchange)

ODE2dchangeitr,ODE,r,sr,simplify:

ODE1opsolveODE2,ⅆⅆrsr

ODE1:=ⅆⅆrsr=18a2Fr

(7)

See Also

dchange

DEtools

dsolve

equinv

odeadvisor

odetest

PDEtools

symgen

symtest

 


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