a package for the Fels-Olver method of moving frames - Maple Help

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GroupActions[MovingFrames] - a package for the Fels-Olver method of moving frames

Calling Sequences

     RightMovingFrame(mu, G, K)

     Invariantization(mu, rho, f)

Parameters

     mu        - a free (left) action of a Lie group G on a manifold M, given as a transformation from M  to M

     G         - a Maple name or string, the name of the initialized coordinate system for the Lie group G

     K         - a list of equations defining a cross-section for the action mu

     rho       - a right moving frame for the action mu

     f         - a Maple expression, defining a function on M

Description

• 

Let G be a Lie group with multiplication * and μ: GM a free (left) action of G on a manifold M. A right moving frame is a map rho:G M such that ρμa, x = rhox*a1 for all a  G and x M.

• 

A cross-section to the action μ: GM   is a submanifold K of M, with codim(K) = dimG, which is transverse to the orbits of μ. The cross-section K has the property that if k1, k2 K and μa, k1 = μa, k2 then k1 = k2.

• 

The Invariantization command will map any function on M to a Ginvariant function.

• 

The commands RightMovingFrame and Invariantization are part of the DifferentialGeometry:-GroupActions:-MovingFrames package. They can be used in the forms RightMovingFrame(...) and Invariantization(...) only after executing the commands with(DifferentialGeometry), with(GroupActions), and with(MovingFrames), but can always be used by executing DifferentialGeometry:-GroupActions:-MovingFrames:-RightMovingFrame(...) and DifferentialGeometry:-GroupActions:-MovingFrames:-Invariantization(...).

• 

References:

[1] M. Fels and P. Olver, Moving Coframes I. A practical algorithm Acta Appl. Math. 51 (1998)

[2] M. Fels and P. Olver, Moving Coframes II. Regularization and theoretical foundations, Acta Appl. Math. 55 (1999) 127-208

Examples

withDifferentialGeometry:withLieAlgebras:

withJetCalculus:withGroupActions:

withMovingFrames:

PreferencesJetNotation,JetNotation2:

 

Example 1.

In this example, we shall use the method of moving frames to construct the fundamental differential invariant for the special affine group (translations, rotations, scaling) in the xy plane.

DGsetupx,y,E,3,verbose

The following coordinates have been protected:

x,y0,y1,y2,y3

The following vector fields have been defined and protected:

D_x,D_y0,D_y1,D_y2,D_y3

The following differential 1-forms have been defined and protected:

dx,dy0,dy1,dy2,dy3

The following type [1,0] biforms have been defined and protected::

Dx

The following type [0,1] biforms (contact 1-forms) have been defined and protected::

Cy0,Cy1,Cy2,Cy3

frame name: E

(2.1)

 

We start with the infinitesimal generators for the action of the special affine group.

E > 

Γ:=evalDGD_x,D_y0,xD_y0y0D_x,y0D_y0+xD_x

Γ:=D_x,D_y0,D_xy0+xD_y0,D_xx+D_y0y0

(2.2)

 

This is a solvable group so we can use the Action command in the GroupAction package to find the action of the special affine group.

E > 

DGsetupa,b,θ,t,G

frame name: G

(2.3)
G > 

μ:=ActionΓ,G

μ:=x=ay0ⅇtsinθ+xⅇtcosθ,y0=b+y0ⅇtcosθ+xⅇtsinθ

(2.4)

 

We use the program Prolong in the JetCalculus package to prolong this action to the 3-jets of E.

E > 

μ3:=simplifyProlongμ,3

μ3:=x=ay0ⅇtsinθ+xⅇtcosθ,y0=b+y0ⅇtcosθ+xⅇtsinθ,y1=cosθy1+sinθsinθy1+cosθ,y2=y2ⅇtcosθ2sinθy13+3cosθ3y12+3cosθ2sinθy1+sinθy13cosθ33cosθy12,y3=sinθy1y3+3y22sinθ+cosθy3ⅇ2tcosθ4sinθy15+5cosθ5y14+10cosθ4sinθy13+2cosθ2sinθy1510cosθ5y1210cosθ3y145cosθ4sinθy110cosθ2sinθy13sinθy15+cosθ5+10cosθ3y12+5cosθy14

(2.5)
E > 

_EnvExplicit:=true

_EnvExplicit:=true

(2.6)

 

We calculate a moving frame for this prolonged action.

E > 

ρ:=RightMovingFrameμ3,G,x=0,y0=0,y1=0,y2=1

Warning, multiple moving frames

ρ:=a=y2y0y1+xy12+12,b=y2xy1y0y12+12,θ=arctany1y12+1,1y12+1,t=lny2y12+13/2,a=y2y0y1+xy12+12,b=y2xy1y0y12+12,θ=arctany1y12+1,1y12+1,t=lny2y12+13/2

(2.7)

 

We use this moving frame to find the fundamental differential invariant on the 3-jet.

E > 

κ:=expandsimplifyInvariantizationμ3,ρ1,y3

κ:=y12y3y223y1+y3y22

(2.8)

See Also

DifferentialGeometry, GroupActions, JetCalculus, LieAlgebras, Action, LieGroup, Prolong


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