|
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 mu: G x M -> M a free (left) action of G on a manifold M. A right moving frame is a map rho: M -> G such that rho(mu(g, x)) = rho(x)*g^(- 1) for all g in G and x in M.
|
|
•
|
A cross-section to the action mu: G x M -> M is a submanifold K of M, with codim(K) = dim(G), which is transverse to the orbits of mu. Thus, if k1, k2 in K and mu(g, k1) = mu(g, k2), then k1 = k2.
|
|
•
|
The Invariantization command will map any function on M to a G invariant function.
|
|
•
|
Further commands for working with moving frames will be added to subsequent releases of the DifferentialGeometry package.
|
|
•
|
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(...).
|
[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
|
|
|
>
|
 
|
|
>
|
 
|
|
>
|
|
|
>
|
|
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.
|
>
|
![DGsetup([x], [y], E, 3, verbose)](/support/helpjp/helpview.aspx?si=6640/file05750/math170.png)
|
We start with the infinitesimal generators for the action of the special affine group.
|
E >
|
![GAMMA := evalDG([D_x, D_y[0], x*D_y[0]-y[0]*D_x, y[0]*D_y[0]+x*D_x])](/support/helpjp/helpview.aspx?si=6640/file05750/math201.png)
|
|
![Gamma := [D_x, D_y[0], -y[0]*D_x+x*D_y[0], x*D_x+y[0]*D_y[0]]](/support/helpjp/helpview.aspx?si=6640/file05750/math204.png)
| (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 >
|
![DGsetup([a, b, theta, t], G)](/support/helpjp/helpview.aspx?si=6640/file05750/math218.png)
|
|
| (2.3) |
|
G >
|
|
|
![mu := [x = a-y[0]*exp(t)*sin(theta)+x*exp(t)*cos(theta), y[0] = b+y[0]*exp(t)*cos(theta)+x*exp(t)*sin(theta)]](/support/helpjp/helpview.aspx?si=6640/file05750/math228.png)
| (2.4) |
We use the program Prolong in the JetCalculus package to prolong this action to the 3-jets of E.
|
E >
|
|
|
| (2.5) |
|
E >
|
|
|
| (2.6) |
We calculate a moving frame for this prolonged action.
|
E >
|
![rho := RightMovingFrame(mu3, G, [x = 0, y[0] = 0, y[1] = 0, y[2] = 1])](/support/helpjp/helpview.aspx?si=6640/file05750/math262.png)
|
|
Warning, multiple moving frames
![rho := [a = -y[2]*(y[0]*y[1]+x)/(1+y[1]^2)^2, b = y[2]*(-y[0]+x*y[1])/(1+y[1]^2)^2, theta = arctan(-y[1]/(1+y[1]^2)^(1/2), 1/(((1+y[1]^2)^(1/2)))), t = ln(y[2]/(1+y[1]^2)^(3/2))], [a = -y[2]*(y[0]*y[1]+x)/(1+y[1]^2)^2, b = y[2]*(-y[0]+x*y[1])/(1+y[1]^2)^2, theta = arctan(y[1]/(1+y[1]^2)^(1/2), -1/(((1+y[1]^2)^(1/2)))), t = ln(-y[2]/(1+y[1]^2)^(3/2))]](/support/helpjp/helpview.aspx?si=6640/file05750/math267.png)
| (2.7) |
We use this moving frame to find the fundamental differential invariant on the 3-jet.
|
E >
|
![kappa := expand(simplify(Invariantization(mu3, rho[1], y[3])))](/support/helpjp/helpview.aspx?si=6640/file05750/math275.png)
|
|
![kappa := -3*y[1]+y[3]/y[2]^2+y[3]*y[1]^2/y[2]^2](/support/helpjp/helpview.aspx?si=6640/file05750/math278.png)
| (2.8) |
|
|
