assign a type (one of projectable, point, contact, differential substitution, generalized differential substitution, generic) to a transformation - Maple Programming Help

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JetCalculus[AssignTransformationType] - assign a type (one of projectable, point, contact, differential substitution, generalized differential substitution, generic) to a transformation

Calling Sequences

     AssignTransformationType(φ)

Parameters

     φ       - a transformation

 

Description

Examples

Description

• 

Let E M and FN be two fiber bundles, and let πk:JkE M , πk:JkFM be the associated bundles of kjets.

[i] A map φ :EF which sends the fibers of E to fibers of F (and hence covers a map φ0:M N) is called a projectable transformation.

[ii] A map φ:EF is called a point transformation.

[iii] A transformation φ :J1E  J1F is called a contact transformation if the fiber dimensions of E and F are 1 and φ pulls back the contact form on J1F to a multiple of the contact form on J1E.

[iv] If φ:JkE  F and φ covers the identity map MN ,then φ is called a differential substitution.

[v] A map φ:JkEF is called a generalized differential substitution.

[vi] A transformation not of one the types [i]--[v] is called generic.

Explicit coordinate formulas for these various types of maps are given in Example 1.

• 

 The command AssignTransformationType(φ ) returns the transformation φ, but with internal representation φ of  changed to encode its transformation type. The type of a transformation and its prolongation order can be determined by the command DGinfo with the keyword "TransformationType".

• 

Any transformation of type [i]--[v] can be prolonged to higher order jet spaces. See Prolong for further information.

• 

The command AssignTransformationType is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form AssignTransformationType(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-AssignTransformationType(...).

Examples

with(DifferentialGeometry): with(JetCalculus):

 

Example 1.

First initialize various jet spaces of one or two independent variables and one dependent variable and prolong them to order 4.

DGsetup([x, y], [u], E, 4): DGsetup([z], [v], F, 4): DGsetup([p, q], [w], K, 4):

 

Case 1. Projectable transformations from E to F.

K > 

Phi1 := Transformation(E, F, [z = A(x, y), v[] = B(x, y, u[])]);

Φ1z=Ax,y,v=Bx,y,u

(2.1)

 

When a transformation is first defined, it is not given a type.

E > 

Tools:-DGinfo(Phi1, "TransformationType");

(2.2)

 

Now assign the transformation Φ1 a type.

E > 

newPhi1 := AssignTransformationType(Phi1);

newPhi1z=Ax,y,v=Bx,y,u

(2.3)
E > 

Tools:-DGinfo(newPhi1, "TransformationType");

projectable,0

(2.4)

 

This indicates that the transformation is a projectable transformation, the 0 indicates that the transformation has not been prolonged to a jet space.

 

Case 2. Point transformations:

E > 

Phi2 := Transformation(E, F, [z = A(x, y, u[]), v[] = B(x, y, u[])]);

Φ2z=Ax,y,u,v=Bx,y,u

(2.5)
E > 

newPhi2 := AssignTransformationType(Phi2);

newPhi2z=Ax,y,u,v=Bx,y,u

(2.6)
E > 

Tools:-DGinfo(newPhi2, "TransformationType");

point,0

(2.7)

 

Case 3. Contact transformations:

E > 

Phi3 := Transformation(E, K, [p = - u[1], q = y, w[] = - u[1]*x + u[], w[1] = x, w[2] = u[2]]);

Φ3p=u1,q=y,w=u1x+u,w1=x,w2=u2

(2.8)
E > 

newPhi3 := AssignTransformationType(Phi3);

newPhi3p=u1,q=y,w=u1x+u,w1=x,w2=u2

(2.9)
E > 

Tools:-DGinfo(newPhi3, "TransformationType");

contact,1

(2.10)

 

By the conventions adopted here, a contact transformation need not be a local diffeomorphism so that, in particular, the dimensions of the bundles E and F  need not coincide.

E > 

Phi4 := Transformation(F, E, [x = z, y = 1, u[] = v[], u[1] = v[1], u[2]=0]);

Φ4x=z,y=1,u=v,u1=v1,u2=0

(2.11)
F > 

newPhi4 := AssignTransformationType(Phi4);

newPhi4x=z,y=1,u=v,u1=v1,u2=0

(2.12)
F > 

Tools:-DGinfo(newPhi3, "TransformationType");

contact,1

(2.13)

 

Case 4. Differential Substitutions:

F > 

vars := x, y, u[], u[1], u[2], u[1,1], u[1, 2], u[2, 2];

varsx,y,u,u1,u2,u1,1,u1,2,u2,2

(2.14)
E > 

Phi5 := Transformation(E, K, [p = x, q = y, w[] = A(vars)]);

Φ5p=x,q=y,w=Ax,y,u,u1,u2,u1,1,u1,2,u2,2

(2.15)
E > 

newPhi5 := AssignTransformationType(Phi5):

E > 

Tools:-DGinfo(newPhi5, "TransformationType");

differentialSubstitution,0

(2.16)

 

Case 5. Generalized Differential Substitutions:

E > 

Phi5 := Transformation(E, F, [z = A(vars), v[] = B(vars)]);

Φ5z=Ax,y,u,u1,u2,u1,1,u1,2,u2,2,v=Bx,y,u,u1,u2,u1,1,u1,2,u2,2

(2.17)
E > 

newPhi5 := AssignTransformationType(Phi5):

E > 

Tools:-DGinfo(newPhi5, "TransformationType");

generalizedDifferentialSubstitution,0

(2.18)

 

Case 6. Generic:

E > 

Phi6 := Transformation(E, F, [z=u[1]*y, v[]=u[2]+x*u[], v[1]=y]);

Φ6z=u1y,v=xu+u2,v1=y

(2.19)
E > 

newPhi6 := AssignTransformationType(Phi6)

F > 

Tools:-DGinfo(newPhi6, "TransformationType");

generic,NA

(2.20)

See Also

DifferentialGeometry

JetCalculus

AssignVectorType

DGinfo

Prolong

Transformation