JetCalculus[AssignTransformationType] - assign a type (one of projectable, point, contact, differential substitution, generalized differential substitution, generic) to a transformation
φ - a transformation
Let E→ M and F→N be two fiber bundles, and let πk:JkE→ M , πk:JkF→M be the associated bundles of k−jets.
[i] A map φ :E→F 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:E→F 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 M→N ,then φ is called a differential substitution.
[v] A map φ:JkE→F 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(...).
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.
Phi1 := Transformation(E, F, [z = A(x, y), v = B(x, y, u)]);
When a transformation is first defined, it is not given a type.
Now assign the transformation Φ1 a type.
newPhi1 := AssignTransformationType(Phi1);
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:
Phi2 := Transformation(E, F, [z = A(x, y, u), v = B(x, y, u)]);
newPhi2 := AssignTransformationType(Phi2);
Case 3. Contact transformations:
Phi3 := Transformation(E, K, [p = - u, q = y, w = - u*x + u, w = x, w = u]);
newPhi3 := AssignTransformationType(Phi3);
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.
Phi4 := Transformation(F, E, [x = z, y = 1, u = v, u = v, u=0]);
newPhi4 := AssignTransformationType(Phi4);
Case 4. Differential Substitutions:
vars := x, y, u, u, u, u[1,1], u[1, 2], u[2, 2];
Phi5 := Transformation(E, K, [p = x, q = y, w = A(vars)]);
newPhi5 := AssignTransformationType(Phi5):
Case 5. Generalized Differential Substitutions:
Phi5 := Transformation(E, F, [z = A(vars), v = B(vars)]);
Case 6. Generic:
Phi6 := Transformation(E, F, [z=u*y, v=u+x*u, v=y]);
newPhi6 := AssignTransformationType(Phi6)
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