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(${\mathbf{φ}}$)

Parameters

$\mathrm{φ}$       - a transformation

Description

 • Let and be two fiber bundles, and let , ${\mathrm{π}}^{k}:{J}^{k}\left(F\right)\to M$ be the associated bundles of $k-$jets.

[i] A map which sends the fibers of to fibers of (and hence covers a map  is called a projectable transformation.

[ii] A map is called a point transformation.

[iii] A transformation  is called a contact transformation if the fiber dimensions of E and $F$ are 1 and $\mathrm{φ}$ pulls back the contact form on ${J}^{1}\left(F\right)$ to a multiple of the contact form on ${J}^{1}\left(E\right)$.

[iv] If  and covers the identity map then is called a differential substitution.

[v] A map 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(${\mathbf{φ}}$ ) returns the transformation $\mathrm{φ}$, but with internal representation ${\mathrm{\phi }}$ 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[])]);
 ${\mathrm{Φ1}}{≔}\left[{z}{=}{A}{}\left({x}{,}{y}\right){,}{{v}}_{\left[\right]}{=}{B}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}\right)\right]$ (2.1)

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

 E > Tools:-DGinfo(Phi1, "TransformationType");
 $\left[\right]$ (2.2)

Now assign the transformation $\mathrm{Φ1}$ a type.

 E > newPhi1 := AssignTransformationType(Phi1);
 ${\mathrm{newPhi1}}{≔}\left[{z}{=}{A}{}\left({x}{,}{y}\right){,}{{v}}_{\left[\right]}{=}{B}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}\right)\right]$ (2.3)
 E > Tools:-DGinfo(newPhi1, "TransformationType");
 $\left[{"projectable"}{,}{0}\right]$ (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[])]);
 ${\mathrm{Φ2}}{≔}\left[{z}{=}{A}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}\right){,}{{v}}_{\left[\right]}{=}{B}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}\right)\right]$ (2.5)
 E > newPhi2 := AssignTransformationType(Phi2);
 ${\mathrm{newPhi2}}{≔}\left[{z}{=}{A}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}\right){,}{{v}}_{\left[\right]}{=}{B}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}\right)\right]$ (2.6)
 E > Tools:-DGinfo(newPhi2, "TransformationType");
 $\left[{"point"}{,}{0}\right]$ (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]]);
 ${\mathrm{Φ3}}{≔}\left[{p}{=}{-}{{u}}_{{1}}{,}{q}{=}{y}{,}{{w}}_{\left[\right]}{=}{-}{{u}}_{{1}}{}{x}{+}{{u}}_{\left[\right]}{,}{{w}}_{{1}}{=}{x}{,}{{w}}_{{2}}{=}{{u}}_{{2}}\right]$ (2.8)
 E > newPhi3 := AssignTransformationType(Phi3);
 ${\mathrm{newPhi3}}{≔}\left[{p}{=}{-}{{u}}_{{1}}{,}{q}{=}{y}{,}{{w}}_{\left[\right]}{=}{-}{{u}}_{{1}}{}{x}{+}{{u}}_{\left[\right]}{,}{{w}}_{{1}}{=}{x}{,}{{w}}_{{2}}{=}{{u}}_{{2}}\right]$ (2.9)
 E > Tools:-DGinfo(newPhi3, "TransformationType");
 $\left[{"contact"}{,}{1}\right]$ (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  need not coincide.

 E > Phi4 := Transformation(F, E, [x = z, y = 1, u[] = v[], u[1] = v[1], u[2]=0]);
 ${\mathrm{Φ4}}{≔}\left[{x}{=}{z}{,}{y}{=}{1}{,}{{u}}_{\left[\right]}{=}{{v}}_{\left[\right]}{,}{{u}}_{{1}}{=}{{v}}_{{1}}{,}{{u}}_{{2}}{=}{0}\right]$ (2.11)
 F > newPhi4 := AssignTransformationType(Phi4);
 ${\mathrm{newPhi4}}{≔}\left[{x}{=}{z}{,}{y}{=}{1}{,}{{u}}_{\left[\right]}{=}{{v}}_{\left[\right]}{,}{{u}}_{{1}}{=}{{v}}_{{1}}{,}{{u}}_{{2}}{=}{0}\right]$ (2.12)
 F > Tools:-DGinfo(newPhi3, "TransformationType");
 $\left[{"contact"}{,}{1}\right]$ (2.13)

Case 4. Differential Substitutions:

 F > vars := x, y, u[], u[1], u[2], u[1,1], u[1, 2], u[2, 2];
 ${\mathrm{vars}}{≔}{x}{,}{y}{,}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}$ (2.14)
 E > Phi5 := Transformation(E, K, [p = x, q = y, w[] = A(vars)]);
 ${\mathrm{Φ5}}{≔}\left[{p}{=}{x}{,}{q}{=}{y}{,}{{w}}_{\left[\right]}{=}{A}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}\right)\right]$ (2.15)
 E > newPhi5 := AssignTransformationType(Phi5):
 E > Tools:-DGinfo(newPhi5, "TransformationType");
 $\left[{"differentialSubstitution"}{,}{0}\right]$ (2.16)

Case 5. Generalized Differential Substitutions:

 E > Phi5 := Transformation(E, F, [z = A(vars), v[] = B(vars)]);
 ${\mathrm{Φ5}}{≔}\left[{z}{=}{A}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}\right){,}{{v}}_{\left[\right]}{=}{B}{}\left({x}{,}{y}{,}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}\right)\right]$ (2.17)
 E > newPhi5 := AssignTransformationType(Phi5):
 E > Tools:-DGinfo(newPhi5, "TransformationType");
 $\left[{"generalizedDifferentialSubstitution"}{,}{0}\right]$ (2.18)

Case 6. Generic:

 E > Phi6 := Transformation(E, F, [z=u[1]*y, v[]=u[2]+x*u[], v[1]=y]);
 ${\mathrm{Φ6}}{≔}\left[{z}{=}{{u}}_{{1}}{}{y}{,}{{v}}_{\left[\right]}{=}{x}{}{{u}}_{\left[\right]}{+}{{u}}_{{2}}{,}{{v}}_{{1}}{=}{y}\right]$ (2.19)
 E > newPhi6 := AssignTransformationType(Phi6)
 F > Tools:-DGinfo(newPhi6, "TransformationType");
 $\left[{"generic"}{,}{"NA"}\right]$ (2.20)