pullback a differential bi-form of type (r, s) by a transformation to a differential bi-form of type (r, s) - Maple Programming Help

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JetCalculus[ProjectedPullback] - pullback a differential bi-form of type (r, s) by a transformation to a differential bi-form of type (r, s)

Calling Sequences

ProjectedPullback(${\mathbf{φ}}$, ${\mathbf{ω}}$)

Parameters

$\mathrm{φ}$    - a transformation between two jet spaces

- a differential bi-form of type defined on the range jet space of $\mathrm{φ}$

Description

 • Let be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let be the $\mathrm{∞}$-th jet bundle of $E$. The space of -forms ${\mathrm{Ω}}^{p}\left({J}^{\mathrm{∞}}\left(E\right)\right)$ decomposes into a direct sum, where is the space of bi-forms of horizontal degree $r$ and vertical degree The precise definition of the space ${\mathrm{Ω}}^{\left(r,s\right)}\left({J}^{\infty }\left(E\right)\right)$is given in the help page for the horizontal exterior derivative. If , then let denote the type component of in the decomposition (*). The command convert/DGbifom calculates the various bi-graded components of a form .Let $F\to N$be another fiber bundle and let . Let be a differential bi-form of type on . Then the projected pullback of is denote by and defined by .
 • Two special cases of this general definition should be noted.

[i]  If $\mathrm{φ}$ is the prolongation of a projectable transformation from to, then the pullback ${\mathrm{φ}}^{*}$ is a bi-degree preserving transformation, that is, if be a differential bi-form of type on ${J}^{\ell }\left(F\right)$, then ${\mathrm{φ}}^{*}\left(\mathrm{η}\right)$ is a differential bi-form of type on Hence ${\mathrm{\phi }}^{†}\left(\mathrm{\eta }\right)$.

[ii] Suppose that  is the prolongation of a point transformation, a contact transformation, a differential substitution or a generalized differential substitution. (See AssignTransformationType for the definitions of these different types of transformations.) Then if is a differential bi-form of type on , where ${\mathrm{\omega }}_{i}$ is a bi-form of degree  on ${J}^{k}\left(E\right)$. In these cases the command ProjectedPullback(${\mathbf{φ}}$$,$ ${\mathbf{ω}}$) returns the type bi-form ${\mathrm{\omega }}_{0}$.

 • Use ProjectedPullback to transform a Lagrangian bi-form to a new Lagrangian bi-form using any of the above transformations.
 • The command ProjectedPullback is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form ProjectedPullback(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-ProjectedPullback(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{JetCalculus}\right):$

First initialize several different jet spaces over bundles ${E}_{1}\to {M}_{1}$, . The dimension of the base spaces are dim(${M}_{1}$) =2, dim(${M}_{2}$) =1, dim(${M}_{3}$) =3.

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],\mathrm{E1},2\right):$$\mathrm{DGsetup}\left(\left[t\right],\left[v\right],\mathrm{E2},2\right):$$\mathrm{DGsetup}\left(\left[p,q,r\right],\left[w\right],\mathrm{E3},2\right):$

Example 1.

Define a transformation ${\mathrm{φ}}_{1}:{E}_{1}\to {E}_{2}$. This transformation is a projectable transformation and therefore pullbacks by the prolongation of ${\mathrm{φ}}_{1}$can be calculated directly using the Pullback command.

 E3 > $\mathrm{Φ1}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E2},\left[t=x,{v}_{[]}={x}^{2}{u}_{[]}\right]\right)$
 ${\mathrm{Φ1}}{:=}\left[{t}{=}{x}{,}{{v}}_{\left[\right]}{=}{{x}}^{{2}}{}{{u}}_{\left[\right]}\right]$ (2.1)
 E1 > $\mathrm{prPhi1}≔\mathrm{Prolong}\left(\mathrm{Φ1},2\right)$
 ${\mathrm{prPhi1}}{:=}\left[{t}{=}{x}{,}{{v}}_{\left[\right]}{=}{{x}}^{{2}}{}{{u}}_{\left[\right]}{,}{{v}}_{{1}}{=}{{x}}^{{2}}{}{{u}}_{{1}}{+}{2}{}{x}{}{{u}}_{\left[\right]}{,}{{v}}_{{1}{,}{1}}{=}{{x}}^{{2}}{}{{u}}_{{1}{,}{1}}{+}{4}{}{x}{}{{u}}_{{1}}{+}{2}{}{{u}}_{\left[\right]}\right]$ (2.2)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{prPhi1},"TransformationType"\right)$
 $\left[{"projectable"}{,}{2}\right]$ (2.3)

Pullback the contact 1-form Cv[1] on to a contact form on -- this can be done with either the Pullback command or the ProjectedPullback command.

 E1 > $\mathrm{Pullback}\left(\mathrm{prPhi1},{\mathrm{Cv}}_{1}\right)$
 ${2}{}{x}{}{{\mathrm{Cu}}}_{\left[\right]}{+}{{x}}^{{2}}{}{{\mathrm{Cu}}}_{{1}}$ (2.4)
 E1 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi1},{\mathrm{Cv}}_{1}\right)$
 ${2}{}{x}{}{{\mathrm{Cu}}}_{\left[\right]}{+}{{x}}^{{2}}{}{{\mathrm{Cu}}}_{{1}}$ (2.5)

Example 2

Define a point transformation ${\mathrm{\phi }}_{1}:{E}_{1}\to {E}_{3}$ and prolong it to a transformation .

 E1 > $\mathrm{Φ2}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E3},\left[p={u}_{[]},q=y,r=1,{w}_{[]}=x\right]\right)$
 ${\mathrm{Φ2}}{:=}\left[{p}{=}{{u}}_{\left[\right]}{,}{q}{=}{y}{,}{r}{=}{1}{,}{{w}}_{\left[\right]}{=}{x}\right]$ (2.6)
 E1 > $\mathrm{prPhi2}≔\mathrm{Prolong}\left(\mathrm{Φ2},1\right)$
 ${\mathrm{prPhi2}}{:=}\left[{p}{=}{{u}}_{\left[\right]}{,}{q}{=}{y}{,}{r}{=}{1}{,}{{w}}_{\left[\right]}{=}{x}{,}{{w}}_{{1}}{=}\frac{{1}}{{{u}}_{{1}}}{,}{{w}}_{{2}}{=}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}{,}{{w}}_{{3}}{=}{0}\right]$ (2.7)

Calculate the projected pullback of the type (1, 0) form $\mathrm{Dp}$.

 E1 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi2},\mathrm{Dp}\right)$
 ${{u}}_{{1}}{}{\mathrm{Dx}}{+}{{u}}_{{2}}{}{\mathrm{Dy}}$ (2.8)

Calculate the projected pullback of the type (1, 1) form .

 E1 > $\mathrm{ω}≔\mathrm{Dp}&wedge\left({\mathrm{Cw}}_{[]}\right)$
 ${\mathrm{ω}}{:=}{\mathrm{Dp}}{}{\bigwedge }{}{{\mathrm{Cw}}}_{\left[\right]}$ (2.9)
 E3 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi2},\mathrm{ω}\right)$
 ${-}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}$ (2.10)

To illustrate the definition of the projected pullback we re-derive this result using the usual Pullback command. First convert $\mathrm{ω}$ from a bi-form to a form ${\mathrm{θ}}_{1}$.

 E1 > $\mathrm{θ1}≔\mathrm{convert}\left(\mathrm{ω},\mathrm{DGform}\right)$
 ${\mathrm{θ1}}{:=}{-}{{w}}_{{2}}{}{\mathrm{dp}}{}{\bigwedge }{}{\mathrm{dq}}{-}{{w}}_{{3}}{}{\mathrm{dp}}{}{\bigwedge }{}{\mathrm{dr}}{+}{\mathrm{dp}}{}{\bigwedge }{}{{\mathrm{dw}}}_{\left[\right]}$ (2.11)

Then pullback using

 E3 > $\mathrm{θ2}≔\mathrm{Pullback}\left(\mathrm{prPhi2},\mathrm{θ1}\right)$
 ${\mathrm{θ2}}{:=}{-}{\mathrm{dx}}{}{\bigwedge }{}{{\mathrm{du}}}_{\left[\right]}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}{}{\mathrm{dy}}{}{\bigwedge }{}{{\mathrm{du}}}_{\left[\right]}$ (2.12)

Then convert back to a bi-form and take the type [1, 1] part.

 E1 > $\mathrm{θ3}≔\mathrm{convert}\left(\mathrm{θ2},\mathrm{DGbiform},\left[1,1\right]\right)$
 ${\mathrm{θ3}}{:=}{-}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}{}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}$ (2.13)

Example 3

Define a differential substitution and prolong it to a transformation .

 E1 > $\mathrm{Φ3}≔\mathrm{Transformation}\left(\mathrm{E2},\mathrm{E1},\left[x={v}_{[]},y={v}_{1},{u}_{[]}={v}_{2}\right]\right)$
 ${\mathrm{Φ3}}{:=}\left[{x}{=}{{v}}_{\left[\right]}{,}{y}{=}{{v}}_{{1}}{,}{{u}}_{\left[\right]}{=}{{v}}_{{2}}\right]$ (2.14)
 E2 > $\mathrm{prPhi3}≔\mathrm{Prolong}\left(\mathrm{Φ3},1\right)$
 ${\mathrm{prPhi3}}{:=}\left[{x}{=}{{v}}_{\left[\right]}{,}{y}{=}{{v}}_{{1}}{,}{{u}}_{\left[\right]}{=}{{v}}_{{2}}{,}{{u}}_{{1}}{=}\frac{{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}}{{{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}}{,}{{u}}_{{2}}{=}\frac{{{v}}_{{1}{,}{1}}{}{{v}}_{{1}{,}{2}}}{{{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}}\right]$ (2.15)

Calculate the projected pullback of the type (1, 0) form

 E2 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi3},2\mathrm{Dx}+3\mathrm{Dy}\right)$
 $\left({3}{}{{v}}_{{1}{,}{1}}{+}{2}{}{{v}}_{{1}}\right){}{\mathrm{Dt}}$ (2.16)

Calculate the projected pullback of the type (1, 0) form ${\mathrm{Cu}}_{\left[\right]}$

 E2 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi3},{u}_{1}{\mathrm{Cu}}_{[]}\right)$
 ${-}\frac{{{v}}_{{1}}^{{2}}{}{{v}}_{{1}{,}{2}}^{{2}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}{}{{\mathrm{Cv}}}_{\left[\right]}{-}\frac{{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}^{{2}}{}{{v}}_{{1}{,}{1}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}{}{{\mathrm{Cv}}}_{{1}}$ (2.17)