prolong a jet space, vector field, transformation, or differential equation to a higher order jet space - Maple Programming Help

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JetCalculus[Prolong] - prolong a jet space, vector field, transformation, or differential equation to a higher order jet space

Calling Sequences

Prolong(k)

Prolong(X, k)

Prolong(k)

Prolong(${\mathbf{Δ}}$ k)

Parameters

k      - a non-negative integer

X      - a vector field defined on a fiber bundle or the jet space of a fiber bundle

$\mathrm{φ}$      - a transformation, defined on a fiber bundle or the jet space of a fiber bundle

$\mathrm{Δ}$      - a differential equation, defined in terms of standard jet space coordinates

Description

 • Let be a fiber bundle, with base dimension and fiber dimension $m$ and let be the $\mathrm{ℓ}$-th jet bundle. The Prolong command will take a geometry object defined, either on $E$ or on and extend or lift that object to a higher order jet space ${J}^{\mathrm{ℓ}}\left(E\right)$. The lifting or prolongation procedures considered here require only algebraic operations and differentiations. There are 4 different types of prolongation which can be performed by the command Prolong.

1. Prolongation of Jet Spaces. Suppose that the command DGsetup has been used to initialize a jet space ${J}^{\mathrm{ℓ}}\left(E\right)$. This means that the standard jet space coordinates , ..., are protected. The coordinate vector fields, coordinate 1-forms, and contact forms to order are initialized and protected. The command Prolong(k), where with extend these protections and definitions to order $k$. The result is same as making a call to DGsetup to initialize the jet space but is slightly faster since Prolong command only needs to define and protect the coordinates,vectors and 1 -forms from order to $k.$

2. Prolongation of Vector Fields. Let $Z$ be a vector field on We say that $Z$ preserves the contact ideal on ${J}^{k}\left(E\right)$ if for any contact form the Lie derivative is also a contact form. Let be a projectable, point, contact, evolutionary, total,or generalized vector field with values in the tangent space E. (See AssignVectorType for the definitions of these types of vector fields.) Then, for each $k$, there is a unique vector field on ${J}^{k}\left(E\right)$ which preserves the contact ideal on and which projects pointwise to $X.$ This vector field Z is called the prolongation of $X$ to order $k$. and is denoted by ${\mathrm{pr}}^{k}\left(X\right)$. The explicit formula for vector field prolongation is given below. The second calling sequence Prolong(X, k) computes the prolongation of the vector field to order $k.$

3. Prolongation of Transformations. Let and be two fiber bundles. We say that a transformation is a generalized contact transformation if for every contact form $\mathrm{Θ}$ on ${J}^{n}\left(F\right)$, the pullback is a contact form on ${J}^{\ell }\left(E\right)$. Let $\mathrm{φ}$ be a projectable transformation, a point transformation, a contact transformation, a differential substitution or a generalized differential substitution. These maps are defined as mappings from ${J}^{p}\left(E\right)$tofor the appropriate values of $p,$$q.$ (See AssignTransformationType for the definitions of these different types of transformations.) Then, for each $k$, there is a unique generalized contact transformation which covers $\mathrm{φ}$. This transformation is called the prolongation of to order and it denoted by ${\mathrm{pr}}^{k}\left(\mathrm{φ}\right)$.The third calling sequence Prolong(k) computes the prolongation of to order $k.$

4. Prolongation of Differential Equations. A system of $\mathrm{ℓ}$-th order differential equations can defined as the zero set of a collection $\mathrm{Δ}$ of functions . The $k-$th order prolongation of denote by ${\mathrm{pr}}^{k}\left(\mathrm{Δ}\right)$is the system of ($\mathrm{ℓ}$ +$k$)-th order differential equations defined as the zero set of the functionsand all their total derivatives  to order $t\le k$. The fourth calling sequence Prolong(Delta, k) computes the prolongation of a system of differential equations to order $k$. Use the command DifferentialEquationData to convert a list of functionsinto a differential equation data structure that can be passed to the Prolong command. The result is a new differential equation data structure representing the prolongation of the differential equations.

 • If a vector field, transformation or differential equation has been prolonged to a certain order using Prolong, then the prolonged objects may themselves be prolonged to a higher order using Prolong.
 • The command Prolong is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form Prolong(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-Prolong(...).
 Details If is a generalized vector field on $E$, then the $k$-th prolongation of X is the vector field where  . For further details see the either of the two books by P. J. Olver.

Examples

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

Example 1. Prolongation of Jet Spaces

Define the jet space ${J}^{1}\left(E\right),$where with coordinates

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],\mathrm{E1},1\right):$

Display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.

 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetVariables"\right)$
 $\left[{x}{,}{y}{,}{{u}}_{{[}{]}}{,}{{u}}_{{1}}{,}{{u}}_{{2}}\right]$ (3.1)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetVectors"\right)$
 $\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{{\mathrm{D_u}}}_{\left[\right]}{,}{{\mathrm{D_u}}}_{{1}}{,}{{\mathrm{D_u}}}_{{2}}\right]$ (3.2)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetForms"\right)$
 $\left[{\mathrm{dx}}{,}{\mathrm{dy}}{,}{{\mathrm{du}}}_{\left[\right]}{,}{{\mathrm{du}}}_{{1}}{,}{{\mathrm{du}}}_{{2}}\right]$ (3.3)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameVerticalBiforms"\right)$
 $\left[{{\mathrm{Cu}}}_{\left[\right]}{,}{{\mathrm{Cu}}}_{{1}}{,}{{\mathrm{Cu}}}_{{2}}\right]$ (3.4)

Prolong the jet space to ${J}^{3}\left(E\right)$.

 E1 > $\mathrm{Prolong}\left(3\right)$
 ${1}$ (3.5)

Again display the jet coordinates, the coordinate vector fields, the 1-forms, and the contact 1-forms.

 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetVariables"\right)$
 $\left[{x}{,}{y}{,}{{u}}_{{[}{]}}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{2}}{,}{{u}}_{{1}{,}{2}{,}{2}}{,}{{u}}_{{2}{,}{2}{,}{2}}\right]$ (3.6)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetVectors"\right)$
 $\left[{\mathrm{D_x}}{,}{\mathrm{D_y}}{,}{{\mathrm{D_u}}}_{\left[\right]}{,}{{\mathrm{D_u}}}_{{1}}{,}{{\mathrm{D_u}}}_{{2}}{,}{{\mathrm{D_u}}}_{{1}{,}{1}}{,}{{\mathrm{D_u}}}_{{1}{,}{2}}{,}{{\mathrm{D_u}}}_{{2}{,}{2}}{,}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{1}}{,}{{\mathrm{D_u}}}_{{1}{,}{1}{,}{2}}{,}{{\mathrm{D_u}}}_{{1}{,}{2}{,}{2}}{,}{{\mathrm{D_u}}}_{{2}{,}{2}{,}{2}}\right]$ (3.7)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetForms"\right)$
 $\left[{\mathrm{dx}}{,}{\mathrm{dy}}{,}{{\mathrm{du}}}_{\left[\right]}{,}{{\mathrm{du}}}_{{1}}{,}{{\mathrm{du}}}_{{2}}{,}{{\mathrm{du}}}_{{1}{,}{1}}{,}{{\mathrm{du}}}_{{1}{,}{2}}{,}{{\mathrm{du}}}_{{2}{,}{2}}{,}{{\mathrm{du}}}_{{1}{,}{1}{,}{1}}{,}{{\mathrm{du}}}_{{1}{,}{1}{,}{2}}{,}{{\mathrm{du}}}_{{1}{,}{2}{,}{2}}{,}{{\mathrm{du}}}_{{2}{,}{2}{,}{2}}\right]$ (3.8)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameVerticalBiforms"\right)$
 $\left[{{\mathrm{Cu}}}_{\left[\right]}{,}{{\mathrm{Cu}}}_{{1}}{,}{{\mathrm{Cu}}}_{{2}}{,}{{\mathrm{Cu}}}_{{1}{,}{1}}{,}{{\mathrm{Cu}}}_{{1}{,}{2}}{,}{{\mathrm{Cu}}}_{{2}{,}{2}}{,}{{\mathrm{Cu}}}_{{1}{,}{1}{,}{1}}{,}{{\mathrm{Cu}}}_{{1}{,}{1}{,}{2}}{,}{{\mathrm{Cu}}}_{{1}{,}{2}{,}{2}}{,}{{\mathrm{Cu}}}_{{2}{,}{2}{,}{2}}\right]$ (3.9)

Example 2. Prolongation of Vector Fields

Define the jet space where  with coordinates .

 E1 > $\mathrm{DGsetup}\left(\left[x\right],\left[u\right],\mathrm{E2},1\right):$

Define an arbitrary point vector field ${X}_{1}$ on ${E}_{2}$.

 E2 > $\mathrm{PDEtools}[\mathrm{declare}]\left(a\left(x,{u}_{[]}\right),b\left(x,{u}_{[]}\right),\mathrm{quiet}\right)$
 E2 > $\mathrm{X1}≔\mathrm{evalDG}\left(a\left(x,{u}_{[]}\right)\mathrm{D_x}+b\left(x,{u}_{[]}\right){\mathrm{D_u}}_{[]}\right)$
 ${\mathrm{X1}}{:=}{a}{}{\mathrm{D_x}}{+}{b}{}{{\mathrm{D_u}}}_{\left[\right]}$ (3.10)

Prolong ${X}_{1}$ to order 1--this agrees with the standard prolongation formula found in all texts.

 E2 > $\mathrm{prX1}≔\mathrm{Prolong}\left(\mathrm{X1},1\right)$
 ${\mathrm{prX1}}{:=}{a}{}{\mathrm{D_x}}{+}{b}{}{{\mathrm{D_u}}}_{\left[\right]}{-}\left({{a}}_{{{u}}_{\left[\right]}}{}{{u}}_{{1}}^{{2}}{+}{{a}}_{{x}}{}{{u}}_{{1}}{-}{{b}}_{{{u}}_{\left[\right]}}{}{{u}}_{{1}}{-}{{b}}_{{x}}\right){}{{\mathrm{D_u}}}_{{1}}$ (3.11)

Define the infinitesimal generator ${X}_{2}$for a rotation in the $x$-$u$ plane.

 E2 > $\mathrm{X2}≔\mathrm{evalDG}\left({u}_{[]}\mathrm{D_x}-x{\mathrm{D_u}}_{[]}\right)$
 ${\mathrm{X2}}{:=}{{u}}_{\left[\right]}{}{\mathrm{D_x}}{-}{x}{}{{\mathrm{D_u}}}_{\left[\right]}$ (3.12)

Prolong ${X}_{2}$to order 1.

 E2 > $\mathrm{pr1X2}≔\mathrm{Prolong}\left(\mathrm{X2},1\right)$
 ${\mathrm{pr1X2}}{:=}{{u}}_{\left[\right]}{}{\mathrm{D_x}}{-}{x}{}{{\mathrm{D_u}}}_{\left[\right]}{-}\left({{u}}_{{1}}^{{2}}{+}{1}\right){}{{\mathrm{D_u}}}_{{1}}$ (3.13)

Prolong ${X}_{2}$to order to 2--we can achieve the same result by prolonging $\mathrm{pr1X2}$.

 E2 > $\mathrm{Prolong}\left(\mathrm{X2},2\right)$
 ${{u}}_{\left[\right]}{}{\mathrm{D_x}}{-}{x}{}{{\mathrm{D_u}}}_{\left[\right]}{-}\left({{u}}_{{1}}^{{2}}{+}{1}\right){}{{\mathrm{D_u}}}_{{1}}{-}{3}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}}{}{{\mathrm{D_u}}}_{{1}{,}{1}}$ (3.14)
 E2 > $\mathrm{Prolong}\left(\mathrm{pr1X2},2\right)$
 ${{u}}_{\left[\right]}{}{\mathrm{D_x}}{-}{x}{}{{\mathrm{D_u}}}_{\left[\right]}{-}\left({{u}}_{{1}}^{{2}}{+}{1}\right){}{{\mathrm{D_u}}}_{{1}}{-}{3}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}}{}{{\mathrm{D_u}}}_{{1}{,}{1}}$ (3.15)

Define the jet space ${J}^{1}\left({E}_{3}\right)$, where with coordinates .

 E2 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],\mathrm{E3},1\right):$

Define a vector field whose flow simultaneously scales the coordinates .

 E3 > $\mathrm{X3}≔\mathrm{evalDG}\left(ax\mathrm{D_x}+by\mathrm{D_y}+c{u}_{[]}{\mathrm{D_u}}_{[]}+d{v}_{[]}{\mathrm{D_v}}_{[]}\right)$
 ${\mathrm{X3}}{:=}{x}{}{a}{}{\mathrm{D_x}}{+}{y}{}{b}{}{\mathrm{D_y}}{+}{{u}}_{\left[\right]}{}{c}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{{v}}_{\left[\right]}{}{d}{}{{\mathrm{D_v}}}_{\left[\right]}$ (3.16)
 E3 > $\mathrm{prX3}≔\mathrm{factor}\left(\mathrm{Prolong}\left(\mathrm{X3},1\right)\right)$
 ${\mathrm{prX3}}{:=}{x}{}{a}{}{\mathrm{D_x}}{+}{y}{}{b}{}{\mathrm{D_y}}{+}{{u}}_{\left[\right]}{}{c}{}{{\mathrm{D_u}}}_{\left[\right]}{+}{{v}}_{\left[\right]}{}{d}{}{{\mathrm{D_v}}}_{\left[\right]}{-}{{u}}_{{1}}{}\left({a}{-}{c}\right){}{{\mathrm{D_u}}}_{{1}}{-}{{u}}_{{2}}{}\left({b}{-}{c}\right){}{{\mathrm{D_u}}}_{{2}}{-}{{v}}_{{1}}{}\left({a}{-}{d}\right){}{{\mathrm{D_v}}}_{{1}}{-}{{v}}_{{2}}{}\left({b}{-}{d}\right){}{{\mathrm{D_v}}}_{{2}}$ (3.17)

Example 3. Prolongation of Transformations

Define the jet space where  with coordinates .

 E3 > $\mathrm{DGsetup}\left(\left[y\right],\left[v\right],F,1\right):$

Define a projectable transformation from ${E}_{2}$ to $F$.

 F > $\mathrm{φ1}≔\mathrm{Transformation}\left(\mathrm{E2},F,\left[y=x,{v}_{[]}=\mathrm{log}\left({u}_{[]}\right)\right]\right)$
 ${\mathrm{φ1}}{:=}\left[{y}{=}{x}{,}{{v}}_{\left[\right]}{=}{\mathrm{ln}}{}\left({{u}}_{\left[\right]}\right)\right]$ (3.18)

Prolong phi1 to order 2.

 E2 > $\mathrm{prphi1}≔\mathrm{Prolong}\left(\mathrm{φ1},2\right)$
 ${\mathrm{prphi1}}{:=}\left[{y}{=}{x}{,}{{v}}_{\left[\right]}{=}{\mathrm{ln}}{}\left({{u}}_{\left[\right]}\right){,}{{v}}_{{1}}{=}\frac{{{u}}_{{1}}}{{{u}}_{\left[\right]}}{,}{{v}}_{{1}{,}{1}}{=}{-}\frac{{{u}}_{{1}}^{{2}}}{{{u}}_{\left[\right]}^{{2}}}{+}\frac{{{u}}_{{1}{,}{1}}}{{{u}}_{\left[\right]}}\right]$ (3.19)

Define a differential substitution from J${}^{2}\left({E}_{2}\right)$ to $F.$

 E2 > $\mathrm{φ2}≔\mathrm{Transformation}\left(\mathrm{E2},F,\left[y={u}_{1},{v}_{[]}={u}_{2}\right]\right)$
 ${\mathrm{φ2}}{:=}\left[{y}{=}{{u}}_{{1}}{,}{{v}}_{\left[\right]}{=}{{u}}_{{2}}\right]$ (3.20)

Prolong phi2 to order 2.

 E2 > $\mathrm{prphi2}≔\mathrm{simplify}\left(\mathrm{Prolong}\left(\mathrm{φ2},2\right)\right)$
 ${\mathrm{prphi2}}{:=}\left[{y}{=}{{u}}_{{1}}{,}{{v}}_{\left[\right]}{=}{{u}}_{{2}}{,}{{v}}_{{1}}{=}\frac{{{u}}_{{1}{,}{2}}}{{{u}}_{{1}{,}{1}}}{,}{{v}}_{{1}{,}{1}}{=}\frac{{{u}}_{{1}{,}{1}{,}{2}}{}{{u}}_{{1}{,}{1}}{-}{{u}}_{{1}{,}{2}}{}{{u}}_{{1}{,}{1}{,}{1}}}{{{u}}_{{1}{,}{1}}^{{3}}}\right]$ (3.21)

Example 4. Prolongation of Differential Equations

Define a second order ode on ${E}_{2}$(coordinates

 E2 > $\mathrm{DE1}≔\mathrm{DifferentialEquationData}\left(\left[{u}_{1,1}={u}_{1}{u}_{[]}\right],\left[{u}_{1,1}\right]\right)$
 ${\mathrm{DE1}}{:=}\left[\left\{{{u}}_{{1}{,}{1}}\right\}{,}\left[{-}{{u}}_{{[}{]}}{}{{u}}_{{1}}{+}{{u}}_{{1}{,}{1}}\right]\right]$ (3.22)

Calculate the second order prolongation of DE1. Note that the list of jet variables to be solved for is also prolonged.

 E2 > $\mathrm{Prolong}\left(\mathrm{DE1},2\right)$
 $\left[\left\{{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}\right\}{,}\left[{-}{{u}}_{{[}{]}}{}{{u}}_{{1}}{+}{{u}}_{{1}{,}{1}}{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{1}{,}{1}}{-}{{u}}_{{1}}^{{2}}{+}{{u}}_{{1}{,}{1}{,}{1}}{,}{-}{{u}}_{{[}{]}}{}{{u}}_{{1}{,}{1}{,}{1}}{-}{3}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}}{+}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}\right]\right]$ (3.23)

Define a system of over-determined partial differential equations in 2 independent variables $x,y$ and 1 dependent variable $u$.

 E2 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],\mathrm{E4},2\right):$
 E4 > $\mathrm{DE2}≔\mathrm{DifferentialEquationData}\left(\left[{u}_{1,1}=x,{u}_{2,2}=xy\right],\left[{u}_{1,1},{u}_{2,2}\right]\right)$
 ${\mathrm{DE2}}{:=}\left[\left\{{{u}}_{{1}{,}{1}}{,}{{u}}_{{2}{,}{2}}\right\}{,}\left[{{u}}_{{1}{,}{1}}{-}{x}{,}{-}{x}{}{y}{+}{{u}}_{{2}{,}{2}}\right]\right]$ (3.24)

The second prolongation of DE2 is an overdetermined system of Frobenius type--all the 4-th order derivatives of which can be solved for.

 E4 > $A≔\mathrm{Prolong}\left(\mathrm{DE2},2\right)$
 ${A}{:=}\left[\left\{{{u}}_{{1}{,}{1}}{,}{{u}}_{{2}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{2}}{,}{{u}}_{{1}{,}{2}{,}{2}}{,}{{u}}_{{2}{,}{2}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{,}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{,}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}\right\}{,}\left[{{u}}_{{1}{,}{1}}{-}{x}{,}{-}{x}{}{y}{+}{{u}}_{{2}{,}{2}}{,}{-}{1}{+}{{u}}_{{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{2}}{,}{-}{y}{+}{{u}}_{{1}{,}{2}{,}{2}}{,}{-}{x}{+}{{u}}_{{2}{,}{2}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{,}{-}{1}{+}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{,}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}\right]\right]$ (3.25)