Prolong - Maple 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 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetVectors"\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]$ (3.2)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameJetForms"\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{1}\right]{,}\left[\left[\left[{5}\right]{,}{1}\right]\right]\right]\right)\right]$ (3.3)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E1},"FrameVerticalBiforms"\right)$
 $\left[{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\right]\right]\right)\right]$