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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(φ , k)

     Prolong(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

     φ      - a transformation, defined on a fiber bundle or the jet space of a fiber bundle

     Δ      - a differential equation, defined in terms of standard jet space coordinates

Description

• 

Let π:E.M be a fiber bundle, with base dimension n and fiber dimension m and let πℓ:JℓE  M be the ℓ-th jet bundle. The Prolong command will take a geometry object defined, either on E or on JE, and extend or lift that object to a higher order jet space JℓE. 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 JℓE. This means that the standard jet space coordinates (xi, uα, uiα, uijα, ..., uij ℓα) are protected. The coordinate vector fields, coordinate 1-forms, and contact forms to order ℓ are initialized and protected. The command Prolong(k), where k l, with extend these protections and definitions to order k. The result is same as making a call to DGsetup to initialize the jet space JkE, but is slightly faster since Prolong command only needs to define and protect the coordinates,vectors and 1 -forms from order ℓ +1 to k.

 2. Prolongation of Vector Fields. Let Z be a vector field on JkE. We say that Z preserves the contact ideal on JkE if for any contact form Θ, the Lie derivative ℒZΘ is also a contact form. Let X 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 Z on JkE which preserves the contact ideal on JkE and which projects pointwise to X. This vector field Z is called the prolongation of X to order k. and is denoted by prkX. The explicit formula for vector field prolongation is given below. The second calling sequence Prolong(X, k) computes the prolongation of the vector field X to order k.

 3. Prolongation of Transformations. Let EM and FN be two fiber bundles. We say that a transformation ψ : JℓE  JnFis a generalized contact transformation if for every contact form Θ on JnF, the pullback ψ*Θ is a contact form on JE. Let 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 JpEto JqFfor 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 ψ : Jp+kE  Jq+kF which covers φ. This transformation Ψ is called the prolongation of φ to order k and it denoted by prkφ. The third calling sequence Prolong(φ , k) computes the prolongation of φ to order k.

 4. Prolongation of Differential Equations. A system of ℓ-th order differential equations can defined as the zero set of a collection Δ of functions Fa:JℓE ℝ . The kth order prolongation of Δ, denote by prkΔis the system of (ℓ +k)-th order differential equations defined as the zero set of the functionsFa and all their total derivatives Di1Di2 DitFa to order tk. The fourth calling sequence Prolong(Delta, k) computes the prolongation of a system of differential equations Δ to order k. Use the command DifferentialEquationData to convert a list of functions Fa:JℓE ℝ into 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 X = Aj    xj +Bα     uα is a generalized vector field on E, then the k-th prolongation of X is the vector field

prkX  = Aj    xi +Bα     uα+ζiα    uiα +ζijα    uijα + +ζi1i2ikα                ui1i2ikα where  ζi1i2iℓ α=Di1Di2Diℓ Bα Aiuiα  ui1i2iℓjαAj.

For further details see the either of the two books by P. J. Olver.

 

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1. Prolongation of Jet Spaces

Define the jet space J1E,where E=ℝ2× ℝ with coordinates x,y,u  x,y.

DGsetupx,y,u,E1,1:

 

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

E1 > 

Tools:-DGinfoE1,FrameJetVariables

x,y,u[],u1,u2

(3.1)
E1 > 

Tools:-DGinfoE1,FrameJetVectors

D_x,D_y,D_u,D_u1,D_u2

(3.2)
E1 > 

Tools:-DGinfoE1,FrameJetForms

dx,dy,du,du1,du2

(3.3)
E1 > 

Tools:-DGinfoE1,FrameVerticalBiforms

Cu,Cu1,Cu2

(3.4)

 

Prolong the jet space J1E to J3E.

E1 > 

Prolong3

1

(3.5)

 

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

E1 > 

Tools:-DGinfoE1,FrameJetVariables

x,y,u[],u1,u2,u1,1,u1,2,u2,2,u1,1,1,u1,1,2,u1,2,2,u2,2,2

(3.6)
E1 > 

Tools:-DGinfoE1,FrameJetVectors

D_x,D_y,D_u,D_u1,D_u2,D_u1,1,D_u1,2,D_u2,2,D_u1,1,1,D_u1,1,2,D_u1,2,2,D_u2,2,2

(3.7)
E1 > 

Tools:-DGinfoE1,FrameJetForms

dx,dy,du,du1,du2,du1,1,du1,2,du2,2,du1,1,1,du1,1,2,du1,2,2,du2,2,2

(3.8)
E1 > 

Tools:-DGinfoE1,FrameVerticalBiforms

Cu,Cu1,Cu2,Cu1,1,Cu1,2,Cu2,2,Cu1,1,1,Cu1,1,2,Cu1,2,2,Cu2,2,2

(3.9)

 

Example 2. Prolongation of Vector Fields

Define the jet space J2E2, where E2 = ℝ ×ℝ  ℝ with coordinates x, u x.

E1 > 

DGsetupx,u,E2,1:

 

Define an arbitrary point vector field X1 on E2.

E2 > 

PDEtools[declare]ax,u[],bx,u[],quiet

E2 > 

X1:=evalDGax,u[]D_x+bx,u[]D_u[]

X1:=aD_x+bD_u

(3.10)

 

Prolong X1 to order 1--this agrees with the standard prolongation formula found in all texts.

E2 > 

prX1:=ProlongX1,1

prX1:=aD_x+bD_uauu12+axu1buu1bxD_u1

(3.11)

 

Define the infinitesimal generator X2for a rotation in the x-u plane.

E2 > 

X2:=evalDGu[]D_xxD_u[]

X2:=uD_xxD_u

(3.12)

 

Prolong X2to order 1.

E2 > 

pr1X2:=ProlongX2,1

pr1X2:=uD_xxD_uu12+1D_u1

(3.13)

 

Prolong X2to order to 2--we can achieve the same result by prolonging pr1X2.

E2 > 

ProlongX2,2

uD_xxD_uu12+1D_u13u1u1,1D_u1,1

(3.14)
E2 > 

Prolongpr1X2,2

uD_xxD_uu12+1D_u13u1u1,1D_u1,1

(3.15)

 

Define the jet space J1E3, where E3 =ℝ2 ×ℝ2  ℝ2 with coordinates x,y,u,vx, y.

E2 > 

DGsetupx,y,u,v,E3,1:

 

Define a vector field X3 whose flow simultaneously scales the coordinates x, y, u,v.

E3 > 

X3:=evalDGaxD_x+byD_y+cu[]D_u[]+dv[]D_v[]

X3:=xaD_x+ybD_y+ucD_u+vdD_v

(3.16)
E3 > 

prX3:=factorProlongX3,1

prX3:=xaD_x+ybD_y+ucD_u+vdD_vu1acD_u1u2bcD_u2v1adD_v1v2bdD_v2

(3.17)

 

Example 3. Prolongation of Transformations

Define the jet space J1F where F =ℝ ×ℝ ℝ with coordinates y, v v.

E3 > 

DGsetupy,v,F,1:

 

Define a projectable transformation from E2 to F.

F > 

φ1:=TransformationE2,F,y=x,v[]=logu[]

φ1:=y=x,v=lnu

(3.18)

 

Prolong phi1 to order 2.

E2 > 

prphi1:=Prolongφ1,2

prphi1:=y=x,v=lnu,v1=u1u,v1,1=u12u2+u1,1u

(3.19)

 

Define a differential substitution from J2E2 to F.

E2 > 

φ2:=TransformationE2,F,y=u1,v[]=u2

φ2:=y=u1,v=u2

(3.20)

 

Prolong phi2 to order 2.

E2 > 

prphi2:=simplifyProlongφ2,2

prphi2:=y=u1,v=u2,v1=u1,2u1,1,v1,1=u1,1,2u1,1u1,2u1,1,1u1,13

(3.21)

 

Example 4. Prolongation of Differential Equations

Define a second order ode on E2(coordinates x, u)

E2 > 

DE1:=DifferentialEquationDatau1,1=u1u[],u1,1

DE1:=u1,1,u[]u1+u1,1

(3.22)

 

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

E2 > 

ProlongDE1,2

u1,1,u1,1,1,u1,1,1,1,u[]u1+u1,1,u[]u1,1u12+u1,1,1,u[]u1,1,13u1u1,1+u1,1,1,1

(3.23)

 

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

E2 > 

DGsetupx,y,u,E4,2:

E4 > 

DE2:=DifferentialEquationDatau1,1=x,u2,2=xy,u1,1,u2,2

DE2:=u1,1,u2,2,u1,1x,xy+u2,2

(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:=ProlongDE2,2

A:=u1,1,u2,2,u1,1,1,u1,1,2,u1,2,2,u2,2,2,u1,1,1,1,u1,1,1,2,u1,1,2,2,u1,2,2,2,u2,2,2,2,u1,1x,xy+u2,2,1+u1,1,1,u1,1,2,y+u1,2,2,x+u2,2,2,u1,1,1,1,u1,1,1,2,u1,1,2,2,u1,1,2,2,1+u1,2,2,2,u2,2,2,2

(3.25)

See Also

DifferentialGeometry, JetCalculus, AssignTransformationType, AssignVectorType, DifferentialEquationData, DGinfo, Transformation


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