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(φ, ω)

Parameters

     φ    - a transformation between two jet spaces

     ω          - a differential bi-form of type r,s defined on the range jet space of φ

 

Description

Examples

Description

• 

Let π:EM be a fiber bundle, with base dimension n and fiber dimension m and let π∞ :J∞E M be the ∞-th jet bundle of E. The space of p -forms ΩpJ∞E decomposes into a direct sum ΩpJ = r+s =p Ωr,sJE, where Ωr,s JE is the space of bi-forms of horizontal degree r and vertical degree s. The precise definition of the space Ωr,sJEis given in the help page for the horizontal exterior derivative. If ω  ΩpJ , then let ωr,s denote the type r,s component of ω in the decomposition (*). The command convert/DGbifom calculates the various bi-graded components of a form ω  ΩpJ .Let FN be another fiber bundle and let φ: JkE  jℓF. Let η be a differential bi-form of type r,s on  JF. Then the projected pullback of η is denote by φω and defined by φη = Φ*ηr,s.

• 

 Two special cases of this general definition should be noted.

[i]  If φ is the prolongation of a projectable transformation from E to F, then the pullback φ* is a bi-degree preserving transformation, that is, if η be a differential bi-form of type r,s on JF, then φ*η is a differential bi-form of type r,s on JF. Hence φη = φ*η.

[ii] Suppose that φ: JkE  jℓF 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 η is a differential bi-form of type r,s on JF ,φ*η= ω0+ω1+ ω2++ ωr, where ωi is a bi-form of degree ri, s+i on JkE. In these cases the command ProjectedPullback(φ, ω) returns the type r,s bi-form ω0. 

• 

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

withDifferentialGeometry:withJetCalculus:

 

First initialize several different jet spaces over bundles E1M1, E2M2, E3M3. The dimension of the base spaces are dim(M1) =2, dim(M2) =1, dim(M3) =3.

DGsetupx,y,u,E1,2:DGsetupt,v,E2,2:DGsetupp,q,r,w,E3,2:

 

Example 1.

Define a transformation φ1:E1E2. This transformation is a projectable transformation and therefore pullbacks by the prolongation of φ1can be calculated directly using the Pullback command.

E3 > 

Φ1TransformationE1,E2,t=x,v=x2u

Φ1:=_DGtransformation,E1,0,E2,0,,1002xu0x2,x,t,x2u,v

(2.1)
E1 > 

prPhi1ProlongΦ1,2

prPhi1:=_DGtransformation,E1,2,E2,2,projectable,2,100000002xu0x2000002xu1+2u02xx200002xu1,1+4u1024x0x200,x,t,x2u,v,x2u1+2xu,v1,x2u1,1+4xu1+2u,v1,1

(2.2)
E1 > 

Tools:-DGinfoprPhi1,TransformationType

projectable,2

(2.3)

 

Pullback the contact 1-form Cv[1] on J2E2 to a contact form on J2E1 -- this can be done with either the Pullback command or the ProjectedPullback command.

E1 > 

PullbackprPhi1,Cv1

_DGbiform,E1,0,1,3,2x,4,x2

(2.4)
E1 > 

ProjectedPullbackprPhi1,Cv1

_DGbiform,E1,0,1,3,2x,4,x2

(2.5)

 

Example 2

Define a point transformation φ1:E1E3 and prolong it to a transformation J1E1  J1E3.

E1 > 

Φ2TransformationE1,E3,p=u,q=y,r=1,w=x

Φ2:=_DGtransformation,E1,0,E3,0,,001010000100,u,p,y,q,1,r,x,w

(2.6)
E1 > 

prPhi2ProlongΦ2,1

prPhi2:=_DGtransformation,E1,1,E3,1,point,1,001000100000000100000001u120000u2u121u100000,u,p,y,q,1,r,x,w,1u1,w1,u2u1,w2,0,w3

(2.7)

 

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

E1 > 

ProjectedPullbackprPhi2,Dp

_DGbiform,E1,1,0,1,u1,2,u2

(2.8)

 

Calculate the projected pullback of the type (1, 1) form Dp Cw1.

E1 > 

ωDp&wedgeCw

ω:=_DGbiform,E3,1,1,1,4,1

(2.9)
E3 > 

ProjectedPullbackprPhi2,ω

_DGbiform,E1,1,1,1,3,−1,2,3,u2u1

(2.10)

 

To illustrate the definition of the projected pullback we re-derive this result using the usual Pullback command. First convert ω from a bi-form to a form θ1.

E1 > 

θ1convertω,DGform

θ1:=_DGform,E3,2,1,2,w2,1,3,w3,1,4,1

(2.11)

 

Then pullback θ1 using pr φ2.

E3 > 

θ2PullbackprPhi2,θ1

θ2:=_DGform,E1,2,1,3,−1,2,3,u2u1

(2.12)

 

Then convert θ2 back to a bi-form and take the type [1, 1] part.

E1 > 

θ3convertθ2,DGbiform,1,1

θ3:=_DGbiform,E1,1,1,1,3,−1,2,3,u2u1

(2.13)

 

Example 3

Define a differential substitution φ3:J2E2E1 and prolong it to a transformation J2E3  J2E1.

E1 > 

Φ3TransformationE2,E1,x=v,y=v1,u=v2

Φ3:=_DGtransformation,E2,1,E1,0,,010001000,v,x,v1,y,v2,u

(2.14)
E2 > 

prPhi3ProlongΦ3,1

prPhi3:=_DGtransformation,E2,2,E1,1,generalizedDifferentialSubstitution,1,010000100000002v12v1,2v12+v1,122+v1,2v12+v1,122v1v1,2v1,1v12+v1,122002v1v1,2v1,1v12+v1,1222v1,12v1,2v12+v1,122+v1,2v12+v1,12,v,x,v1,y,v2,u,v1v1,2v12+v1,12,u1,v1,1v1,2v12+v1,12,u2

(2.15)

 

Calculate the projected pullback of the type (1, 0) form 2Dx +3 Dy

E2 > 

ProjectedPullbackprPhi3,2Dx+3Dy

_DGbiform,E2,1,0,1,3v1,1+2v1

(2.16)

 

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

E2 > 

ProjectedPullbackprPhi3,u1Cu

_DGbiform,E2,0,1,2,v12v1,22v12+v1,122,3,v1v1,22v1,1v12+v1,122

(2.17)

See Also

DifferentialGeometry

JetCalculus

DGinfo

Prolong

Pullback

PushforwardTotalVector

Transformation