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JetCalculus[GeneratingFunctionToContactVector] - find the contact vector field defined by a generating function

Calling Sequences

     GeneratingFunctionToContactVector(S)

Parameters

     S         - a Maple expression

 

Description

Examples

Description

• 

 Let π:E  M be a fiber bundle with 1-dimensional fiber and let π1: J1E  M be 1st order jet space of E. In terms of the usual coordinates xi, u, ui on J1E, the contact form on J1E is C = du  uidxi. A vector field X on J1E which preserves the contact form C, in the sense that ℒXC= λ C, is called an infinitesimal contact transformation or a contact vector field. There is a formula which assigns to each locally defined real-valued function S on J1E a contact vector field XS. The function S is called the generating function for the contact vector field XS. In terms of the local coordinates xi, u, ui , we have S = Sxi, u, ui and

XS = S ui   xi + S  uiS ui  u + Sxi + ui Su   ui .

  For further details see P. J. Olver,  Equivalence, Invariants and Symmetry, page 131.

• 

The command GeneratingFunctionToContactVector(S) returns the contact vector field defined by the function S.

• 

The command GeneratingFunctionToContactVector is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form GeneratingFunctionToContactVector(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-GeneratingFunctionToContactVector(...).

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

The formula for the contact vector field in terms of the generating function with 1 independent variable.

DGsetupx,u,J11,1:

J11 > 

PDEtools[declare]Sx,u[],u1,quiet

J11 > 

GeneratingFunctionToContactVectorSx,u[],u1

Su1D_x+u1Su1+SD_u+Sx+u1SuD_u1

(2.1)

 

The formula for the contact vector field in terms of the generating function with 2 independent variables.

J11 > 

DGsetupx,y,u,J21,1:

J21 > 

PDEtools[declare]Sx,y,u[],u1,u2,quiet

J21 > 

GeneratingFunctionToContactVectorSx,y,u[],u1,u2

Su1D_xSu2D_y+u2Su2u1Su1+SD_u+Sx+u1SuD_u1+Sy+u2SuD_u2

(2.2)

 

The formula for the contact vector field in terms of the generating function with 3 independent variables.

J21 > 

DGsetupx,y,z,u,J31,1:

J31 > 

PDEtools[declare]Sx,y,z,u[],u1,u2,u3,quiet

J31 > 

GeneratingFunctionToContactVectorSx,y,z,u[],u1,u2,u3

Su1D_xSu2D_ySu3D_z+u3Su3u2Su2u1Su1+SD_u+Sx+u1SuD_u1+Sy+u2SuD_u2+Sz+u3SuD_u3

(2.3)

 

Example 2.

We choose some specific generating functions and calculate the resulting contact vector fields.

J31 > 

ChangeFrameJ21:

J21 > 

Sx+3y:

J21 > 

GeneratingFunctionToContactVectorS

x+3yD_u+D_u1+3D_u2

(2.4)
J21 > 

Su[]:

J21 > 

GeneratingFunctionToContactVectorS

uD_u+u1D_u1+u2D_u2

(2.5)
J21 > 

Sau1+bu2:

J21 > 

GeneratingFunctionToContactVectorS

aD_xbD_y

(2.6)

 

Example 3.

Check the properties of the vector field X obtained from  S = uuy2.

J21 > 

Su[]u22:

J21 > 

XGeneratingFunctionToContactVectorS

X2uu2D_yuu22D_u+u1u22D_u1+u23D_u2

(2.7)

 

X preserves the contact 1-form.

J21 > 

LieDerivativeX,Cu[]

u22Cu

(2.8)

 

X is the prolongation of its projection to the space of independent and dependent variables.

J21 > 

ΦProjectionTransformation1,0

Φx=x,y=y,u=u

(2.9)
J21 > 

YPushforwardΦ,X

Y2uu2D_yuu22D_u

(2.10)
J21 > 

Y1ProlongY,1

Y12uu2D_yuu22D_u+u1u22D_u1+u23D_u2

(2.11)
J21 > 

Y1 &minus X

0D_x

(2.12)

 

Example 4.

We use the commands GeneratingFunctionToContactVector and Flow to find a contact transformation.

J21 > 

Su12+x2:

J21 > 

XGeneratingFunctionToContactVectorS

X2u1D_x+u12+x2D_u+2xD_u1

(2.13)
J21 > 

ΦFlowX,t

Φx=u1sin2t+xcos2t,y=y,u=u12cos2tsin2t4+t2+u1cos2t2xx2cos2tsin2t4+t2+u12cos2tsin2t4+t2+x2cos2tsin2t4+t2u1x+u,u1=u1cos2t+xsin2t,u2=u2

(2.14)

 

Check that Φ is a contact transformation.

J21 > 

PullbackΦ,du[]u1dxu2dy

u1dxu2dy+du

(2.15)

 

We note that Φ takes on a simple form for t = π4 and that it linearizes the Monge-Ampere equation uxxuyy  uxy2 = 1.

J21 > 

Φ1Φt=π4|Φt=π4

Φ1x=u1,y=y,u=u1x+u,u1=x,u2=u2

(2.16)
J21 > 

Φ2ProlongΦ1,2

Φ2x=u1,y=y,u=u1x+u,u1=x,u2=u2,u1,1=1u1,1,u1,2=u1,2u1,1,u2,2=u1,22u1,1+u2,2

(2.17)
J21 > 

ΔPullbackΦ2,u1,1u2,2u1,221

Δu1,22u1,1+u2,2u1,1u1,22u1,121

(2.18)
J21 > 

simplifyΔ

u2,2+u1,1u1,1

(2.19)

See Also

DifferentialGeometry

JetCalculus

Flow

LieDerivative

ProjectionTransformation

Prolong

Pullback

Pushforward

AssignVectorType

 


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