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JetCalculus[HigherEulerOperators] - apply the higher Euler operators to a function or a differential bi-form

Calling Sequences

     HigherEulerOperators(F)

     HigherEulerOperators(ω)

Parameters

     F         - a function on the jet space of a fiber bundle

     ω         - a differential bi-form on the jet space a fiber bundle

Description

• 

Let π:EM  be a fiber bundle, with base dimension n and fiber dimension m and let πk:JkE M  be the k-th jet bundle. Introduce local coordinates (xi, uα, uiα, uijα, ..., uij  ℓα , ...) where, as usual, if s:ME is a section and σ=jksx:ME is the k-jet of s, then

uij  ℓασ = k sα xxi xixℓ   and 1ijℓ dimM.

• 

The higher Euler operators are generalizations of the Euler-Lagrange operators and arise in many formulas in the variational calculus for higher order variational problems. They can be defined as follows. Let F be a function on JkE. Let I = i1i2ir be a multi-index. Then the r-th order higher Euler operator is defined by

 

EαIF = FuIα  r+11DhF uIhα +r+12Dhi F   uIhiα r+33DhijF     uIhijα +  .

 

If ω is a differential bi-form on JkE, then the Euler operators EαIω are defined by

 

 EαIω = ι αIω  r+1rDh ι αIhω +r+22Dhi ιαIhiω   r+33 Dhij ιαhijω + ,  

where  ι αij denotes interior product with the vector field             uijα .

• 

The first calling sequence HigherEulerOperators(F) returns a list of the higher Euler operators of the function F. Each element of the list is a function on jet spaces. The length of the list equals the fiber dimension of the jet bundle JkE, where k is the order of F.

• 

The second calling sequence HigherEulerOperators(ω) returns a list of the higher Euler operators of ω. Each element of the list is a differential form on jet space. The length of the list equals the fiber dimension of the jet bundle on which ω is defined.

• 

Higher Euler operators are studied in detail in S. J. Aldersley Higher Euler operators and some of their applications, J. Math Phys. 20 (1979) 522-531. We mention two important properties. First, if F and G are two functions on jet space, the product rule for the Euler-Lagrange operator is given in terms of the higher Euler operators by

EαFG = |I| 0 EαIFDIG+  DIFEαIG.

Second, a function F on jet space may be expressed as an r-fold total derivative if and only if EαIF = 0 for all multi-indices with length I r+1.

• 

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

Examples

with(DifferentialGeometry): with(JetCalculus):

 

Example 1.

Create the jet space J2ℝ2, ℝwith independent variables x,y and dependent variable u.

DGsetup([x, y], [u], E1, 2):

E1 > 

F := u[1]*u[2,2]^2;

Fu1u2,22

(2.1)

 

Apply the higher Euler operators to F.

E1 > 

EulerF := expand(HigherEulerOperators(F));

EulerF0,0,4u2,2,2u1,2+2u1u2,2,2,2,u2,22,4u2,2u1,24u1u2,2,2,0,0,2u1u2,2

(2.2)

 

To interpret this result we first list the current jet coordinates.

E1 > 

Vars := Tools:-DGinfo(E1, "FrameJetVariables");

Varsx,y,u,u1,u2,u1,1,u1,2,u2,2

(2.3)

 

Then the various components of the higher Euler operators for F will be labeled by these jet coordinates as:

E1 > 

Eu[0, 0] := EulerF[3]; Eu[1, 0] := EulerF[4]; Eu[0, 1] := EulerF[5]; Eu[2, 0] := EulerF[6]; Eu[1, 1] := EulerF[7]; Eu[0, 2] := EulerF[8];

Eu0,04u2,2,2u1,2+2u1u2,2,2,2

Eu1,0u2,22

Eu0,14u2,2u1,24u1u2,2,2

Eu2,00

Eu1,10

Eu0,22u1u2,2

(2.4)

 

Example 2.

Create the jet space J2ℝ2, ℝ2with independent variables x,y and dependent variables u, v.

E1 > 

DGsetup([x, y], [u, v], E2, 1):

E2 > 

G := u[1]*v[2]^2;

Gu1v22

(2.5)

 

Apply the higher Euler operators to G.

E2 > 

EulerG := expand(HigherEulerOperators(G));

EulerG0,0,2v2v1,2,2v2u1,22u1v2,2,v22,0,0,2u1v2

(2.6)

 

To interpret this result we first list the current jet coordinates.

E2 > 

Vars := Tools:-DGinfo(E2, "FrameJetVariables");

Varsx,y,u,v,u1,u2,v1,v2

(2.7)

 

Then the various components of the higher Euler operators for G will be labeled by these jet coordinates as:

E2 > 

Eu[0, 0] := EulerG[3]; Ev[0, 0] := EulerG[4]; Eu[1, 0] := EulerF[5]; Eu[0, 1] := EulerF[6]; Ev[1, 0] := EulerF[7]; Ev[0, 1] := EulerF[8];

Eu0,02v2v1,2

Ev0,02v2u1,22u1v2,2

Eu1,04u2,2u1,24u1u2,2,2

Eu0,10

Ev1,00

Ev0,12u1u2,2

(2.8)

 

Example 3.

Create the jet space J3,  with independent variable x and dependent variable u.

E2 > 

DGsetup([x], [u], E3, 3):

E3 > 

H := TotalDiff(u[]*u[1]^2, [1,1,1]);

H2u1,12+2u1u1,1,1u1+10u1u1,1+2uu1,1,1u1,1+5u12+4uu1,1u1,1,1+2uu1u1,1,1,1

(2.9)

 

Because H is a 3-fold total derivative, the first 3 Euler operators will vanish.

E3 > 

EulerG := expand(HigherEulerOperators(H));

EulerG0,0,0,0,2uu1,1u12,2uu1

(2.10)

 

Example 4.

Create the jet space J22, with independent variables x,y and dependent variable u.

E3 > 

DGsetup([x, y], [u], E1, 2):

 

Calculate the higher Euler operators for ω1.

E1 > 

omega1 := evalDG(Cu[1] &w Cu[2, 2]);

ω1Cu1Cu2,2

(2.11)
E1 > 

HigherEulerOperators(omega1);

0Cu,0Cu,2Cu1,2,2,Cu2,2,2Cu1,2,0Cu,0Cu,Cu1

(2.12)

 

Calculate the higher Euler operators for ω2.

E1 > 

omega2 := evalDG(Cu[1] &w Cu[2, 2] &w Dx);

ω2DxCu1Cu2,2

(2.13)
E1 > 

HigherEulerOperators(omega2);

0DxCu,0DxCu,2DxCu1,2,2,DxCu2,2,2DxCu1,2,0DxCu,0DxCu,DxCu1

(2.14)

See Also

DifferentialGeometry, JetCalculus, DGinfo, Prolong, Pullback, TotalDiff, Transformation


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