The higher Euler operators E^J are generalizations of the Euler-Lagrange operators E which arise in many formulas in the variational calculus for higher order variational problems.  They can be defined as follows.  First we recall the multi-index notation used in the JetCalculus package.  If the base manifold or space of independent variables has dimension n and J is the n-tuple J = [j_1, j_2, ... j_n], then D_J denotes repeated differentiation j_1 times with respect to the first independent variable, j_2 times with respect to the second independent variable and so on.  The length of J is |J| = j_1 + j_2 + ... j_n.

Let F be a function on J^k(R^n, R^m).  Then the higher Euler operators E^J(F) can be defined by the identity E(gF) = D_J(g) E^J(F).  Here g is an arbitrary function of the independent variables and there is an implicit summation over all n-tuples J with |J| <= k.  We call |J| the order of the higher Euler operator E^J.  The higher Euler operators of order 0 are just the ordinary Euler-Lagrange operators.

Let omega be a differential bi-form on J^k(R^n, R^m).  Then the higher Euler operators of omega are defined in a similar fashion using the integration by parts operator in place of the Euler-Lagrange operator.

The first calling sequence HigherEulerOperator(F) returns a list of the higher Euler operators of F.  Each element of the list is a function on jet spaces.  The length of the list equals the dimension of the jet bundle J^k(R^n, R^m), where k is the order of F.

The second calling sequence HigherEulerOperator(omega) returns a list of the higher Euler operators of omega.  Each element of the list is a differential form on jet spaces.  The length of the list equals the fiber dimension of the jet bundle J^k(R^n, R^m) -> R^n.

For a sample application, we mention the theorem that a function F on a jet space may be expressed as an r-fold total derivative if and only if E^J(F) = 0 for all J, |J| <= r + 1.

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

with(DifferentialGeometry): with(JetCalculus):

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

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

EulerF := expand(HigherEulerOperators(F));

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

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];

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

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

EulerG := expand(HigherEulerOperators(G));

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

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];

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

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

EulerG := expand(HigherEulerOperators(H));

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

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

HigherEulerOperators(omega1);

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

HigherEulerOperators(omega2);