Let $\mathrm{\π}\:E\to M$ be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let ${\mathrm{\π}}^k\:J^k\left(E\right) \to M$ be the $k$-th jet bundle. Introduce local coordinates $\(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}$, ..., $u_{\mathrm{ij} \cdot \cdot \cdot \mathrm{\ℓ}}^{\mathrm{\α}} \, ..\.\)$where, as usual, if $s\:M\to E$ is a section and$\mathrm{\σ}\=j^k\left(s\right)\left(x\right)\:M\to E$is the $k$-jet of $s\,$then

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 $J^k\left(E\right)\.$Let $I \= i_1{i}_2\cdot \cdot \cdot i_r$ be a multi-index. Then the $r$-th order higher Euler operator is defined by

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 $J^k\left(E\right)$, where $k$is the order of F.

The second calling sequence HigherEulerOperators($\mathrm{\ω}$) returns a list of the higher Euler operators of $\mathrm{\ω}$. 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 $\mathrm{\ω}$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

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(...).

with(DifferentialGeometry): with(JetCalculus):

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

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

$F≔u_1{u}_{2,2}^2$

EulerF := expand(HigherEulerOperators(F));

$\mathrm{EulerF}≔\left[0\,0\,4u_{2,2,2}{u}_{1,2}+2u_1{u}_{2,2,2,2}\,u_{2,2}^2\,-4u_{2,2}{u}_{1,2}-4u_1{u}_{2,2,2}\,0\,0\,2u_1{u}_{2,2}\right]$

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

$\mathrm{Vars}≔\left[x\,y\,u_{\left[\right]}\,u_1\,u_2\,u_{1,1}\,u_{1,2}\,u_{2,2}\right]$

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];

${\mathrm{Eu}}_{0,0}≔4u_{2,2,2}{u}_{1,2}+2u_1{u}_{2,2,2,2}$

${\mathrm{Eu}}_{1,0}≔u_{2,2}^2$

${\mathrm{Eu}}_{0,1}≔-4u_{2,2}{u}_{1,2}-4u_1{u}_{2,2,2}$

${\mathrm{Eu}}_{2,0}≔0$

${\mathrm{Eu}}_{1,1}≔0$

${\mathrm{Eu}}_{0,2}≔2u_1{u}_{2,2}$

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

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

$G≔u_1{v}_2^2$

EulerG := expand(HigherEulerOperators(G));

$\mathrm{EulerG}≔\left[0\,0\,-2v_2{v}_{1,2}\,-2v_2{u}_{1,2}-2u_1{v}_{2,2}\,v_2^2\,0\,0\,2u_1{v}_2\right]$

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

$\mathrm{Vars}≔\left[x\,y\,u_{\left[\right]}\,v_{\left[\right]}\,u_1\,u_2\,v_1\,v_2\right]$

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];

${\mathrm{Eu}}_{0,0}≔-2v_2{v}_{1,2}$

${\mathrm{Ev}}_{0,0}≔-2v_2{u}_{1,2}-2u_1{v}_{2,2}$

${\mathrm{Eu}}_{1,0}≔-4u_{2,2}{u}_{1,2}-4u_1{u}_{2,2,2}$

${\mathrm{Eu}}_{0,1}≔0$

${\mathrm{Ev}}_{1,0}≔0$

${\mathrm{Ev}}_{0,1}≔2u_1{u}_{2,2}$

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

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

$H≔\left(2u_{1,1}^2+2u_1{u}_{1,1,1}\right)u_1+\left(10u_1{u}_{1,1}+2u_{\left[\right]}{u}_{1,1,1}\right)u_{1,1}+\left(5u_1^2+4u_{\left[\right]}{u}_{1,1}\right)u_{1,1,1}+2u_{\left[\right]}{u}_1{u}_{1,1,1,1}$

EulerG := expand(HigherEulerOperators(H));

$\mathrm{EulerG}≔\left[0\,0\,0\,0\,-2u_{\left[\right]}{u}_{1,1}-u_1^2\,2u_{\left[\right]}{u}_1\right]$

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

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

$\mathrm{\ω1}≔{\mathrm{Cu}}_1\bigwedge {\mathrm{Cu}}_{2,2}$

HigherEulerOperators(omega1);

$\left[0{\mathrm{Cu}}_{\left[\right]}\,0{\mathrm{Cu}}_{\left[\right]}\,-2{\mathrm{Cu}}_{1,2,2}\,{\mathrm{Cu}}_{2,2}\,2{\mathrm{Cu}}_{1,2}\,0{\mathrm{Cu}}_{\left[\right]}\,0{\mathrm{Cu}}_{\left[\right]}\,-{\mathrm{Cu}}_1\right]$

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

$\mathrm{\ω2}≔\mathrm{Dx}\bigwedge {\mathrm{Cu}}_1\bigwedge {\mathrm{Cu}}_{2,2}$

HigherEulerOperators(omega2);

$\left[0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\,0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\,2\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{1,2,2}\,-\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{2,2}\,-2\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{1,2}\,0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\,0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\,\mathrm{Dx}\bigwedge {\mathrm{Cu}}_1\right]$