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

Calling Sequences

HigherEulerOperators(F)

HigherEulerOperators(${\mathbf{ω}}$)

Parameters

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

$\mathrm{ω}$         - a differential bi-form on the jet space a fiber bundle

Description

 • Let  be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let  be the $k$-th jet bundle. Introduce local coordinates , ..., where, as usual, if $s:M\to E$ is a section andis the $k$-jet of then

and dim$\left(M\right)$.

 • 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 be a function on Let  be a multi-index. Then the $r$-th order higher Euler operator is defined by

If is a differential bi-form on then the Euler operators are defined by

,

where denotes interior product with the vector field  .

 • 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 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 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, ifand $G$ are two functions on jet space, the product rule for the Euler-Lagrange operator is given in terms of the higher Euler operators by



Second, a function $F$ on jet space may be expressed as an $r$-fold total derivative if and only if ${E}_{\mathrm{\alpha }}^{I}\left(F\right)$ = 0 for all multi-indices with length

 • 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 with independent variables $\left(x,y\right)$ and dependent variable $u$.

 > DGsetup([x, y], [u], E1, 2):
 E1 > F := u[1]*u[2,2]^2;
 ${F}{≔}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}^{{2}}$ (2.1)

Apply the higher Euler operators to F.

 E1 > EulerF := expand(HigherEulerOperators(F));
 ${\mathrm{EulerF}}{≔}\left[{0}{,}{0}{,}{4}{}{{u}}_{{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{2}}{+}{2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}{,}{{u}}_{{2}{,}{2}}^{{2}}{,}{-}{4}{}{{u}}_{{2}{,}{2}}{}{{u}}_{{1}{,}{2}}{-}{4}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}{,}{2}}{,}{0}{,}{0}{,}{2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}\right]$ (2.2)

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

 E1 > Vars := Tools:-DGinfo(E1, "FrameJetVariables");
 ${\mathrm{Vars}}{≔}\left[{x}{,}{y}{,}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}\right]$ (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];
 ${{\mathrm{Eu}}}_{{0}{,}{0}}{≔}{4}{}{{u}}_{{2}{,}{2}{,}{2}}{}{{u}}_{{1}{,}{2}}{+}{2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}$
 ${{\mathrm{Eu}}}_{{1}{,}{0}}{≔}{{u}}_{{2}{,}{2}}^{{2}}$
 ${{\mathrm{Eu}}}_{{0}{,}{1}}{≔}{-}{4}{}{{u}}_{{2}{,}{2}}{}{{u}}_{{1}{,}{2}}{-}{4}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}{,}{2}}$
 ${{\mathrm{Eu}}}_{{2}{,}{0}}{≔}{0}$
 ${{\mathrm{Eu}}}_{{1}{,}{1}}{≔}{0}$
 ${{\mathrm{Eu}}}_{{0}{,}{2}}{≔}{2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}$ (2.4)

Example 2.

Create the jet space with independent variables $\left(x,y\right)$ and dependent variables

 E1 > DGsetup([x, y], [u, v], E2, 1):
 E2 > G := u[1]*v[2]^2;
 ${G}{≔}{{u}}_{{1}}{}{{v}}_{{2}}^{{2}}$ (2.5)

Apply the higher Euler operators to G.

 E2 > EulerG := expand(HigherEulerOperators(G));
 ${\mathrm{EulerG}}{≔}\left[{0}{,}{0}{,}{-}{2}{}{{v}}_{{2}}{}{{v}}_{{1}{,}{2}}{,}{-}{2}{}{{v}}_{{2}}{}{{u}}_{{1}{,}{2}}{-}{2}{}{{u}}_{{1}}{}{{v}}_{{2}{,}{2}}{,}{{v}}_{{2}}^{{2}}{,}{0}{,}{0}{,}{2}{}{{u}}_{{1}}{}{{v}}_{{2}}\right]$ (2.6)

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

 E2 > Vars := Tools:-DGinfo(E2, "FrameJetVariables");
 ${\mathrm{Vars}}{≔}\left[{x}{,}{y}{,}{{u}}_{\left[\right]}{,}{{v}}_{\left[\right]}{,}{{u}}_{{1}}{,}{{u}}_{{2}}{,}{{v}}_{{1}}{,}{{v}}_{{2}}\right]$ (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];
 ${{\mathrm{Eu}}}_{{0}{,}{0}}{≔}{-}{2}{}{{v}}_{{2}}{}{{v}}_{{1}{,}{2}}$
 ${{\mathrm{Ev}}}_{{0}{,}{0}}{≔}{-}{2}{}{{v}}_{{2}}{}{{u}}_{{1}{,}{2}}{-}{2}{}{{u}}_{{1}}{}{{v}}_{{2}{,}{2}}$
 ${{\mathrm{Eu}}}_{{1}{,}{0}}{≔}{-}{4}{}{{u}}_{{2}{,}{2}}{}{{u}}_{{1}{,}{2}}{-}{4}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}{,}{2}}$
 ${{\mathrm{Eu}}}_{{0}{,}{1}}{≔}{0}$
 ${{\mathrm{Ev}}}_{{1}{,}{0}}{≔}{0}$
 ${{\mathrm{Ev}}}_{{0}{,}{1}}{≔}{2}{}{{u}}_{{1}}{}{{u}}_{{2}{,}{2}}$ (2.8)

Example 3.

Create the jet space  with independent variable and dependent variable $u$.

 E2 > DGsetup([x], [u], E3, 3):
 E3 > H := TotalDiff(u[]*u[1]^2, [1,1,1]);
 ${H}{≔}\left({2}{}{{u}}_{{1}{,}{1}}^{{2}}{+}{2}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}{,}{1}}\right){}{{u}}_{{1}}{+}\left({10}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}}{+}{2}{}{{u}}_{\left[\right]}{}{{u}}_{{1}{,}{1}{,}{1}}\right){}{{u}}_{{1}{,}{1}}{+}\left({5}{}{{u}}_{{1}}^{{2}}{+}{4}{}{{u}}_{\left[\right]}{}{{u}}_{{1}{,}{1}}\right){}{{u}}_{{1}{,}{1}{,}{1}}{+}{2}{}{{u}}_{\left[\right]}{}{{u}}_{{1}}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}$ (2.9)

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

 E3 > EulerG := expand(HigherEulerOperators(H));
 ${\mathrm{EulerG}}{≔}\left[{0}{,}{0}{,}{0}{,}{0}{,}{-}{2}{}{{u}}_{\left[\right]}{}{{u}}_{{1}{,}{1}}{-}{{u}}_{{1}}^{{2}}{,}{2}{}{{u}}_{\left[\right]}{}{{u}}_{{1}}\right]$ (2.10)

Example 4.

Create the jet space with independent variables $\left(x,y\right)$ and dependent variable $u$.

 E3 > DGsetup([x, y], [u], E1, 2):

Calculate the higher Euler operators for ${{\mathrm{ω}}}_{{1}}$.

 E1 > omega1 := evalDG(Cu[1] &w Cu[2, 2]);
 ${\mathrm{ω1}}{≔}{{\mathrm{Cu}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}{,}{2}}$ (2.11)
 E1 > HigherEulerOperators(omega1);
 $\left[{0}{}{{\mathrm{Cu}}}_{\left[\right]}{,}{0}{}{{\mathrm{Cu}}}_{\left[\right]}{,}{-}{2}{}{{\mathrm{Cu}}}_{{1}{,}{2}{,}{2}}{,}{{\mathrm{Cu}}}_{{2}{,}{2}}{,}{2}{}{{\mathrm{Cu}}}_{{1}{,}{2}}{,}{0}{}{{\mathrm{Cu}}}_{\left[\right]}{,}{0}{}{{\mathrm{Cu}}}_{\left[\right]}{,}{-}{{\mathrm{Cu}}}_{{1}}\right]$ (2.12)

Calculate the higher Euler operators for ${{\mathrm{ω}}}_{{2}}$.

 E1 > omega2 := evalDG(Cu[1] &w Cu[2, 2] &w Dx);
 ${\mathrm{ω2}}{≔}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}{,}{2}}$ (2.13)
 E1 > HigherEulerOperators(omega2);
 $\left[{0}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{,}{0}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{,}{2}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{2}{,}{2}}{,}{-}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}{,}{2}}{,}{-}{2}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{2}}{,}{0}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{,}{0}{}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{\left[\right]}{,}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}}\right]$ (2.14)