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

For the first calling sequence, EulerLagrange(L) returns the list of functions $\left[E_1\left(L\right)\, E_2\left(L\right)\, ..\. \, E_m\left(L\right)\right]$on $J^{2k}\left(E\right)$.

The differential forms on the jet spaces $J^k\left(E\right)$ can be bi-graded by their horizontal and vertical/contact degree. A differential form of horizontal degree $n$and vertical degree 0 is called a Lagrangian form or Lagrangian bi-form. In terms of local coordinates on $J^k\left(E\right)$, a Lagrangian bi-form $\mathrm{\λ}$ can be expressed as

The third calling sequence EulerLagrange($\mathrm{\ω}$) returns a list of $m$ differential bi-forms of vertical degree 1 less than the vertical degree of $\mathrm{\ω}$. Here the partial derivatives with respect to the jets of dependent variables $u_{\mathrm{ij} \cdot \cdot \cdot \ell }^{\mathrm{\alpha }}$ in the usual formula for the Euler-Lagrange operator acting on functions are replaced by interior products of the corresponding vector fields, that is,

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

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{JetCalculus}\right)\:$

$\mathrm{DGsetup}\left(\left[t\right]\,\left[u\,v\,w\right]\,E\,2\right)\:$

$L≔\frac{1\left(u_1^2\+v_1^2\+w_1^2\right)}{2}-V\left(u_{\[\]}\,v_{\[\]}\,w_{\[\]}\right)$

$L≔\frac{u_1^2}{2}+\frac{v_1^2}{2}+\frac{w_1^2}{2}-V\left(u_{\left[\right]}\,v_{\left[\right]}\,w_{\left[\right]}\right)$

$\mathrm{EL}≔\mathrm{EulerLagrange}\left(L\right)$

$\mathrm{EL}≔\left[-V_{u_{\left[\right]}}-u_{1,1}\,-V_{v_{\left[\right]}}-v_{1,1}\,-V_{w_{\left[\right]}}-w_{1,1}\right]$

$\mathrm{convert}\left(\mathrm{EL}\,\mathrm{DGdiff}\right)$

$\left[-{\mathrm{D}}_1\left(V\right)\left(u\left(t\right)\,v\left(t\right)\,w\left(t\right)\right)-u_{t,t}\,-{\mathrm{D}}_2\left(V\right)\left(u\left(t\right)\,v\left(t\right)\,w\left(t\right)\right)-v_{t,t}\,-{\mathrm{D}}_3\left(V\right)\left(u\left(t\right)\,v\left(t\right)\,w\left(t\right)\right)-w_{t,t}\right]$

$\mathrm{\λ}≔L\&mult\mathrm{Dt}$

$\mathrm{\lambda }≔\left(\frac{u_1^2}{2}+\frac{v_1^2}{2}+\frac{w_1^2}{2}-V\left(u_{\left[\right]}\,v_{\left[\right]}\,w_{\left[\right]}\right)\right)\mathrm{Dt}$

$\mathrm{EulerLagrange}\left(\mathrm{\λ}\right)$

$\left(V_{u_{\left[\right]}}+u_{1,1}\right)\mathrm{Dt}\bigwedge {\mathrm{Cu}}_{\left[\right]}+\left(V_{v_{\left[\right]}}+v_{1,1}\right)\mathrm{Dt}\bigwedge {\mathrm{Cv}}_{\left[\right]}+\left(V_{w_{\left[\right]}}+w_{1,1}\right)\mathrm{Dt}\bigwedge {\mathrm{Cw}}_{\left[\right]}$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[u\right]\,E\,2\right)\:$

$\mathrm{L2}≔F\left(x\,u_{\[\]}\,u_1\,u_{1\,1}\right)\:$

${\mathrm{PDEtools}}_{\mathrm{declare}}\left(F\left(x\,u_{\[\]}\,u_1\,u_{1\,1}\right)\,\mathrm{quiet}\right)$

$\mathrm{Eul1}≔\mathrm{EulerLagrange}\left(\mathrm{L2}\right)$

$\mathrm{Eul1}≔\left[F_{u_{\left[\right]}}-F_{x,u_1}-F_{u_{\left[\right]},u_1}{u}_1-F_{u_1,u_1}{u}_{1,1}-F_{u_1,u_{1,1}}{u}_{1,1,1}+F_{x,x,u_{1,1}}+F_{x,u_{\left[\right]},u_{1,1}}{u}_1+F_{x,u_1,u_{1,1}}{u}_{1,1}+F_{x,u_{1,1},u_{1,1}}{u}_{1,1,1}+\left(F_{u_{\left[\right]},u_{\left[\right]},u_{1,1}}{u}_1+F_{u_{\left[\right]},u_1,u_{1,1}}{u}_{1,1}+F_{u_{\left[\right]},u_{1,1},u_{1,1}}{u}_{1,1,1}+F_{x,u_{\left[\right]},u_{1,1}}\right)u_1+\left(F_{u_{\left[\right]},u_1,u_{1,1}}{u}_1+F_{u_1,u_1,u_{1,1}}{u}_{1,1}+F_{u_1,u_{1,1},u_{1,1}}{u}_{1,1,1}+F_{u_{\left[\right]},u_{1,1}}+F_{x,u_1,u_{1,1}}\right)u_{1,1}+\left(F_{u_{\left[\right]},u_{1,1},u_{1,1}}{u}_1+F_{u_1,u_{1,1},u_{1,1}}{u}_{1,1}+F_{u_{1,1},u_{1,1},u_{1,1}}{u}_{1,1,1}+F_{u_1,u_{1,1}}+F_{x,u_{1,1},u_{1,1}}\right)u_{1,1,1}+F_{u_{1,1},u_{1,1}}{u}_{1,1,1,1}\right]$

$\mathrm{P0}\,\mathrm{P1}\,\mathrm{P2}≔\frac{\partial }{\partial u_{\[\]}}\mathrm{L2}\,\frac{\partial }{\partial u_1}\mathrm{L2}\,\frac{\partial }{\partial u_{1\,1}}\mathrm{L2}$

$\mathrm{P0},\mathrm{P1},\mathrm{P2}≔F_{u_{\left[\right]}},F_{u_1},F_{u_{1,1}}$

$\mathrm{Eul2}≔\mathrm{P0}-\mathrm{TotalDiff}\left(\mathrm{P1}\,\left[1\right]\right)\+\mathrm{TotalDiff}\left(\mathrm{P2}\,\left[1\,1\right]\right)$

$\mathrm{Eul2}≔F_{u_{\left[\right]}}-F_{x,u_1}-F_{u_{\left[\right]},u_1}{u}_1-F_{u_1,u_1}{u}_{1,1}-F_{u_1,u_{1,1}}{u}_{1,1,1}+F_{x,x,u_{1,1}}+F_{x,u_{\left[\right]},u_{1,1}}{u}_1+F_{x,u_1,u_{1,1}}{u}_{1,1}+F_{x,u_{1,1},u_{1,1}}{u}_{1,1,1}+\left(F_{u_{\left[\right]},u_{\left[\right]},u_{1,1}}{u}_1+F_{u_{\left[\right]},u_1,u_{1,1}}{u}_{1,1}+F_{u_{\left[\right]},u_{1,1},u_{1,1}}{u}_{1,1,1}+F_{x,u_{\left[\right]},u_{1,1}}\right)u_1+\left(F_{u_{\left[\right]},u_1,u_{1,1}}{u}_1+F_{u_1,u_1,u_{1,1}}{u}_{1,1}+F_{u_1,u_{1,1},u_{1,1}}{u}_{1,1,1}+F_{u_{\left[\right]},u_{1,1}}+F_{x,u_1,u_{1,1}}\right)u_{1,1}+\left(F_{u_{\left[\right]},u_{1,1},u_{1,1}}{u}_1+F_{u_1,u_{1,1},u_{1,1}}{u}_{1,1}+F_{u_{1,1},u_{1,1},u_{1,1}}{u}_{1,1,1}+F_{u_1,u_{1,1}}+F_{x,u_{1,1},u_{1,1}}\right)u_{1,1,1}+F_{u_{1,1},u_{1,1}}{u}_{1,1,1,1}$

$\mathrm{Eul2}-{\mathrm{Eul1}}_1$

$0$

$\mathrm{Preferences}\left(''JetNotation''\,''JetNotation2''\right)$

$''JetNotation1''$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[u\right]\,E\,2\right)\:$

$\mathrm{L2}≔F\left(x\,u_0\,u_1\,u_2\right)\:$

${\mathrm{PDEtools}}_{\mathrm{declare}}\left(F\left(x\,u_{\[\]}\,u_1\,u_2\right)\,\mathrm{quiet}\right)$

$\mathrm{Eul1}≔\mathrm{EulerLagrange}\left(\mathrm{L2}\right)$

$\mathrm{Eul1}≔\left[F_{u_0}-F_{x,u_1}-F_{u_0,u_1}{u}_1-F_{u_1,u_1}{u}_2-F_{u_1,u_2}{u}_3+F_{x,x,u_2}+F_{x,u_0,u_2}{u}_1+F_{x,u_1,u_2}{u}_2+F_{x,u_2,u_2}{u}_3+\left(F_{u_0,u_0,u_2}{u}_1+F_{u_0,u_1,u_2}{u}_2+F_{u_0,u_2,u_2}{u}_3+F_{x,u_0,u_2}\right)u_1+\left(F_{u_0,u_1,u_2}{u}_1+F_{u_1,u_1,u_2}{u}_2+F_{u_1,u_2,u_2}{u}_3+F_{u_0,u_2}+F_{x,u_1,u_2}\right)u_2+\left(F_{u_0,u_2,u_2}{u}_1+F_{u_1,u_2,u_2}{u}_2+F_{u_2,u_2,u_2}{u}_3+F_{u_1,u_2}+F_{x,u_2,u_2}\right)u_3+F_{u_2,u_2}{u}_4\right]$

$\mathrm{Preferences}\left(''JetNotation''\,''JetNotation1''\right)$

$''JetNotation2''$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\left[u\right]\,E\,1\right)\:$

$\mathrm{L3}≔\frac{1\left(u_1^2\+u_2^2\+u_3^2\right)}{2}$

$\mathrm{L3}≔\frac{u_1^2}{2}+\frac{u_2^2}{2}+\frac{u_3^2}{2}$

$\mathrm{E3}≔\mathrm{EulerLagrange}\left(\mathrm{L3}\right)$

$\mathrm{E3}≔\left[-u_{1,1}-u_{2,2}-u_{3,3}\right]$

$\mathrm{convert}\left({\mathrm{E3}}_1\,\mathrm{DGdiff}\right)$

$-u_{x,x}-u_{y,y}-u_{z,z}$

$\mathrm{\λ3}≔\mathrm{evalDG}\left(\mathrm{L3}\left(\mathrm{Dx}\&w\mathrm{Dy}\right)\&w\mathrm{Dz}\right)$

$\mathrm{\λ3}≔\left(\frac{u_1^2}{}\right)$