The celebrated theorem of E. Noether provides a formula for the calculation of a first integral or conservation law for any symmetry of the Lagrangian. This formula, which is very complicated for high order Lagrangians is easily implemented using the horizontal homotopy for the variational bicomplex.

Within the framework of the JetCalculus package conservation laws are represented by differential forms of degree $n-1$, where $n$ is the dimension of $M$, whose horizontal or total exterior derivative vanishes by virtue of the Euler Lagrange equations.

The vector field $X$ is a symmetry of the Lagrangian $\mathrm{\λ}$ if the Lie derivative of l with respect to the prolongation of $X$ (to the order of λ) vanishes.

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

with(DifferentialGeometry): with(JetCalculus):

DGsetup([t], [x,y], E, 2):

L := 1/2*(x[1]^2 + y[1]^2 -1/sqrt(x[]^2 + y[]^2));

$L≔\frac{x_1^2}{2}+\frac{y_1^2}{2}-\frac{1}{2\sqrt{x_{\left[\right]}^2+y_{\left[\right]}^2}}$

EL := EulerLagrange(L);

$\mathrm{EL}≔\left[\frac{x_{\left[\right]}}{2{\left(x_{\left[\right]}^2+y_{\left[\right]}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-x_{1,1}\,\frac{y_{\left[\right]}}{2{\left(x_{\left[\right]}^2+y_{\left[\right]}^2\right)}^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-y_{1,1}\right]$

Eq := solve(EL, {x[1,1], y[1,1]});

$\mathrm{Eq}≔\left\{x_{1,1}=\frac{\sqrt{x_{\left[\right]}^2+y_{\left[\right]}^2}{x}_{\left[\right]}}{2\left(x_{\left[\right]}^4+2x_{\left[\right]}^2{y}_{\left[\right]}^2+y_{\left[\right]}^4\right)}\,y_{1,1}=\frac{\sqrt{x_{\left[\right]}^2+y_{\left[\right]}^2}{y}_{\left[\right]}}{2\left(x_{\left[\right]}^4+2x_{\left[\right]}^2{y}_{\left[\right]}^2+y_{\left[\right]}^4\right)}\right\}$

lambda := L &mult Dt;

$\mathrm{\lambda }≔\frac{x_1^2\sqrt{x_{\left[\right]}^2+y_{\left[\right]}^2}+y_1^2\sqrt{x_{\left[\right]}^2+y_{\left[\right]}^2}-1}{2\sqrt{x_{\left[\right]}^2+y_{\left[\right]}^2}}\mathrm{Dt}$

X:= x[]*D_y[] - y[]*D_x[];

$X≔-{\mathrm{D\_x}}_{\left[\right]}{y}_{\left[\right]}+{\mathrm{D\_y}}_{\left[\right]}{x}_{\left[\right]}$

X1:= Prolong(X,1);

$\mathrm{X1}≔-y_{\left[\right]}{\mathrm{D\_x}}_{\left[\right]}+x_{\left[\right]}{\mathrm{D\_y}}_{\left[\right]}-y_1{\mathrm{D\_x}}_1+x_1{\mathrm{D\_y}}_1$

LieDerivative(X, lambda);

$0\mathrm{Dt}$

F :=Noether(X, lambda);

$F≔-y_1{x}_{\left[\right]}+x_1{y}_{\left[\right]}$

tdF := TotalDiff(F, t);

$\mathrm{tdF}≔-x_{\left[\right]}{y}_{1,1}+y_{\left[\right]}{x}_{1,1}$

eval(tdF, Eq);

$0$

with(GroupActions):

DGsetup([x, y, t], [u], J, 1);

$\mathrm{frame\; name:\; J}$

lambda := evalDG((u[1]^2 + u[2]^2 - u[3]^2)*Dx &w Dy &w Dt);

$\mathrm{\lambda }≔\left(u_1^2+u_2^2-u_3^2\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge \mathrm{Dt}$

Gamma := InfinitesimalSymmetriesOfGeometricObjectFields([lambda], output="list");

$\mathrm{\Γ}≔\left[-2x\mathrm{D\_x}-2y\mathrm{D\_y}-2t\mathrm{D\_t}+u_{\left[\right]}{\mathrm{D\_u}}_{\left[\right]}\,{\mathrm{D\_u}}_{\left[\right]}\,t\mathrm{D\_x}+x\mathrm{D\_t}\,t\mathrm{D\_y}+y\mathrm{D\_t}\,\mathrm{D\_t}\,-y\mathrm{D\_x}+x\mathrm{D\_y}\,\mathrm{D\_y}\,\mathrm{D\_x}\right]$

X1 := D_u[];

$\mathrm{X1}≔{\mathrm{D\_u}}_{\left[\right]}$

omega1 := Noether(X1, lambda);

$\mathrm{\ω1}≔2u_3\mathrm{Dx}\bigwedge \mathrm{Dy}+2u_2\mathrm{Dx}\bigwedge \mathrm{Dt}-2u_1\mathrm{Dy}\bigwedge \mathrm{Dt}$

HorizontalExteriorDerivative(omega1);

$-\left(2u_{1,1}+2u_{2,2}-2u_{3,3}\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge \mathrm{Dt}$

X2 := evalDG(x*D_x +y*D_y +t*D_t -1/2*u[]*D_u[]);

$\mathrm{X2}≔x\mathrm{D\_x}+y\mathrm{D\_y}+t\mathrm{D\_t}-\frac{u_{\left[\right]}}{2}{\mathrm{D\_u}}_{\left[\right]}$

omega2 := Noether(X2, lambda);

HorizontalExteriorDerivative(omega1);

$\mathrm{\ω2}≔-\left(t{u}_1^2+t{u}_2^2+t{u}_3^2+2x{u}_1{u}_3+2y{u}_2{u}_3+u_{\left[\right]}{u}_3\right)\mathrm{Dx}\bigwedge \mathrm{Dy}-\left(2t{u}_2{u}_3+2x{u}_1{u}_2-y{u}_1^2+y{u}_2^2+y{u}_3^2+u_{\left[\right]}{u}_2\right)\mathrm{Dx}\bigwedge \mathrm{Dt}+\left(2t{u}_1{u}_3+x{u}_1^2-x{u}_2^2+x{u}_3^2+2y{u}_1{u}_2+u_{\left[\right]}{u}_1\right)\mathrm{Dy}\bigwedge \mathrm{Dt}$

$-\left(2u_{1,1}+2u_{2,2}-2u_{3,3}\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge \mathrm{Dt}$

factor(HorizontalExteriorDerivative(omega2));

X3 := evalDG(x*D_t +t*D_x);

$\mathrm{X3}≔t\mathrm{D\_x}+x\mathrm{D\_t}$

omega3 := Noether(X3, lambda);

$\mathrm{\ω3}≔-\left(2t{u}_1{u}_3+x{u}_1^2+x{u}_2^2+x{u}_3^2\right)\mathrm{Dx}\bigwedge \mathrm{Dy}-2u_2\left(t{u}_1+x{u}_3\right)\mathrm{Dx}\bigwedge \mathrm{Dt}+\left(t{u}_1^2-t{u}_2^2+t{u}_3^2+2x{u}_1{u}_3\right)\mathrm{Dy}\bigwedge \mathrm{Dt}$

factor(HorizontalExteriorDerivative(omega3));