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with(DifferentialGeometry): with(JetCalculus):
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Example 1.
We set a simple single integral problem with 2 dependent variables and compute the Euler-Lagrange equations
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DGsetup([t], [x,y], E, 2):
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E >
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L := 1/2*(x[1]^2 + y[1]^2 -1/sqrt(x[]^2 + y[]^2));
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| (2.1) |
E >
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EL := EulerLagrange(L);
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| (2.2) |
E >
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Eq := solve(EL, {x[1,1], y[1,1]});
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| (2.3) |
The Lagrangian is invariant under rotations in the plane. Let us check this. To be technically correct we should work with the differential 1-form defined by .
E >
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lambda := L &mult Dt;
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| (2.4) |
E >
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X:= x[]*D_y[] - y[]*D_x[];
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| (2.6) |
E >
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LieDerivative(X, lambda);
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Now we find the first integral associated to the symmetry :
E >
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F :=Noether(X, lambda);
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To check that this is indeed a first integral, take the total derivative of with respect to and substitute from the Euler-Lagrange equations.
E >
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tdF := TotalDiff(F, t);
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Example 2.
We use the command InfinitesimalSymmetriesOfGeometricObjectFields to find the symmetries of the Lagrangian for the wave equation in (2+1) dimensions.
We then use the command Noether to calculate the associated conservation laws.
E >
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DGsetup([x, y, t], [u], J, 1);
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J >
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lambda := evalDG((u[1]^2 + u[2]^2 - u[3]^2)*Dx &w Dy &w Dt);
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| (2.12) |
J >
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Gamma := InfinitesimalSymmetriesOfGeometricObjectFields([lambda], output="list");
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| (2.13) |
Let us find the conservation law associated to the symmetry of infinitesimal translations in the dependent variable . We check the horizontal exterior derivative of vanishes on solutions to the 2+1 wave equation.
J >
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omega1 := Noether(X1, lambda);
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| (2.15) |
J >
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HorizontalExteriorDerivative(omega1);
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| (2.16) |
Let us find the conservation law associated to the symmetry of infinitesimal scaling of the independent and dependent variables. We check the horizontal exterior derivative of vanishes on solutions to the 2+1 wave equation.
J >
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X2 := evalDG(x*D_x +y*D_y +t*D_t -1/2*u[]*D_u[]);
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| (2.17) |
J >
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omega2 := Noether(X2, lambda);
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J >
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HorizontalExteriorDerivative(omega1);
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| (2.18) |
J >
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factor(HorizontalExteriorDerivative(omega2));
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Finally, let us find the conservation law associated to the symmetry of infinitesimal boosts in the independent variables and . We check that the horizontal exterior derivative of vanishes on solutions to the 2+1 wave equation.
J >
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X3 := evalDG(x*D_t +t*D_x);
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J >
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omega3 := Noether(X3, lambda);
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| (2.21) |
J >
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factor(HorizontalExteriorDerivative(omega3));
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