find the conservation law for the Euler-Lagrange equations from a given symmetry of the Lagrangian - Maple Programming Help

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JetCalculus[Noether] - find the conservation law for the Euler-Lagrange equations from a given symmetry of the Lagrangian

Calling Sequences

     Noether(X, l)

Parameters

     X         -  a vector field representing a symmetry of the Lagrangian λ

     λ         - a Lagrangian for a variational principle, defined as a top degree horizontal form of the jet space JkE of a bundle π:EM

 

Description

Examples

Description

• 

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 n1, 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 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(...).

Examples

withDifferentialGeometry:withJetCalculus:

 

Example 1.

We set a simple single integral problem with 2 dependent variables and compute the Euler-Lagrange equations

DGsetupt,x,y,E,2:

E > 

L1x12+y121x[]2+y[]22

Lx122+y12212x2+y2

(2.1)
E > 

ELEulerLagrangeL

ELx2x2+y232x1,1,y2x2+y232y1,1

(2.2)
E > 

EqsolveEL,x1,1,y1,1

Eqx1,1=x2+y2x2x4+2x2y2+y4,y1,1=x2+y2y2x4+2x2y2+y4

(2.3)

 

The Lagrangian L is invariant under rotations in the xy plane. Let us check this. To be technically correct we should work with the differential 1-form λ defined by L.

E > 

λL &mult Dt

λx12x2+y2+y12x2+y212x2+y2Dt

(2.4)
E > 

Xx[]D_y[]y[]D_x[]

XD_xy+D_yx

(2.5)
E > 

X1ProlongX,1

X1yD_x+xD_yy1D_x1+x1D_y1

(2.6)
E > 

LieDerivativeX,λ

0Dt

(2.7)

Now we find the first integral associated to the symmetry X:

E > 

FNoetherX,λ

Fy1x+x1y

(2.8)

 

To check that this is indeed a first integral, take the total derivative of F with respect to t and substitute from the Euler-Lagrange equations.

E > 

tdFTotalDiffF,t

tdFxy1,1+yx1,1

(2.9)
E > 

evaltdF,Eq

0

(2.10)

 

 

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 > 

withGroupActions:

E > 

DGsetupx,y,t,u,J,1

frame name: J

(2.11)
J > 

λevalDGu12+u22u32Dx &w Dy &w Dt

λu12+u22u32DxDyDt

(2.12)
J > 

ΓInfinitesimalSymmetriesOfGeometricObjectFieldsλ,output=list

Γ2xD_x2yD_y2tD_t+uD_u,D_u,tD_x+xD_t,tD_y+yD_t,D_t,yD_x+xD_y,D_y,D_x

(2.13)

 

Let us find the conservation law ω1 associated to the symmetry of infinitesimal translations in the dependent variable u. We check the horizontal exterior derivative of ω1 vanishes on solutions to the 2+1 wave equation.

J > 

X1D_u[]

X1D_u

(2.14)
J > 

ω1NoetherX1,λ

ω12u3DxDy+2u2DxDt2u1DyDt

(2.15)
J > 

HorizontalExteriorDerivativeω1

2u1,1+2u2,22u3,3DxDyDt

(2.16)

 

Let us find the conservation law ω2 associated to the symmetry of infinitesimal scaling of the independent and dependent variables. We check the the horizontal exterior derivative of ω2 vanishes on solutions to the 2+1 wave equation.

J > 

X2evalDGxD_x+yD_y+tD_t1u[]D_u[]2

X2xD_x+yD_y+tD_tu2D_u

(2.17)
J > 

ω2NoetherX2,λ

J > 

HorizontalExteriorDerivativeω1

ω2tu12+tu22+tu32+2xu1u3+2yu2u3+uu3DxDy2tu2u3+2xu1u2yu12+yu22+yu32+uu2DxDt+2tu1u3+xu12xu22+xu32+2yu1u2+uu1DyDt

2u1,1+2u2,22u3,3DxDyDt

(2.18)
J > 

factorHorizontalExteriorDerivativeω2

u1,1+u2,2u3,32u3t+2u1x+2u2y+uDxDyDt

(2.19)

 

Finally, let us find the conservation law ω3 associated to the symmetry of infinitesimal boosts in the independent variables x and t. We check that the horizontal exterior derivative of ω3 vanishes on solutions to the 2+1 wave equation.

J > 

X3evalDGxD_t+tD_x

X3tD_x+xD_t

(2.20)
J > 

ω3NoetherX3,λ

ω32tu1u3+xu12+xu22+xu32DxDy2u2tu1+xu3DxDt+tu12tu22+tu32+2xu1u3DyDt

(2.21)
J > 

factorHorizontalExteriorDerivativeω3

2u1,1+u2,2u3,3tu1+xu3DxDyDt

(2.22)

 

See Also

DifferentialGeometry

JetCalculus

EulerLagrange

HorizontalExteriorDerivative

InfinitesimalSymmetriesOfGeometricObjectFields

Prolong

TotalDiff

 


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