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DifferentialGeometry

  

DGsetup

  

set up a coordinate system, a frame, a Lie algebra, define a set of abstract forms

 

Calling Sequence

Parameters

Description

Examples

Calling Sequence

DGsetup(varlist1, framename, options)

DGsetup(varlist1, varlist2, framename, options)

DGsetup(varlist1, varlist2, framename, jetorder, options)

DGsetup(framedata, options)

DGsetup(Liealgebradata, options)

DGsetup(alg, rho, V)

DGsetup(abstractforms, streqn, framename)

 

Parameters

varlist1

-

a list of unassigned Maple names

varlist2

-

a list of unassigned Maple names

framename

-

an unassigned Maple name or a string

jetorder

-

a positive integer

framedata

-

the structure equations for an anholonomic frame, as calculated by the procedure FrameData

Liealgebradata

-

the structure equations for a Lie algebra, as calculated by the procedure LieAlgebraData

abstractforms

-

a list specifying a set of abstract forms, without reference to any underlying set of coordinates

streqn

-

a list of structure equations for the exterior derivatives and interior products of an abstract form

options

-

a list of frame labels, a list of co-frame labels, the keyword 'quiet' or 'verbose'

Description

• 

All computational sessions with the DifferentialGeometry package begin with a call to the DGsetup command.  This command fixes the coordinate names for the manifold being defined; specifies the vectors and 1-forms to be used as local frames and co-frames on the manifold; and stores all the computational rules needed to work with the given frame or co-frame.  It is also used by the JetCalculus package to initialize a jet space to any given order and by the LieAlgebras package to prepare for computations with Lie algebras.

• 

The following table summarizes the different calling sequences for DGsetup.

Ex 1.

Create a two-dimensional manifold M with local coordinates x,y.

 

    > DGsetup([x, y], M)

Ex 2.

Create a fiber bundle E with fiber coordinates u,v over a three-dimensional base space with coordinates x,y,z.

 

    > DGsetup([x, y, z], [u,v], M);

Ex 3.

Create the 3rd order jet space J  for 2 independent variables x,y  and 1 dependent variable u.

 

    > DGsetup([x, y], [u], J, 3);

Ex 4a.

Perform calculations on a three-dimensional manifold N in terms of an anholonomic frame F. The command FrameData is used to calculate the structure equations for the frame and this is passed to DGsetup.

 

    > DGsetup([x, y, z], M);

 

    > F := evalDG([D_x, x*D_x + D_y, y D_x + D_z]);

 

    > FD := FrameData(F, N);

 

    > DGsetup(FD);

Ex 4b.

Perform calculations on a three-dimensional manifold N in terms of an anholonomic frame F. Label the frame vectors X,Y,Z and the co-frame 1-forms α,β,σ.

 

    > DGsetup(FD, [X, Y, Z], [alpha, beta, sigma]);

Ex 5.

Perform calculations on a three-dimensional manifold N in terms of an anholonomic co-frame Ω. The command FrameData is used to calculate the structure equations for the frame and this is passed to DGsetup.

 

    > DGsetup([x, y, z], M);

 

    > Omega := evalDG([dx, x*dx + dy, y*dx + dz]);

 

    > FD := FrameData(Omega, N);

 

    > DGsetup(FD);

 Ex 6a.

Initialize a Lie algebra alg1 defined by a set of 3 matrices A=M1,M2,M3. Use LieAlgebraData to calculate the structure equations for the Lie algebra and pass this result to DGsetup.

 

    > A := [Matrix([[1, 0], [0, 0]]), Matrix([[0, 1], [0, 0]]), Matrix([[0, 0],[0, 1]])];

 

    > LD := LieAlgebraData(A, alg1);

 

    > DGsetup(LD);

 Ex 6b.

Initialize a Lie algebra alg1 defined by a set of 3 matrices. Label the basis elements for the Lie algebra as f1,f2,f3 and the dual 1-forms by ξ1,ξ2,ξ3

 

    > DGsetup(LD, [f], [xi]):

 Ex 7.

Initialize a Lie algebra alg1 defined by a set of 3 vector fields Γ=X1,X2,X3. Use LieAlgebraData to calculate the structure equations for the Lie algebra and passed this result to DGsetup.

 

    > DGsetup([x, y], M);

 

    > Gamma : = evalDG([D_x, D_y, y*D_x - x*D_y]);

 

    > LD := LieAlgebraData(A, alg1):

 

    > DGsetup(LD);

 Ex 8.

Initialize a Lie algebra alg1 from a set of structure equations. For other ways to initialize a Lie algebra, see LieAlgebraData.

 

    > B := [x, y, h];

 

    > S := [[h, x] = 2*x, [h, y] = -2*y, [x, y] = h];

 

    > LD := LieAlgebraData( S, B, alg);

 

    > DGsetup(LD);

 Ex 9.

Initialize a classical simple Lie algebra, say sl3, the Lie algebra of trace-free matrices. See SimpleLieAlgebraData.

 

    > LD := SimpleLieAlgebraData(sl(3), alg);

 

    > DGsetup(LD);

Ex 10.

Initialize a Lie algebra with coefficients in a representation.

 

    > LD := LieAlgebraData( [h,x] = 2x, [h,y] = -2y, [x,y] =h], [x, y, h], alg);

 

    > DGsetup(LD);

 

    > DGsetup([x1, x2, x3], V);

 

    > A := Adjoint();

 

    > rho := Representation(alg, V, Adjoint());

 

    > DGsetup(alg, rho, R):

Ex 11.

Initialize a set of abstract differential forms with a given set of structure equations.

 

    > DGsetup([f = dgform(0), alpha = dgform(1), beta = dgfom(2)], [d(beta) = alpha &w beta], M)

Ex 12.

Initialize an abstract co-frame and set of abstract differential forms with a given set of structure equations.

 

    > DGsetup([[alpha, beta], f = dgform(0),  sigma = dgform(2)], [d(alpha) = f sigma, hook(D_alpha, sigma) = beta], M):

Examples

withDifferentialGeometry:withLieAlgebras:

 

Example 1.  Use the first calling sequence to set up a coordinate system on a manifold.

In this first example, we create a two-dimensional manifold M with coordinates [x, y].

DGsetupx,y,M,verbose

The following coordinates have been protected:

x,y

The following vector fields have been defined and protected:

_DGvector,M,,1,1,_DGvector,M,,2,1

The following differential 1-forms have been defined and protected: