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DifferentialGeometry[DGsetup] - set up a coordinate system, a frame, a Lie algebra, define a set of abstract forms

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 2-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 3-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 3-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 3-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 3-dimensional manifold N in terms of an anholonomic co-frame Ω. The command  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 up a coordinate system on a manifold.

In this first example, we create a 2-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:

`*`D_x,`*`D_y

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

`*`dx,`*`dy

frame name: M

(1)

The coordinates are now protected names which cannot be assigned values.

The standard coordinate vectors D_x, D_y and dual 1 forms dx, dy have been defined and protected.  We display the internal representation of these vectors just to show that they have been assigned values. (Knowledge of this internal representation of various objects used in the DifferentialGeometry package is not required of the user).

lprintD_x:

_DG([["vector", M, []], [[[1], 1]]])

lprintdx

_DG([["form", M, 1], [[[1], 1]]])

Without the option 'verbose', the information concerning the definition and protection of variable is suppressed.  This behavior of the DGsetup command can be controlled globally with the Preferences command.

DGsetupu,v,N

frame name: N

(2)

 

Example 2.  Use the second calling sequence to set up coordinates on a vector or fiber bundle.

In many situations in differential geometry, one deals with vector bundles or fiber bundles pi: E -> M.  For these applications, use the second calling sequence to DGsetup.  The

first argument varlist1 should be coordinates for the basis manifold M and the second argument should be the list of coordinates for the fiber of the bundle E.  Here is the command

for creating a rank-2 bundle over a 3-dimensional space.  The base coordinates are [x, y, z] and the fiber coordinates are [u, v].

DGsetupx,y,z,u,v,P,verbose

The following coordinates have been protected:

x,y,z,u,v

The following vector fields have been defined and protected:

`*`D_x,`*`D_y,`*`D_z,`*`D_u,`*`D_v

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

`*`dx,`*`dy,`*`dz,`*`du,`*`dv

frame name: P

(3)

This particular use of DGsetup could be used, for example, to study a 3-dimensional manifold imbedded in a 5-dimensional Riemannian manifold, where the vectors D_u and D_v define a local basis for the normal bundle.

 

Example 3.  Use the third calling sequence to set up coordinates on a jet bundle. The JetCalculus subpackage of the DifferentialGeometry package contains an extensive set of commands for symbolic computations on jet spaces.  These spaces provide the natural mathematical setting for the geometric approach to the calculus of variations and to differential equations.  The third calling sequence for the DGsetup command will create a jet space over a bundle pi: E -> M with prescribed lists of independent variables (coordinates on M) and dependent variables (fiber coordinates on E). In addition to the coordinate vectors and coordinate differential forms on jet space, DGsetup also defines and protects the contact forms on jet space.  See the JetCalculus overview page for additional information regarding these differential forms. The user may specify which of two notational conventions are to be used to denote the jet coordinates.  See the Preferences command for details.

First we initialize the 3rd order jet space for 1 independent and 2 dependent variables.

DGsetupt,x,y,J12,3,verbose

The following coordinates have been protected:

t,x[],y[],x1,y1,x1,1,y1,1,x1,1,1,y1,1,1

The following vector fields have been defined and protected:

`*`D_t,`*`D_x,`*`D_y,`*`D_x1,`*`D_y1,`*`D_x1,1,`*`D_y1,1,`*`D_x1,1,1,`*`D_y1,1,1

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

`*`dt,`*`dx,`*`dy,`*`dx1,`*`dy1,`*`dx1,1,`*`dy1,1,`*`dx1,1,1,`*`dy1,1,1

The following type [1,0] biforms have been defined and protected::

`*`Dt

The following type [0,1] biforms (contact 1-forms) have been defined and protected::

`*`Cx,`*`Cy,`*`Cx1,`*`Cy1,`*`Cx1,1,`*`Cy1,1,`*`Cx1,1,1,`*`Cy1,1,1

frame name: J12

(4)

Here is the initialization of the 3rd order jet space for 2 independent and 1 dependent variable.

DGsetupx,y,U,J21,3,verbose

The following coordinates have been protected:

x,y,U[],U1,U2,U1,1,U1,2,U2,2,U1,1,1,U1,1,2,U1,2,2,U2,2,2

The following vector fields have been defined and protected:

`*`D_x,`*`D_y,`*`D_U,`*`D_U1,`*`D_U2,`*`D_U1,1,`*`D_U1,2,`*`D_U2,2,`*`D_U1,1,1,`*`D_U1,1,2,`*`D_U1,2,2,`*`D_U2,2,2

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

`*`dx,`*`dy,`*`dU,`*`dU1,`*`dU2,`*`dU1,1,`*`dU1,2,`*`dU2,2,`*`dU1,1,1,`*`dU1,1,2,`*`dU1,2,2,`*`dU2,2,2

The following type [1,0] biforms have been defined and protected::

`*`Dx,`*`Dy

The following type [0,1] biforms (contact 1-forms) have been defined and protected::

`*`CU,`*`CU1,`*`CU2,`*`CU1,1,`*`CU1,2,`*`CU2,2,`*`CU1,1,1,`*`CU1,1,2,`*`CU1,2,2,`*`CU2,2,2

frame name: J21

(5)

 

Example 4.  Use the fourth calling sequence to set up an anholonomic frame on a manifold.

It is extremely important, for many calculations in DifferentialGeometry, to work with frames and co-frames other than the coordinate frame (e.g. D_x, D_y, ...) and the coordinate co-frame (e.g. dx, dy ...). For example, curvature tensor calculations are often greatly simplified by working in an orthonormal frame. To set up a frame for subsequent computations with the DifferentialGeometry package, first use the FrameData command to calculate the structure equations for the frame or co-frame.  Then pass the output of the FrameData command to DGsetup.

DGsetupx,y,z,M

frame name: M

(6)

The three vectors in the list F are the vectors that will be used as a frame.

F:=evalDGD_x,D_y+xD_x,D_z+yD_x

F:=`*`D_x,xD_x+`*`D_y,yD_x+`*`D_z

(7)

The command FrameData calculates the structure equations for this frame. These structure equations are displayed with the convention that the first element of the frame is E1, the second element is E2, and so on.  The name of the manifold with coordinates [x, y, z] and anholonomic frame F is M1.

StructureEquations:=FrameDataF,M1

StructureEquations:=E1,E2=E1,E2,E3=1yE1

(8)

Pass the output of FrameData to the command DGsetup.  Now in place of the coordinate vectors D_x, D_y, D_z and coordinate form dx, dy, dz, we shall use the vectors F[1], F[2], F[3] and the dual 1 forms.  The default labels for the frame elements are E1, E2, E3 and the default labels for the co-frame elements are Theta1, Theta2, Theta3.

DGsetupStructureEquations,verbose

The following coordinates have been protected:

x,y,z

The following vector fields have been defined and protected:

`*`E1,`*`E2,`*`E3

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

`*`Θ1,`*`Θ2,`*`Θ3

frame name: M1

(9)

As sample calculations on the manifold M1 we have:

LieBracketE2,E3

1+yE1

(10)

ExteriorDerivativeΘ1

`*`Θ1Θ2+1+yΘ2Θ3

(11)

We can specify the labels for the frame and the co-frame as optional arguments to DGsetup.  The second argument is a list of names for the frame elements and the third argument is a list of names for the co-frame elements.  If a list with just one name is given, then that name is used as the kernel name for the list of frame or co-frame elements.  For example:

DGsetupStructureEquations,X,Y,Z,ω,verbose

The following coordinates have been protected:

x,y,z

The following vector fields have been defined and protected:

`*`X,`*`Y,`*`Z

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

`*`ω1,`*`ω2,`*`ω3

frame name: M1

(12)

 

Example 5.  Use the fourth calling sequence to set up an anholonomic co-frame on a manifold.

The same commands used in the previous example can be used to create an anholonomic co-frame on a manifold.

DGsetupx,y,z,M

frame name: M

(13)

The three 1-foms in the list Omega are the 1-forms that will be used as a co-frame.

Ω:=evalDGdx,xdx+dy,ydx+dz

Ω:=`*`dx,xdx+`*`dy,ydx+`*`dz

(14)

The command FrameData calculates the structure equations for this co-frame.  These structure equations are displayed with the convention that the first element of the frame is Θ1, the second element is Θ2, and so on.  The name of the manifold with coordinates [x, y, z] and anholonomic frame Omega is M1.

StructureEquations:=FrameDataΩ,M1

StructureEquations:=dΘ1=0,dΘ2=0,dΘ3=Θ1Θ2

(15)

Pass the output of FrameData to the command DGsetup.  Now in place of the coordinate vectors D_x, D_y, D_z and coordinate form dx, dy, dz, we shall use the 1-forms Omega[1], Omega[2],  Omega[3] and the dual vector fields.  The default labels for the frame elements are E1, E2, E3 and the default labels for the co-frame elements are Theta1, Theta2, Theta3.

DGsetupStructureEquations,verbose

The following coordinates have been protected:

x,y,z

The following vector fields have been defined and protected:

`*`E1,`*`E2,`*`E3

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

`*`Θ1,`*`Θ2,`*`Θ3

frame name: M1

(16)

 

Example 6.  Use the fifth calling sequence to initialize a Lie algebra, defined by a set of matrices

The DGsetup command can also be used to initialize a Lie algebra the same way that DGsetup is used to initialize an anholonomic frame on a manifold.  Typically, one first obtains the structure equations for the Lie algebra using the LieAlgebraData command and then passes these structure equations to DGsetup.  Structure equations for a large number of Lie algebras can also be retrieved from the database of Lie algebras found in the Library package.

M:=Matrix1,0,0,0,Matrix0,1,0,0,Matrix0,0,0,1

M:=1000,0100,0001

(17)

Calculate the structure equations for the matrix Lie algebra defined by the matrices M.  Here e1 refers to the first matrix in M, e2 the second matrix, ....

StructureEquations:=LieAlgebraDataM,Alg1

StructureEquations:=e1,e2=e2,e2,e3=e2

(18)

The default names for the basis elements for the Lie algebra are e1, e2, e3....  The default names for the basis elements for the dual elements of the Lie algebra are theta1, theta2, ....

DGsetupStructureEquations,verbose

The following vector fields have been defined and protected:

`*`e1,`*`e2,`*`e3

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

`*`