EXCALC
A System for Doing Calculations in the Calculus
of
Modern Differential Geometry
USER's MANUAL
Eberhard Schruefer
April 1986
Acknowledgments
This program was developed over several years. I would like to express
my deep gratitude to Dr. Anthony Hearn for his continuous interest in
this work, and especially for his hospitality and support during a
visit in 1984/85 at the Rand Corporation, where substantial progress
on this package could be achieved. The Heinrich Hertz-Stiftung sup-
ported this visit. Many thanks are also due to Drs. F.W. Hehl, University
of Cologne, and J.D. McCrea, University College Dublin, for their
suggestions and work on testing this program.
Table of Contents
Introduction
1. Declarations
2. Exterior Multiplication
3. Partial Differentiation
4. Exterior Differentiation
5. Inner Product
6. Lie Derivative
7. Hodge-* Duality Operator
8. Variational Derivative
9. Handling of Indices
10. Metric Structures
11. Riemannian Connections
12. Ordering and Structuring
13. Appendix
A1. Summary of Operators and Commands
A2. Examples
INTRODUCTION
EXCALC is designed for easy use by all who are familiar with the
calculus of Modern Differential Geometry. Its syntax is kept as close
as possible to standard textbook notations. Therefore, no great
experience in writing computer algebra programs is required. It is
almost possible to input to the computer the same as what would have
been written down for a handcalculation. For example, the statement
f*x^y + u_|(y^z^x)
would be recognized by the program as a formula involving exterior
products and an inner product. The program is currently able to
handle scalar-valued exterior forms, vectors and operations between
them, as well as non-scalar valued forms (indexed forms). With this,
it should be an ideal tool for studying differential equations,
doing calculations in general relativity and field theories, or doing
simple things as calculating the Laplacian of a tensor field for
an arbitrary given frame. With the increasing popularity of this
calculus, this program should have an application in almost any field
of physics and mathematics.
Since the program is completely embedded in REDUCE, all features and
facilities of REDUCE are available in a calculation. Even for those
who are not quite comfortable in this calculus, there is a good chance
of learning it by just playing with the program.
This is still a very experimental version, and changes of the syntax
are to be expected. The performance of the program can still be
increased considerably.
Complaints and comments are appreciated and should be sent to:
Dr. Eberhard Schruefer
Institut fuer methodische Grundlagen (F1-P)
Gesellschaft fuer Mathematik und Datenverarbeitung mbH
Postfach 1240
Schloss Birlinghoven
D-5205 St. Augustin 1
WEST GERMANY
If the use of this program leads to a publication, this document should
be cited, and a copy of the article should be sent to the above address.
1. Declarations
________________
Geometrical objects like exterior forms or vectors are introduced to the
system by declaration commands. The declarations can appear anywhere in
a program, but must, of course, be made prior to the use of the object.
Everything that has no declaration is treated as a constant; therefore
zero-forms must also be declared.
An exterior form is introduced by
PFORM declaration1,declaration2,....;
where
declaration::=<name>=<number>|<identifier>|<expression>
<name>::=<identifier>|<identifier>(<arguments>)
For example
PFORM U=K,V=4,F=0,W=DIM-1;
would declare U to be an exterior form of degree K, V to be a form of
degree 4, F to be a form of degree 0 (a function), and W to be a form of
degree DIM-1.
If the exterior form should have indices, the declaration would be
PFORM CURV(A,B)=2,CHRIS(A,B)=1;
The name of the indices is arbitrary.
The declaration of vectors is similar. The command TVECTOR takes a list
of names.
Example:
To declare X as a vector and COMM as a vector with two indices, one
would say
TVECTOR X,COMM(A,B);
If a declaration of an already existing name is made, the old
declaration is removed, and the new one is taken.
2. Exterior Multiplication
___________________________
Exterior multiplication between exterior forms is carried out with the
nary infix operator ^ (wedge). Factors are ordered according to the
usual ordering in REDUCE using the commutation rule for exterior
products.
Examples:
PFORM U=1,V=1,W=K;
U^V;
U^V
V^U;
- U^V
U^U;
0
W^U^V;
K
( - 1) *U^V^W
(3*U-A*W)^(W+5*V)^U;
A*(5*U^V^W - U^W^W)
It is possible to declare the dimension of the underlying space by
SPACEDIM <number>|<identifier>;
If an exterior product has a degree higher than the dimension of the
space, it is replaced by 0:
SPACEDIM 4;
PFORM U=2,V=3;
U^V;
0
3. Partial Differentiation
___________________________
Partial differentiation is denoted by the operator @. Its capability is
the same as the REDUCE DF operator.
Examples:
@(SIN X,X);
COS(X)
@(F,X);
0
An identifier can be declared to be a function of certain variables.
This is done with the command _F_D_O_M_A_I_N. The following would tell the
partial differentiation operator that F is a function of the variables X
and Y and that H is a function of X.
FDOMAIN F=F(X,Y),H=H(X);
Applying @ to F and H would result in
@(F,X);
@ F
X
@(X*F,X);
F + X*@ F
X
@(H,Y);
0
The partial derivative symbol can also be an operator with a single
argument. It then represents a natural base element of a tangent
vector.
Example:
A*@ X + B*@ Y;
A*@ + B*@
X Y
4. Exterior Differentiation
____________________________
Exterior differentiation of exterior forms is carried out by the
operator d. Products are normally differentiated out, i.e.
PFORM X=0,Y=K,Z=M;
D(X * Y);
X*d Y + d X^Y
D(R*Y);
R*d Y
D(X*Y^Z);
K
( - 1) *X*Y^d Z + X*d Y^Z + d X^Y^Z
This expansion can be suppressed by the command _N_O_X_P_N_D _D.
NOXPND D;
D(Y^Z);
d(Y^Z)
To obtain a canonical form for an exterior product when the expansion is
switched off, the operator D is shifted to the right if it appears in
the leftmost place.
D Y ^ Z;
K
- ( - 1) *Y^d Z + d(Y^Z)
Expansion is performed again when the command _X_P_N_D _D is executed.
Functions which are implicitly defined by the FDOMAIN command are
expanded into partial derivatives:
PFORM X=0,Y=0,Z=0,F=0;
FDOMAIN F=F(X,Y);
D F;
@ F*d X + @ F*d Y
X Y
If an argument of an implicitly defined function has further
dependencies the chain rule will be applied e.g.
FDOMAIN Y=Y(Z);
D F;
@ F*d X + @ F*@ Y*d Z
X Y Z
Expansion into partial derivatives can be inhibited by _N_O_X_P_N_D @ and
enabled again by _X_P_N_D @.
The operator is of course aware of the rules that a repeated application
always leads to zero and that there is no exterior form of higher degree
than the dimension of the space.
D D X;
0
PFORM U=K;
SPACEDIM K;
D U;
0
5. Inner Product
_________________
The inner product between a vector and an exterior form is represented
by the diphthong _| (underscore or-bar), which is the notation of many
textbooks. If the exterior form is an exterior product, the inner
product is carried through any factor
Example:
PFORM X=0,Y=K,Z=M;
TVECTOR U,V;
U_|(X*Y^Z);
K
X*(( - 1) *Y^U_|Z + U_|Y^Z)
In repeated applications of the inner product to the same exterior form
the vector arguments are ordered e.g.
(U+X*V)_|(U_|(3*Z));
- 3*U_|V_|Z
The duality of natural base elements is also known by the system, i.e.
PFORM X=0,Y=0;
(A*@ X+B*@(Y))_|(3*D X-D Y);
3*A - B
6. Lie Derivative
__________________
The Lie derivative can be taken between a vector and an exterior form or
between two vectors. It is represented by the infix operator |_ . In
the case of Lie differentiating, an exterior form by a vector, the Lie
derivative is expressed through inner products and exterior
differentiations, i. e.
PFORM Z=K;
TVECTOR U;
U |_ Z;
U_|d Z + d(U_|Z)
If the arguments of the Lie derivative are vectors, the vectors are
ordered using the anticommutivity property, and functions (zero forms)
are differentiated out.
Example:
TVECTOR U,V;
V |_ U;
- U|_V
PFORM X=0,Y=0;
(X*U)|_(Y*V);
- U*Y*V_|d X + V*X*U_|d Y + X*Y*U|_V
7. Hodge-* Duality Operator
____________________________
The Hodge-* duality operator maps an exterior form of degree K to an
exterior form of degree N-K, where N is the dimension of the space. The
double application of the operator must lead back to the original
exterior form up to a factor. The following example shows how the factor
is chosen here
SPACEDIM N;
PFORM X=K;
# # X;
2
(K + K*N)
( - 1) *X*SGN
The indeterminate SGN in the above example denotes the sign of the deter
minant of the metric. It can be assigned a value or will be
automatically set if more of the metric structure is specified (via
COFRAME), i.e. it is then set to g/|g|, where g is the determinant of
the metric. If the Hodge-* operator appears in an exterior product of
maximal degree as the leftmost factor, the Hodge-* is shifted to the
right according to
PFORM X=K,Y=K;
# X ^ Y;
2
(K + K*N)
( - 1) *X^# Y
More simplifications are performed if a coframe is defined.
8. Variational Derivative
__________________________
The function VARDF returns as its value the variation of a given
Lagrangian n-form with respect to a specified exterior form (a field of
the Lagrangian). In the shared variable BNDEQ!*, the expression is
stored that has to yield zero if integrated over the boundary.
Syntax:
VARDF(<Lagrangian n-form>,<exterior form>)
Example:
SPACEDIM 4;
PFORM L=4,A=1,J=3;
L:=-1/2*D A ^ # D A - A^# J$ %Lagrangian of the e.m. field
VARDF(L,A);
- (# J + d # d A) %Maxwell's equations
BNDEQ!*;
- 'A^# d A %Equation at the boundary
Restrictions:
In the current implementation, the Lagrangian must be built up by the
fields and the operations d, #, and @. Variation with respect to indexed
quantities is currently not allowed.
For the calculation of the conserved currents induced by symmetry
operators (vector fields), the function NOETHER is provided. It has the
syntax:
NOETHER(<Lagrangian n-form>,<field>,<symmetry generator>)
Example:
pform l=4,a=1,f=2;
spacedim 4;
l:= -1/2*d a^#d a; %Free Maxwell field;
tvector x(k); %An unspecified generator;
noether(l,a,x(-k));
( - 2*d(X _|A)^# d A - (X _|d A)^# d A + d A^(X _|# d A))/2
K K K
Comment:The above expression would be the canonical energy-momentum
3-forms of the Maxwell field, if X is interpreted as a translation;
9. Handling of Indices
_______________________
Exterior forms and vectors may have indices. On input, the indices are
given as arguments of the object. A positive argument denotes a
superscript and a negative argument a subscript. On output, the indexed
quantity is displayed two dimensionally if NAT is on. Indices may be
identifiers or numbers. However, zero is currently not allowed to be an
index.
Example:
PFORM OM(K,L)=M,E(K)=1;
E(K)^E(-L);
K
E ^E
L
OM(4,-2);
4
OM
2
In the current release, full simplification is performed only if an
index range is specified. It is hoped that this restriction can be re-
moved soon. If the index range (the values that the indices can obtain)
is specified, the given expression is evaluated for all possible index
values, and the summation convention is understood.
Example:
INDEXRANGE T,R,PH,Z;
PFORM E(K)=1,S(K,L)=2;
W := E(K)*E(-K);
T R PH Z
W := E *E + E *E + E *E + E *E
T R PH Z
S(K,L):=E(K)^E(L);
T T
S := 0
R T T R
S := - E ^E
PH T T PH
S := - E ^E
.
.
.
If the expression to be evaluated is not an assignment, the values of
the expression are displayed as an assignment to an indexed variable
with name NS. This is done only on output, i.e. no actual binding to
the variable NS occurs.
E(K)^E(L);
T T
NS := 0
R T T R
NS := - E ^E
.
.
.
It should be noted, however, that the index positions on the variable NS
can sometimes not be uniquely determined by the system (because of
possible reorderings in the expression). Generally it is advisable to
use assignments to display complicated expressions.
In certain cases, one would like to inhibit the summation over specified
index names, or at all. For this the command _N_O_S_U_M <indexname1>,..; and
the switch NOSUM are available. The command NOSUM has the effect that
summation is not performed over those indices which had been listed.
The command _R_E_N_O_S_U_M enables summation again. The switch NOSUM, if on,
inhibits any summation.
It is possible to declare an indexed quantity completely antisymmetric
or completely symmetric by the command _A_N_T_I_S_Y_M_M_E_T_R_I_C <name1>,..; or
_S_Y_M_M_E_T_R_I_C <name1>,...;. If applicable, these commands should be issued,
since great savings in memory and execution time result. Only strict
components are printed.
10. Metric Structures
______________________
A metric structure is defined in EXCALC by specifying a set of basis one-
forms (the coframe) together with the metric.
Syntax:
COFRAME <identifier><(index1)>=<expression1>,
<identifier><(index2)>=<expression2>,
.
.
<identifier><(indexn)>=<expressionn>
WITH METRIC <name>=<expression>;
This statement automatically sets the dimension of the space and the
index range. The clause WITH METRIC can be omitted if the metric is
Euclidean and the shorthand _W_I_T_H _S_I_G_N_A_T_U_R_E <diagonal elements> can be
used in the case of a pseudo-Euclidean metric. The splitting of a metric
structure in its metric tensor coefficients and basis one-forms is
completely arbitrary including the extrems of an orthonormal frame and a
coordinate frame.
Examples:
COFRAME E R=D R, E(PH)=R*D PH
WITH METRIC G=E(R)*E(R)+E(PH)*E(PH); %Polar coframe;
COFRAME E(R)=D R,E(PH)=R*D(PH); %Same as before;
COFRAME O(T)=D T, O X=D X
WITH SIGNATURE -1,1; %A Lorentz coframe;
COFRAME B(XI)=D XI, B(ETA)=D ETA
WITH METRIC W=-1/2*(B(XI)*B(ETA)+B(ETA)*B(XI)); %A lightcone coframe;
COFRAME E R=D R, E PH=D PH %Polar coordinate
WITH METRIC G=E R*E R+R**2*E PH*E PH; %basis;
Individual elements of the metric can be accessed just by calling them
with the desired indices. The value of the determinant of the covariant
metric is stored in the variable DETM!*. The metric is not needed for
lowering or raising of indices as the system performs this
automatically, i.e. no matter in what index position values were
assigned to an indexed quantity, the values can be retrieved for any
index position just by writing the indexed quantity with the desired
indices.
Example:
COFRAME E T=D T,E X=D X,E Y=D Y
WITH SIGNATURE -1,1,1;
PFORM F(K,L)=0;
ANTISYMMETRIC F;
F(-T,-X):=EX$ F(-X,-Y):=B$ F(-T,-Y):=0$
ON NERO;
F(K,-L):=F(K,-L);
X
F := - EX
T
T
F := - EX
X
Y
F := - B
X
X
F := B
Y
Any expression containing differentials of the coordinate functions will
be transformed into an expression of the basis one-forms.The system also
knows how to take the exterior derivative of the basis one-forms.
Example (spherical coordinates):
COFRAME E(R)=D(R), E(TH)=R*D(TH), E(PH)=R*SIN(TH)*D(PH);
D R^D TH;
R TH
(E ^E )/R
D(E(TH));
R TH
(E ^E )/R
PFORM F=0;
FDOMAIN F=F(R,TH,PH);
FACTOR E;
ON RAT;
D F; %The "gradient" of F in spherical coordinates;
R TH PH
E *@ F + (E *@ F)/R + (E *@ F)/(R*SIN(TH))
R TH PH
The frame dual to the frame defined by the COFRAME command can be
introduced by _F_R_A_M_E <identifier>. This command causes the dual property
to be recognized, and the tangent vectors of the coordinate functions
are replaced by the frame basis vectors.
Example:
COFRAME B R=D R,B PH=R*D PH,E Z=D Z; %Cylindrical coframe;
FRAME X;
ON NERO;
X(-K)_|B(L);
R
NS := 1
R
PH
NS := 1
PH
Z
NS := 1
Z
X(-K) |_ X(-L); %The commutator of the dual frame;
NS := X /R
PH R PH
NS := ( - X )/R %I. e. it is not a coordinate base;
R PH PH
As a convenience, the frames can be displayed at any point in a program
by the command _D_I_S_P_L_A_Y_F_R_A_M_E; .
The Hodge-* duality operator returns the explicitly constructed dual
element if applied to coframe base elements. The metric is properly
taken into account.
The total antisymmetric Levi-Cevita tensor _E_P_S is also avaiable. The
value of EPS with an even permutation of the indices in a covariant
position is taken to be +1.
11. Riemannian Connections
___________________________
The command RIEMANNCONX is provided for calculating the connection 1
forms. The values are stored on the name given to RIEMANNCONX. This
command is far more efficient than calculating the connection from the
differential of the basis one-forms and using inner products.
Example (Calculate the connection 1-form and curvature 2-form on S(2)):
COFRAME E TH=R*D TH,E PH=R*SIN(TH)*D PH;
RIEMANNCONX OM;
OM(K,-L); %Display the connection forms;
TH
NS := 0
TH
PH PH
NS := (E *COS(TH))/(SIN(TH)*R)
TH
TH PH
NS := ( - E *COS(TH))/(SIN(TH)*R)
PH
PH
NS := 0
PH
PFORM CURV(K,L)=2;
CURV(K,-L):=D OM(K,-L) + OM(K,-M)^OM(M-L); %The curvature forms
TH
CURV := 0
TH
PH TH PH 2
CURV := ( - E ^E )/R %Of course it was a sphere with
TH %radius R.
TH TH PH 2
CURV := (E ^E )/R
PH
PH
CURV := 0
PH
12. Ordering and Structuring
_____________________________
The ordering of an exterior form or vector can be changed by the command
_F_O_R_D_E_R. In an expression, the first identifier or kernel in the
arguments of FORDER is ordered ahead of the second, and so on, and
ordered ahead of all not appearing as arguments. This ordering is done
on the internal level and not only on output. The execution of this
statement can therefore have tremendous effects on computation time and
memory requirements. _R_E_M_F_O_R_D_E_R brings back standard ordering for those
elements that are listed as arguments.
Another ordering command is _I_S_O_L_A_T_E. It takes one argument. The system
attempts to shift out this identifier or kernel to the leftmost
position, utilizing commutation and derivative rules. _R_E_M_I_S_O_L_A_T_E
restores normal ordering.
Example:
PFORM U=K,V=L,W=M;
U^D(V)^W;
U^d V^W
FORDER V;
U^D(V)^W;
(K*L + K)
( - 1) *d V^U^W
ISOLATE V;
U^D(V);
(K*L + K) L
( - 1) *(d(V^U) - ( - 1) *V^d U)
An expression can be put in a more structured form by renaming a
subexpression. This is done with the command KEEP which has the syntax
KEEP <name1>=<expresson1>,<name2>=<expression2>,...
The effect is that rules are set up for simplifying <name> without
introducing its definition in an expression. In an expression the system
also tries by reordering to generate as many instances of <name> as
possible.
Example:
PFORM X=0,Y=0,Z=0,F=0,J=3;
KEEP J=D X^D Y^D Z;
J;
J
D J;
0
J^D X;
0
FDOMAIN F=F(X);
D F^D Y^D Z;
@ F*J
X
The capabilities of KEEP are currently very limited. Only exterior
products should occur as righthand sides in KEEP.
13. Appendix
_____________
A1. Summary of Operators and Commands
^ Exterior Multiplication 2
@ Partial Differentiation 3
@ Tangent Vector 3
# Hodge-* Operator 7
_| Inner Product 5
|_ Lie Derivative 6
ANTISYMMETRIC Declares completely antisymmetric indexed 9
quantities
COFRAME Declaration of a coframe 10
d Exterior differentiation 2
DISPLAYFRAME Displays the frame 10
EPS Levi-Civita tensor 10
FDOMAIN Declaration of implicit dependencies 3
FORDER Ordering command 12
FRAME Declares the frame dual to the coframe 10
INDEXRANGE Declaration of indices 9
ISOLATE Ordering command 12
KEEP Structuring command 12
METRIC Clause of COFRAME to specify a metric 10
NOETHER Calculates the Noether current 8
NOSUM Inhibits summation convention 9
NOXPND d Inhibits the use of product rule for d 4
NOXPND @ Inhibits expansion into partial derivatives 4
PFORM Declaration of exterior forms 1
REMFORDER Clears ordering 12
REMISOLATE Clears ISOLATE command 12
RENOSUM Enables summation convention 9
RIEMANNCONX Calculation of a Riemannian Connection 11
SIGNATURE Clause of COFRAME to specify a pseudo-
Euclidean metric 10
SPACEDIM Command to set the dimension of a space 2
SYMMETRIC Declares completely symmetric indexed 9
quantities
TVECTOR Declaration of vectors 1
VARDF Variational derivative 8
XPND d Enables the use of product rule for d 4
(is set on default)
XPND @ Enables expansion into partial derivatives 4
(is set on default)
A2. Examples
The following examples should illustrate the use of EXCALC. It is not
intended to show the most efficient or most elegant way of stating the
problems; rather the variety of syntactic constructs are exemplified.
The examples are on a test file distributed with EXCALC.
%Problem: Calculate the PDE's for the isovector of the heat equation.
%--------
% (c.f. B.K. Harrison, f.B. Estabrook, "Geometric Approach...",
% J. Math. Phys. 12, 653, 1971);
%The heat equation @ psi = @ psi is equivalent to the set of exterior
% xx t
%equations (with u=@ psi, y=@ psi):
% T x
pform psi=0,u=0,x=0,y=0,t=0,a=1,da=2,b=2;
a:=d psi - u*d t - y*d x;
da:=- d u^d t - d y^d x;
b:=u*d x^d t - d y^d t;
%Now calculate the PDE's for the isovector;
tvector v;
pform vpsi=0,vt=0,vu=0,vx=0,vy=0;
fdomain vpsi=vpsi(psi,t,u,x,y),vt=vt(psi,t,u,x,y),vu=vu(psi,t,u,x,y),
vx=vx(psi,t,u,x,y),vy=vy(psi,t,u,x,y);
v:=vpsi*@ psi + vt*@ t + vu*@ u + vx*@ x + vy*@ y;
factor d;
on rat;
i1:=v |_ a - l*a;
pform o=1;
o:=ot*d t + ox*d x + ou*d u + oy*d y;
fdomain f=f(psi,t,u,x,y);
i11:=v_|d a - l*a + d f;
let vx=-@(f,y),vt=-@(f,u),vu=@(f,t)+u*@(f,psi),vy=@(f,x)+y*@(f,psi),
vpsi=f-u*@(f,u)-y*@(f,y);
factor ^;
i2:=v |_ b - xi*b - o^a + zet*da;
let ou=0,oy=@(f,u,psi),ox=-u*@(f,u,psi),
ot=@(f,x,psi)+u*@(f,y,psi)+y*@(f,psi,psi);
i2;
let zet=-@(f,u,x)-@(f,u,y)*u-@(f,u,psi)*y;
i2;
let xi=-@(f,t,u)-u*@(f,u,psi)+@(f,x,y)+u*@(f,y,y)+y*@(f,y,psi)+@(f,psi);
i2;
let @(f,u,u)=0;
i2; % These PDE's have to be solved;
clear a,da,b,v,i1,i11,o,i2,xi,t;
remfdomain f;
clear @(f,u,u);
%Problem:
%--------
%Calculate the integrability conditions for the system of PDE's:
%(c.f. B.F. Schutz, "Geometrical Methods of Mathematical Physics"
%Cambridge University Press, 1984, p. 156)
% @ z /@ x + a1*z + b1*z = c1
% 1 1 2
% @ z /@ y + a2*z + b2*z = c2
% 1 1 2
% @ z /@ x + f1*z + g1*z = h1
% 2 1 2
% @ z /@ y + f2*z + g2*z = h2
% 2 1 2 ;
pform w(k)=1,integ(k)=4,z(k)=0,x=0,y=0,a=1,b=1,c=1,f=1,g=1,h=1,
a1=0,a2=0,b1=0,b2=0,c1=0,c2=0,f1=0,f2=0,g1=0,g2=0,h1=0,h2=0;
fdomain a1=a1(x,y),a2=a2(x,y),b1=b1(x,y),b2=b2(x,y),
c1=c1(x,y),c2=c2(x,y),f1=f1(x,y),f2=f2(x,y),
g1=g1(x,y),g2=g2(x,y),h1=h1(x,y),h2=h2(x,y);
a:=a1*d x+a2*d y$
b:=b1*d x+b2*d y$
c:=c1*d x+c2*d y$
f:=f1*d x+f2*d y$
g:=g1*d x+g2*d y$
h:=h1*d x+h2*d y$
%The equivalent exterior system:;
factor d;
w(1) := d z(-1) + z(-1)*a + z(-2)*b - c;
w(2) := d z(-2) + z(-1)*f + z(-2)*g - h;
indexrange 1,2;
factor z;
%The integrability conditions:;
integ(k) := d w(k) ^ w(1) ^ w(2);
clear a,b,c,f,g,h,w(k),integ(k);
%Problem:
%--------
%Calculate the PDE's for the generators of the d-theta symmetries of
%the Lagrangian system of the planar Kepler problem.
%c.f. W.Sarlet, F.Cantrijn, Siam Review 23, 467, 1981;
%Verify that time translation is a d-theta symmetry and calculate the
%corresponding integral;
pform t=0,q(k)=0,v(k)=0,lam(k)=0,tau=0,xi(k)=0,et(k)=0,theta=1,f=0,
l=0,glq(k)=0,glv(k)=0,glt=0;
tvector gam,y;
indexrange 1,2;
fdomain tau=tau(t,q(k),v(k)),xi=xi(t,q(k),v(k)),f=f(t,q(k),v(k));
l:=1/2*(v(1)**2+v(2)**2)+m/r$ %The Lagrangian;
pform r=0;
fdomain r=r(q(k));
let @(r,q 1)=q(1)/r,@(r,q 2)=q(2)/r,q(1)**2+q(2)**2=r**2;
lam(k):=-m*q(k)/r; %The force;
gam:=@ t + v(k)*@(q(k)) + lam(k)*@(v(k))$
et(k) := gam _| d xi(k) - v(k)*gam _| d tau$
y :=tau*@ t + xi(k)*@(q(k)) + et(k)*@(v(k))$ %Symmetry generator;
theta := l*d t + @(l,v(k))*(d q(k) - v(k)*d t)$
factor @;
s := y |_ theta - d f$
glq(k):=@(q k)_|s;
glv(k):=@(v k)_|s;
glt:=@(t)_|s;
%Translation in time must generate a symmetry;
xi(k) := 0;
tau := 1;
glq k;
glv k;
glt;
%The corresponding integral is of course the energy;
integ := - y _| theta;
clear l,lam k,gam,et k,y,theta,s,glq k,glv k,glt,t,q k,v k,tau,xi k;
remfdomain r,f;
%Problem:
%--------
%Calculate the "gradient" and "Laplacian" of a function and the "curl"
%and "divergence" of a one-form in elliptic coordinates;
coframe e u=sqrt(cosh(v)**2-sin(u)**2)*d u,
e v=sqrt(cosh(v)**2-sin(u)**2)*d v,
e ph=cos u*sinh v*d ph;
pform f=0;
fdomain f=f(u,v,ph);
factor e,^;
on rat,gcd;
order cosh v, sin u;
%The gradient:;
d f;
factor @;
%The Laplacian:;
# d # d f;
%Another way of calculating the Laplacian:
-#vardf(1/2*d f^#d f,f);
remfac @;
%Now calculate the "curl" and the "divergence" of a one-form;
pform w=1,a(k)=0;
fdomain a=a(u,v,ph);
w:=a(-k)*e k;
%The curl:;
x := # d w;
factor @;
%The divergence;
y := # d # w;
remfac @;
clear x,y,w,u,v,ph,e k,a k;
remfdomain a,f;
%Problem:
%--------
%Calculate in a spherical coordinate system the Navier Stokes equations;
coframe e r=d r,e th=r*d th,e ph=r*sin th*d ph;
frame x;
fdomain v=v(t,r,th,ph),p=p(r,th,ph);
pform v(k)=0,p=0,w=1;
%We first calculate the convective derivative;
w := v(-k)*e(k)$
factor e; on rat;
cdv := @(w,t) + (v(k)*x(-k)) |_ w - 1/2*d(v(k)*v(-k));
%next we calculate the viscous terms;
visc := nu*(d#d# w - #d#d w) + nus*d#d# w;
%finally we add the pressure term and print the components of the
%whole equation;
pform nasteq=1,nast(k)=0;
nasteq := cdv - visc + 1/rho*d p$
factor @;
nast(-k) := x(-k) _| nasteq;
remfac @,e;
clear v k,x k,nast k,cdv,visc,p,w,nasteq;
remfdomain p,v;
%Problem:
%--------
%Calculate from the Lagrangian of a vibrating rod the equation of
% motion and show that the invariance under time translation leads
% to a conserved current;
pform y=0,x=0,t=0,q=0,j=0,lagr=2;
fdomain y=y(x,t),q=q(x),j=j(x);
factor ^;
lagr:=1/2*(rho*q*@(y,t)**2-e*j*@(y,x,x)**2)*d x^d t;
vardf(lagr,y);
%The Lagrangian does not explicitly depend on time; therefore the
%vector field @ t generates a symmetry. The conserved current is
pform c=1;
factor d;
c := noether(lagr,y,@ t);
%The exterior derivative of this must be zero or a multiple of the
%equation of motion (weak conservation law) to be a conserved current;
remfac d;
d c;
%i.e. it is a multiple of the equation of motion;
clear lagr,c;
%Problem:
%--------
%Show that the metric structure given by Eguchi and Hanson induces a
%self-dual curvature.
%c.f. T. Eguchi, P.B. Gilkey, A.J. Hanson, "Gravitation, Gauge Theories
% and Differential Geometry", Physics Reports 66, 213, 1980;
for all x let cos(x)**2=1-sin(x)**2;
pform f=0,g=0;
fdomain f=f(r), g=g(r);
coframe o(r) =f*d r,
o(theta) =(r/2)*(sin(psi)*d theta-sin(theta)*cos(psi)*d phi),
o(phi) =(r/2)*(-cos(psi)*d theta-sin(theta)*sin(psi)*d phi),
o(psi) =(r/2)*g*(d psi+cos(theta)*d phi);
frame e;
pform gamma1(a,b)=1,curv2(a,b)=2;
antisymmetric gamma1,curv2;
factor o;
gamma1(-a,-b):=-(1/2)*( e(-a)_|(e(-c)_|(d o(-b)))
-e(-b)_|(e(-a)_|(d o(-c)))
+e(-c)_|(e(-b)_|(d o(-a))) )*o(c)$
curv2(-a,b):=d gamma1(-a,b) + gamma1(-c,b)^gamma1(-a,c)$
factor ^;
curv2(a,b):= curv2(a,b)$
let f=1/g;
let g=sqrt(1-(a/r)**4);
pform chck(k,l)=2;
antisymmetric chck;
%The following has to be zero for a self-dual curvature;
chck(k,l):=1/2*eps(k,l,m,n)*curv2(-m,-n)+curv2(k,l);
clear gamma1(a,b),curv2(a,b),f,g,chck(a,b),o(k),e(k);
remfdomain f,g;
%Problem:
%--------
%Calculate for a given coframe and given torsion the Riemannian part and
%the torsion induced part of the connection. Calculate the curvature.
%For a more elaborate example see E.Schruefer, F.W. Hehl, J.D. McCrea,
%"Exterior Calculus on the Computer: The REDUCE-Package EXCALC Applied
%to General Relativity and to the Poincare Gauge Theory", GRG, vol. 19,
%1987, pp. 197-218
pform ff=0, gg=0;
fdomain ff=ff(r), gg=gg(r);
coframe o(4)=d u+2*b0*cos(theta)*d phi,
o(1)=ff*(d u+2*b0*cos(theta)*d phi)+ d r,
o(2)=gg*d theta,
o(3)=gg*sin(theta)*d phi
with metric g=-o(4)*o(1)-o(4)*o(1)+o(2)*o(2)+o(3)*o(3);
frame e;
pform tor(a)=2,gwt(a)=2,gam(a,b)=1,
u1=0,u3=0,u5=0;
antisymmetric gam;
fdomain u1=u1(r),u3=u3(r),u5=u5(r);
tor(4):=0$
tor(1):=-u5*o(4)^o(1)-2*u3*o(2)^o(3)$
tor(2):=u1*o(4)^o(2)+u3*o(4)^o(3)$
tor(3):=u1*o(4)^o(3)-u3*o(4)^o(2)$
gwt(-a):=d o(-a)-tor(-a)$
%The following is the combined connection;
%The Riemannian part could have equally well been calculated by the
%RIEMANNCONX statement;
gam(-a,-b):=(1/2)*( e(-b)_|(e(-c)_|gwt(-a))
+e(-c)_|(e(-a)_|gwt(-b))
-e(-a)_|(e(-b)_|gwt(-c)) )*o(c);
pform curv(a,b)=2;
antisymmetric curv;
factor ^;
curv(-a,b):=d gam(-a,b) + gam(-c,b)^gam(-a,c);
showtime;
end;