C o m m u r d c ~ s

ELSEVIER Co'npmer Physics Communications 10O (1997) 177-I94

EDS: A REDUCE package fc, r exterior differential systems David Hartley !,2

Institute for Algori'.hms and Scientific Computing, GMD - Gernum National Research Cenler fi.'r ~nfarnration Technology. D-53754 St. Augustin, Germany

Received 13 May 1996

Abstract

The EDS package for sy,abolic analysis of partial differential equations using the geometrical approach of exterior dif- ferential systems is described. The package implements much of exterior diiferential systems th~.~ory, including pro[ong~iofl and involution analysis, and has been optimised for iarge, nonlinear ln-ob!ems. Besides the de,-cription of the facilities in the package, two nontrivial sample calculations arising from general rela:Mty are presented.

!.091 lffSC: 58AI5; 68Q40; 35A30 Keywords: Exterior differential systems; Partial differential equations; Symbolic computation

P R O G R A M S U M M A R Y

Title of program: EDS

Comlogue identifier: ADEV

Program obtainable from: CPC Program Library, Queen's Univer- sity of Belfast, N. ireland

Licensing provisions: none

Computer for which the program is designed and others on which it is operable: Any system capable of running REDUCE 3.6

Operating systems under which the program has been tested: SunOS5, OSF3, HP-UX, MS-Windows 3.1

Programming language used: RLISP

Memory required to execute with typical data: 8 MB

No. o[ bytes in distrilnaed progrum, including test data. etc.: 979047

Distribution fi)rn~t: ASCII

Keywords: Exterior differential systems, partial differential equa- tions, symbolic computatiwa

Nature of physical problem Symbolic analysis of PDE systems.

Method of solution REDUCE functions for formulation ~md analysis of P.DE systems in the language of exterior differentia,! systeras,

Restrictioas on the complexity elf "he pro.biota High!y nonlinear systems may no: b-' tr~ ~tabl~.

New address from July ~996: Dept. of Physics ~ad Math. Physics, University of Ade)ai~ SA 5005, Australia. 2 E-maih DHm't!ey@physi~.ade~de.edu.au

0010-4655/97/$17.00 Copyright ~ 1997 Elsevier Science B.V. All rights reserved. PIIS0010.4655(96)00138-5

[78 D. Harlley/Computer Phy¢ics Communications IO0 (1997) 177-194

T2,~Tical running time Secomls to hours; depends unpredictably on dam.

Unusual features of the program Use of REDUCE 3 rr);oe data types.

L O N G ~ 1 t l T E - U P

L Introduction

Exterior differential systems give a geometrical framework for partial differential equations. The geometrical formulation has several advantages stemming from its coordinate-independence, including superior treatment of nonlinear and global problems. Many physical theories, general relativity being the best example, ha~/~ a natural geometrical character, itt which case exterior differential systems can be used to very good effect. Space does not permit a proper introduction to exterior differential systems beyond the scant details given in Section 2, but there are a number of tex~ on the subject (e.g. [2,15]).

The EDS package [8] , written for the REDUCE computer algobra system, provides a number of tools for se~.fi.ng up and manipulating exterior differential syst.'ms and implements many features of the theory. Its main strengths are the ability to use anholonomic or moving frames and the care taken with non!iaear problems.

There has long been interest in implementing the theory of exterior differential systems in a computer algebra system (e.g. [ 1,7,9] ). The EDS package owes much to these earlier efforts, and also to related packages for PDE analysis (e.g. [ 12-14]) , as well as to earlier versions of EDS pruduccd at Lancaster university with R.W. Tucker and P.A. Tuckey. EDS uses the exterior calculus package EXCALC of E. Schrtifer [3] and the exterior algebraic ideals package XIDEAL [ I0].

The paper is laid out as follows. Section 2 discusses the way in which the mathematical quantities used a, ex~.eri6r differential systems are stored in EDS. The commands available for creating and analysing exterior systems are sketched in Section 3: a note on the complete set of command tables appears in Append/.". A. Section 4 gives two longer examples showing how nontriviai calculations (in this case from general relativity) can be performed with EDS.

2, EDS objects ~.mi c o ~ p t s

This sectio~ ~scrihes the various ways exterior systems quantities are represented in REDUCE within EDS. These repre:~nttations are seminal to the scope of the package. Particularly important is the normal form (Sec.~.tion 2.10) ir~to which an exterior system must be brought before most of the matber~,aticai analysis uf EDS can proceed.

Z i . Variables and expressions

EDS makes use of thee exterior calculus package EXCALC [3] to provide the operations of exterior algebra and calculus. "Ihus, coordi_r, ates and functions must he declared as 0-forms using an EXCALC p f o r g declaration. Such variables are dewoted (O-form variable? here, aii other variables are regarded as constant parameters. Similarly. a (p-form variable) is a variable so declared. Expressions involving such variables are denoted <p-form).

2.2. Caframings

W/thin the context of EDS, a coframing means a real finite-dimensional differentiahle manifold with a given global cobasis. The information about a coframing required by EDS is kept in a (coframing) object. The cobasis

D. Hartley/ Computer Physics CommunicaGions I O0 (1997) 177-194

Table I Commands for constructing EDS objecL~

|79

Comm~nd Function

coframing(cob,crd,rsx ,dry)

coframing (S)

eda(S,D,M)

contact (r, M, N)

pde2eds (pde, dep, ind)

- ~ e t . c o f r a m i n g ( M )

setoeoframing(E)

set -col taming ()

constructs a ~coframing) with the given coba.~is cob, coordinat.es crd. restrictions rsx ~.J structure equations dry: crd, rsx and dry are optional

construcis a (coframing) capable of supporting the given (system)

constructs a simple (EDS) object with given system slid independence condition: if ,~,! is nca supplied, it is deduced from the rest

constructs the (EDS) for the contact system of the jet bundle jr( M. N)

converts a PDE system to an EDS: dependent and independent variables are deduced if they are not specified (variable dependencies are re~lmved)

sets backgreund coframing and returns previous one

clear:; background coframing and returns previous one

is the identifying element of an EDS (coframing): distinct cobases for the same differentiable manifo|d are treated as distinct (coframing) objects in EDS. The cobasis may be either holonomic or anholonomic, allowing some manifolds with nontrivial topology (eg. group manifolds) to be treated.

In addition to the co,.,=~,~,̂ '---:" an EDS (cofiaming) can be given coom~,,a,~:;,~'- "'~" s," . . . . . . . . . . . . . """ . . . . . . ~ . . . . . . . . . . . . - ' :~"o - "d restrictions. The coordinates may be an incomplete or ovetcomnlete set. The structure equations express the ex te r i~ derivalive of the coordinates and cobasis elements as needed. All coordinate, differentials must be expressed in terms of the given cobasis, but not all cobasis differentials need be known. The restrictions are a set of inequalities (at present using just ~ ) describing point sets not in the manifold.

A (coframing) object is represented by the co f ram±ng operator (see Table 1 ). (See Appendix A for the argument variables.)

The (coframing) object is, of course, by no means a full description of a differentiable manitbld. For example, there is no topology and there are no charts. However, the (coframhtg) object carries sufficient information about the underlying manifold to allow a range of exterior systems calculations to be carried out. ,~s such, it is convenient to accept an abuse of language and think of the (coframing) object as a manifold.

Examples - R 3 with cobasis {dx, dy, dz } and coordinates {x, y, z }. - ]R2\{0} with cobasis {e I , e 2 }, a single coordinate {r}, "structure equations" {dr = e t, de I = 0, de 2 = e z AeZ/r}

and restr:'ctions {r ~ 0}. - R2\{G} with cobasis {dx, dy}, coordinates { x , y } and restrictions { x 2 + y2 :~ 0}. - S I with cobasis {oJ} and structure equations {d~ = 0}. - S 2 cannot be encapsulated by an EDS (coframing) since there is no global cobasis.

2.3. Background coframing

The rules and orderings encapsulat~:,¢ in a (coframing) object are usually inactive. However, when ope r~ iom are [erformed on a (coframing~ or :~ (EDS) ob j~ t , this intbrmation is activated for the duration of those operations. It is possible to activate the intbrmation in a (coframing) object globally, by making it the background coframing us i ,g the command s o t _ c o f r a m i n g . All subsequent EXCALC operations will be govern~:d by t ~ rules. Operations on (EDS) objects are unaffected, since their coframings are siill a':tivated locally.

~80

Z 4. Systems

D. Hartley/Computer Phyz;cs Communications !00 (1997) 177-194

In EDS, the label (system) refers to a llst

{~-fo.~,...} of differential forms. Tais is distinct from an (EDS), which has additional structure. However, many EDS operators will accept either an (EDS) or a (system) as arguments. In the latter case, any extra information which is required is taken from the background coframing.

ZS. Exterior 4ifferential systems

A simple (EDS), or exterior differential system, is a triple (S, /2,M), where M is a (coframing), S is a ~system) on M, and [1 is an independence condition: either a decomposable (p-form) or a (~ystem) of I-forms on M (exterior differential systems without independence condition are not treated by EDS).

A simple (EDS) is represented by the ed-~ operator (see Table 1 ). More generally, an (EDS) is a list of simple (EDS) objects where the various coframings are all disjoint.

This last requirement in not strictly enforced within EDS unless the e d s d i s j o i n t switch is on. These more general (ED,S~ objects can he used to repr~.,ent, for example, the pullback of a nonlinear exterior system onto a variety All commands which take an (EDS) argument accept both simple and compound types.

The trivial (EDS), describing an inconsistent problem with no solutions, is defined to be ({I} , {}, {}).

Z6. Integral elements

An i~tegral element of an exterior system (S,/2, M) is a subspace P C TpM of the tangent space at some p~nt p E M such that all forms in S vanish when evaluated on vectors from P. In addition, no nonzero vector in P may ar,,nr[ every f~.rm in/2.

b3ternativei?', an integral element P C TpM can be represented by its annihilator P± C T~M, comprising those 1-forms .',t p which annul every vector in P. This can also be understood as a maximal set of l-forms at p such that S _~ 0 (rrw:! P-J-) and the rank o f / 2 is preserved modulo P±. This is the representation used by EDS. Further, the referer~ce to the point p is omitted, so an (integral element) in EDS is a distritnJtion on M, specified as a (system) of l-forms.

In specifying an integral element for a particular (EDS), it is [~assible to omit the Pfaffian component of the (EDS), since these l-forms must he part of any integ'al element.

Examples - With M = R 3 = { ( x , y , z ) } , S = {dxAdz} and /2 = {dx, dy}, the integral element P = {0x + 0z,0y} is

equally determir~ed by it5 annihilator P± = {dz - dx}. - For S = {dz - y dx} and /2 = {dx}, the integral element P = {ax + yaz } can he specified simply as {dy}.

2.Z Properties

For large problems, it c ~ require a gnat deal of computation to establish whether, for example, a system is closed or not. In order to save recomw:~ing such properties, an (EDS) object carries a list of (properties) of the tbrm

where (keyword) is one of c losed , qua~, i l inear , p f a f f i a n or i n v o l u t i v e , and (value) is either 0 (false) oz 1 (true). These properties are suppressed when an (EDS) is printed, unless the n a t switch is o f f .

D. Hartley /Compurer Physics Cotmnunications 100¢1997) 177-194 |8[

Properties are usually generated automatically by EDS as required, and may be examined using the p r o p e r l ; i e s command. If a property is not yet present on the list, it is not yet known, and must be checked explicitly if required.

In addition to the properties just described, an (EDS) object carries a number of hidden properties w h ~ record the results of previous ealeulatiocs, such as the closure or information about the prolongation ~ the system. These hidden properties speed up mat6' operations which con',ain common subealculation~. The h'~den properties are stored using internal LISP data structures and so are not available for inspection.

2.8. Maps

Within EDS, a map f : M - : N is given as a (map) objec,, a list

{(O-form "oar~ab~e) = (O-for~),...,(O-form) neq (O-forra),...}

of substitutions and restrictions. The substitutions express coordinates on the target manifold N in terms of on the source manifold M. The restrictions describe point sets not contained in the so~,r~ manifoki M. ordering of substitutions and restrictions in the list is unimportant. It is not neeessary that the resU'ictions and fight-hand sides of the subs'dtutions be written entirely in M coordinates, but it must be possible by substitution to produce expressions on M (see the examples below). Any denominators in the substitutions are automatically added to the list of restrictions. It is not necessary to include trivial equations for coordina~ which are present on both M aad N. Note that projections cannot ~ represented ir this fashion.

Examples - The map R2\{0} ---, R 3, ( x , y ) ~-. ( x , y , z = x 2 + y 2 ) is represented {z = x 2 + y 2 , z ~ 0}.

{ x = u + v, y = u - v } might represent the coordinate change R 3 ---, R 3, ( u, v, : ) ~ ( x = u + v, y = u - v, z ). - { x = u + v , y = 2 u - x } is the same map again. - {x = 2v + y , y = 2u - x} is unacceptable since x and y cannot be eliminat.ed from the fight-hand sides by

repeated substitution.

2.9. Cobasis transformations

A cobasis transformation is given in EDS by a (transform), a list

{ ( l - f o r m ~ a r ( t a b l e ) = ( l - f o r m ) , . . . }

of substitutions. When applying a transformation to a (p-form) or Isystem), it is necessary to specify the forward

transformation just as for a sub substitution. For (EDS) and (coframing) objects, it is also possible to specify the inverse of the desired substitution: EDS will automatically invert the transformatioa as required. For ~ p a r t ~ change of cobasi°~, it i~ not necessary to include h'Svial equalities.

Examples - {w t = x d y - y dx, w 2 ~, x dx + y dy} give:+ a transformation between Cartesian and polar cobases on R2\{O}. - On J: (R2,R) with c¢~basis {du.dp, dq, (~r, ds, dt, dx, d v~, the list {01 = du - p d x - qdy , O 2 = dp - r d x -

sdy, O 3 = dq - s d x - ,~dy} specifies a new cobasis in which the contact system is simply {0t,02,03}.

2.10. Normal f o rm

Parts of the theory of exterior differential systems apply only at points on the underlying manifold where the system is in some sense nonsingular. To ensure the theory applies. EDS automatically works all exterior systems (5,/2, M) into a normal f o rm in which

182 D. ltartley/Computer Physics Communications I00 (1997) 177-194

O) The Pfaffian (degree I) component of S i:~ in solved form, where each expression has a distinguished term wihta coefficient I, unique to that expression.

(it) "f'ne independence condition/2 is also in solved form. (iii) The distinguished terms from the i-forms in S have been eliminated from the rest of S and from/2. (iv) Any l-forms in S whici~ vanish modulo the independence condition are removed from the system and

their coefficients are appended as 0-forms. Conditions (i) and (it) ensure the l-form subsystem has constant rank, while (iii) is convenient for many

tests and calculaiioas. In bringing the system into solved form, divisions will be made only by coefficients wkich are constants, parameters or functions which are nowhere zero on the manifold. The test for nowhere- zero functions uses the restrictions component of the (coframing) structure and is still primitive: facts such as x 2 + 1 ~ 0 on a real manifold are overlooked unless explicitly recorded.

This "normal form" has, of course, nothing to do with the various normal forms (e.g. Goursat) into which some exterior systems may be brought by cobasis transformations and choices of generators.

Examples - On M = {(u ,v ,w) E/R 3 l u ~ v}, the Pfaffian system

{udu + vdv + dw, (u 2 ~ u - v2)du + udv + dw}

has the solved form

{dr + (u + v)du, dw + ( - u v + u - v)du}.

- Since the independence condition is defined only modulo the system, the system

S = { d u - d x - u r d y } , / 2 = d x A dy

has an equivalent normal form

S - - - { d x - d u + u r d y } , / 2 = d u / ' Oy.

2.11, Standard cobasis

Given an (EDS) (S , /2 ,M) in normal form, the cobasis of the (coframb~g) M can be decomposed into three sets: {0~}, the distinguished tenns from the l-forms in S, {to/}, the distinguished terms from the 1-forms in /2, and the remainder {~rp}. Within EDS, {Oa,TrP, to i} is called the standard cobasis, and all expressions are ordered so that 0" > zr a > toi. The ordering within the three sets is determined by the REDUCE variable ordering.

E.ramples - For the system S = {du-dx-u~. dy}, /2 = dxA dy, the decomposed standard cobasis is {du}U{du:.}U{dx, dy}. - For the contact system

[ ' du - ux dx - u ~. dy , S = { dux - u ~ x d x - u ~ v d v , D = d x A dy ,

( duy - Uxy dx - Uyl,. dy ,

the standard cobasis is {du, dux, duy } U {duxx, duxy, dusy } U {dx, dy}.

D. Hm~ley/Computer Physics Communications I00 (1997) 1~ 7-194 |83

2.12. Tableatoc

For a Pfaffian exterior differential system ({0°}, {¢o'}, M) which is quasilinear, the tableau A = [ ~ / | is a matrix of 1-forms such that

d 0 ~ + ~ / a o , ~ 0 (rood {0° , ,o~^oJ}) .

The ~ are not unique: if {0 a, 7rp, M} is a standard cobasis for the system, the EDS (tableau) is a ~ x containing linear combinations of the ~r p only. Zero rows are omit~ed.

3. EDS functions

This section outli~es the commands available to the user within the EDS package. These range fr<~n t ~ basic requirements of setting up a problem and manipulating it into a desired form to tim more sophisticated mathematical analysis of a problem. The latter commands constitute a constructive implementation of some of the most powerful theorems from exterior systems theory.

3.1. Constructing and manipulating EDS objects

The first requirement is the ability to specify an exterior system within REDUCE. As described in Section 2, EDS stores the information using (ED~ and (coframing) objects. The basic commands for constructing ~ e objects are eds and coframing. For convenience, EDS also supplies a command eon tacz for ~ a g the commonly used jet bundle contact systems, and a command pde2cds for converting a PDE system in REDUCE df or EXCALC @ notation into an (EDS). Table i ~hows the syntax of these commands, as wel| as the commands for setting the background coframing (Section 2.3).

Having generated an EDS object using the commands in Table 1 or by further manipulation, it is often to inspect or extract the various components. The commands in Table 2 -,dlow EDS objects to he disassembled into their constituent parts. For example, cobasi~, coorclina~es, r e s t r i c t i o n s and s t z ' u c z u r ¢ ~ g i o ~ return the parts of a (coframing) object. Applied to an (.EDS), these same commands refer to ~e (coframing) part of the (EDS). The parts of an (EDS) are accessed with syszem, independence and c o f r a m ~ In addition, it is often convenient to be able to extract the 0-forms or l-forms from a system, so the zero._fonas and one_forms are provided.

C::mbir, iag the constructors and selectors, it is possible to alter parts of an EDS object "by hand". For example, to extend the independence condition of an (EDS) E with a n ~ (l-form) omega, the syntax is

E := eds(system(E),omega . independence(E),cofram~ng(E)) ;

The next set of commands, listed in Table 3, are low-level t ~ t . l o n s rot ma-nipuiating EDS objects. mainly perform, some surgery on the underlying (coframing) object. Tnus t r a n s f o r m performs a transformation, T;allback can he used to pull back to another manifold (or perform a coordinate change), aad r e ~ t r i c z c~a~ be used to exclude singular point sets from the working space. Typical applications for these commands include creating the proper coframing for a complicated problem (see Section 4.2 for an example), adapting a cobasis for some (system), or testing a solution of a~ (EDS).

3.2. Analysing and testing ED$ objects

The main commands for mathematical analysis of exte~or systems are given in Tables 4 and 5. Some of the operations (,e.g. cauchy_,~yste~ c losure , d e r i v e d . ~ y s t ~ t a b l e a u ) implement fairly simple maLhematica~

[ 84 D. Hartley / ~omputer Physics Communh'ations I O0 (1997) 177- I ~4

Commands fc~r inspecting EDS ob~cts

Command ~ n c ~ n

c o f r ~ . i n g (~E)

cofraZLigg 0

cob~sis ( ~ ) cobasis (:~)

co~rdgnat.~'~ ( M ) coord/nate s (E)

struct~q.e quat icn~ (M) s t ructv.re .eouat ions (E)

r e s ~ ' r i c t i o ~ tM) r e s ~ r i c t i ~ (E)

sys tem(E)

~eFendence ( F- )

propez-~iea(E)

c~e~or~s (E) one_f or~s (S)

z e r o - f o ~ (E) zero_for~s (S)

extracts the underlying {coframing)

returns the current background coframing

extracts the underlying eobasis

extracts hhe coordinates

extracts the rules for exterior derivatives for cobasis and coordinates

2,':.:.~ "..~.7. ~ ".-_':g "_~1:~2: a_.~scribin~ !he restrictions in ti~e {coframin(,~

extracts the (system) par of E

exuacts the independence condition from E as a Fq'ar~an {system)

returns the currently known pro~nies of E as a list of equations (keyword)=(value)

.~lects the I-forms from a system

~lects am 0-fo:'ms from a system

Table 3 Cow-.ands for manipulating EDS objects

Command Function

a u ~ e n t ( E , S)

M c r o s s N E c r o s s N

pu l lback ( E , / ) p~Lllback (S, f ) pu i lback (fl=

pu l lback ( M, j ;

r e s t r i c t (E, f ) r e s g r i c t ( S , / ) restrict ( O, D

r e s t r i c t ( M, f )

era~usfora (M, X) t r a n s f o r m (E,X) transform(S, X) t r ~ f o r m ( ~ l , X)

lifg(E)

appends the extra for;~s in S to flit system in E

forms the direct prodvcc of two coframings: an (EDS) E is lifted to the extended space

pulls back the first argument using the ~map) f

gctums a (coframing) N suitable as the source for f : N --, M

restricts the first argument to the points specified by the (map) J

acids the resmctions in f to M

applies the change of cobasis X to the first m'gument: for a (coframing) M or an (EDS) E, X may be specified in either the forward or p.~vers¢ direction

eliminates any 0-forr~ in E by solving and pulling back

D. Hartley/ Computer Physics Communications 100 (i997) t77-i94

Table 4 Commands for analysing exterior systents

185

Command Function

caz~an.syst em (E) caftan_system (S) caz'~an_system(/~)

cauchy.syst em (E) cauchy.system (S) cauchy.sys1:em (/2)

characters(E) characters(T)

characters(E, P)

closu~'e (E)

derived-sys~ (K) derived-system (S)

dim_gr as sm~rLn. _vat i n~.y (E) dim_gr assmann_variet y ( E, P )

dim(M) dim(E)

involution(E)

linsarise (E, P)

integraLelement (E)

prolong(E)

tableau(E)

torsion (E)

grassmann_variety (E)

calculates th~ Caftan system (as~iated Pfaff system, retracting space): no differentiates are performed

ca!ca!a'~,~ !he Ca-chy sys:em: !he Ca.~,an system of the closure under extet~,or diffe~ent~io¢

calculates the (reduced) Carman characters {s~ ..... sn} (E quasilincar)

Canan characters for a nonlinear E at integral element P

calculates the closure of E under exterior differentiation

calculates the first derived system of the Pfaffian system E or S

dimension of the Gre~ssmann ~ndle variety of integral elements: for nonlinear E. the b ~ element P must he given

returns the manifold dimension

repeatedly prolongs E to involution (or inconsistency)

lineatis¢ the (nonlinear) EDS E with res~ct to tl~_ integral clcn'~nt P

find :, random (integral element) of E

prolongs E, and projects back down to a subvariety of tl~ original manifold if integrabilRy conditions ari~

calculates the (tableau) of the quasilincar Pfaffian (EDS) E

returns a (system) of 0-f~rms specifying 1he inte~.-'.~bili,.y conditions for the ~mi l~ar or qaasiliaear Pfaffian (EDS) E

returns tim contact (EDS) fer the Grassmann bundle of n-planes over the manifold of E, aug..,p~-n*.ed by Lh~-- 0-forms specifying !he vafie~ of integral elements of E

definitions. Others, such as the commands concerning involution ( c h a r a c t e r s , p r o l o n g , i n v o l u e i o n ) , re la~ to more sophisticated parts of the theory.

To date, the emphasis in EDS has I~en ;aid on involution analysis. The main tools here arc th~ Caftan ch&-actcrs - - a p,,ntn,g,ti,~n n f * n ~ v t . , . ; c ~ r tv~tom "['he nroicm~.:~.*i~n nf ~ e m to the vmicty of its i m e g ~ ca l ,~ . v . w . . . . . ~ . . . . . . . . . . . . . . . . .F . . . . . . . . . . . • , ~ . . . . . . .

elements is carried out by the p r o l o n g c o n t m ~ d . If some component of the ~'~-iety does not surjec~ ¢n '~ the original manifold (e.g. when there ~ e fi~:-order i n t ~ b i l i t y condit ions) , then p r o l o n g pulls the o t ~ g i ~ system back onto the projected image of tha~ component. The generic REDUCE s o l v e package is used ha decomposing the integral eleincnt variety into submanifolds of the Gcassmaan bundle. For special p r o b ~ the generic approach may not he satisfactory. In such cases, the g r a s ~ a n n _ v a r i e t y command can he u s ~ to supply the equations describing the variety, which can tben be .solved by some more specialised t e c h n i c . The, solutions can then Ix: used to generate the prolongation "by hand" using p a . l l b a e k .

In previous work [9] , the Cart, an charact.ers were ~mbl i sbed using the a l ~ ' . h m supplied by the C ~ K ~ l c r theorem. In the current EDS, the c h a r a c C e r s command calculates the reduced C m a a c b ~ which is usually much faster. The char~tcrs arc relurned as a list {sz . . . . . sn]-; the z , c ~ h character so is mg returned, since it is simply the numhet of !-forms in the system. The definition used for the last character, s,, = (dim M - n) - (so + sl + ... + s n - i ) , allows Cattan's test to be used even when Cauchy c h a r a c t c r i ~

i86 D. Harlley/Computer Physics Communications 100 (1997) 177-194

Tz~ble 5 Conunands for tesling exterior systems

Conm,,a~ Function

, . l o s e r ( E ) checks foc closure under exterior difl~:rentiation c l o s e d ( S ) closed(~)

i .u~oiutive(E)

p f - .~ffian(E)

qua~ili~:ear (E)

se~iline~r (E)

E ¢qu iv E'

applies Cartan's test for involution

checks if E is generated by !-forms and thei.r exterior derivatives

tests if the ch,sure of E can be generated by forms at most linear in the complement of the independence condition

tests if the closure of E is quasilinear and, in addition, the coefficients of the linear terms contain only in~=penden,' -;~ri~les or constants

checks whether E and E' are algebraically equivalent

iabie 6 Experimental funr*!ons (unsL~le)

Command Function

poi-care (/D

~var ian ts~E,crd) invarian~,s(S,crd)

syLhol..relal:ions ( E, or)

s y m b o l ~ a t r i x ( E, ~)

charac~;er i s z i c _ v a r i e t t ( E, ~)

ealcul~=s d~e bonm~opy intcg~.,~' from the proof of PoincarCg lemma: if .fl is exact, then the result is a form whose exterior derivative gives back f l

calculates the invari.-.nts (first integrals) of a completely i,teg~b!= Pfaffian system using the inductive proof of the~ Frohenius theorem: the optional second argument specifies the order in which lhe coordinates are to he projected away

returns relations between the entries of the tableau matrix, represented by 2-indexed (1-form) variables l f i

returns the symbol ro .~x for a quasilinear (EDS) E as a function of (O-form) variables ~i

returns equations describing the characteristic variety of E in terms of (O-fi~rm) variables ~i

are present. For a nonlinear system, the characters can vary from one component of the integral element variety to the next

EDS caters for this by allowing the specification of a base point for the calculation. The i n t e g r a l _ e l e m e n t command helps here by generating a random integral element for use as the base point. The algorithm currently employed here is the Cartan-Ki-ihler construction, as adapted by Wahlquist [ 16] to make random choices. This procedure automalically generates the Cartan characters as a side-bene~:, but may fail to find any integral etements on noninvolutive systems.

The i n v o l u t i v e test applies Cartan's test for involution to an (EDS). The same can be done manually using c h a x a c e e r s and d i m ~ a s s m a n n _ v a r i e t y to establish the dimension of the prolongation. For quasilinc~r sys/~'ras, the latter can be considerably faster than act.ua)ly calculating the prolongation.

A ~ imponam aspect of exterior differential systems is the study of characteristics, The basic functions for Cauchy characteristics, car~.-an.zyszem and cauchy.system, are provided. For quasilinear systems, the tableau can be produced, as well as the symbol matrix and symbol relations. The equations specifying the ci~-acterislic variety can thus be generated and analysed using the algebraic facilities of REDUCE. The com- mands s y m b o l . ~ a Z r i g , s y m b o l . . r e l a t i o n s a n d c h a r a c z e r i s t i c _ v a r i e t y a re sti l l at a ve ry e x p e r i m e n t a l

shage ( s e e Tab l e 6 ) .

D. Hartley/Computer Physics Comm:mications i00 G997) 177-194 ~7

Table 7 Switches (all off by default)

Switch Fenc:~on

edsverbose

edsdebug

edssloppy

edsdisjoint

ranpos genpos

if on. <lisp!ays additional inform...a!ion ~ calculz-:.ioiis pl-ogress

if on. produces copious quantities of rrmstly internal inform~ion, in addition to tha~ produ¢~ by edsverboso

if on, allows EDS to divide by expressions not known tcJ bc nonzero and ~ quas~i~¢a¢ ~y~tems as semilinear

if on, forces varieties to be decomposed into di~oint components

if on, u~s a random or generic flag of integral elements when calculating CaNan c ~ : otherwi~ the independence condition as pre~nted guides ~h¢ choice of flag

3.3. Switches

As usual ill REDUC_, some aspec!s of the be_haviour of EDS can be controlled by switches which can be turned on or o f f . Table 7 lists those available in EDS.

Normally, EDS does not of itself divide by any expression it cannot verify to be nonzero. This is particularly important in bringing an (EDS) into normal form (Section 2.10). If the switch ¢ d s s l o l ~ y is on, otber divisors are allowed as a last resort, and are automatically appended to the (coframing) r e s ~ as a record. Furthermore, quasilinear problems are treated as if they were semilinear. This can improve perfommm:e when onty the principal part of the system is of interest.

The switch s d s d i s j o i n l ; controls whether EDS enforces the decomposition of varieties into a disjoint of (coframing) objects. While this ,-'s Lheoretically desirable, it often leads to myriad singular pieces, so die switch is usually off. The d i s j o i n command (Table 8) can be used to manually break up a decomposition.

Finally, the r a n p o s and genpos switches govern the way the c h a r a c e e r s command works. The characters are defined by the maximal ranks of the successive polar systems for integral elements. This set is guaranteed if genpos (for general position) is on. However, this setting is extremely slow for all but simplest problems. Switching r a n p o s on chooses a random fl,ag of polar systems, usually yielding the correct characters. The default setting (both switches o f f ) uses a flag based on the independence condition 'and variable ordering, and is much faster than the other two.

3.4. AuM!iary functions

EDS provides a number of auxiliary functions designed to ease working with exterior forms and exterior systems (listed in Table 8). Among the commands available a:.:e: index_expand for explicitly listing indexed expressions, linear_divisors for extracting the l-form factors from a (p-form), disjoin to decompose a iis~ of (map) objects into a disjoint union, and another form of the con-anand atructure~quaZions to find e~terior derivatives for the left-hand s!des of a (tr,,-~zform) where ~'~ose of the right-hand sides are known. For example,

{{x = 0}, {y = 0}} ( l )

decomposes to

{{x = O,y #o), {~ = O,y = o}, {y = o,~ ~o}}, (2)

and

185

T ~ 8 Aexi~ry functions

D. Hartley/Compuler Physics Communications 100 (1997) 177-194

Command Function

coordinates (S)

i n v e r t (X)

structure~quatio~s (X) strncC~e-equations (X, X - I)

iin~r~%-isors Cl2)

exfacZors (/2)

~u:lex~.xpemd (any)

p'le2j e t (pde, dep, ind)

d is jo in ({/,g, ...})

cle~J:up CE) cle~mupfM) reorder (E) reorder (M)

scans the expressions in S for coordinates

returns the inverse (tra,:sfi~rm) X-l

returns exterior derivatives of Ihs (X)

calculate~ a b~,~ fer '..he space of I -fo,."m factors of 11

as for l ineaz~livisors, but v, ith ~fie additional (nonlinear) factor

returns a list of copies of its argument, with free EXCALC indices replaced by successive values from the relevant index range

cenverts a PDE system into jet bundle notation, replacing derivatives by jet bundle coordinates (variable dependencies are not affected)

rcstOlCo variable depcr.dcnc~e~ destreyed by pde2eds

decompo~s the variety specified by the given (map) variables into a disjoint union

returns a fresh copy of E or M with all properties and stored results removed

returns a f~sh copy of E or M, conforming to the prevailing REDUCE kernel order

0 = x d y - y d x , ( 3 )

~ = x d y + y d x ( 4 )

impl ies the exter ior derivat ives

dO =4" ^ O/(xy) , (5)

d~b = 0 . ( 6 )

Bes ides these facilities, there are some experimental commands whose implementa t ion is still unstable. For example , i n v a x i a n z s calculates the invariants or first integrals o f a completely integrable Pfaffian sys tem by so lv ing a series o f O D E systems.

4. Examples

4.1. Ricci-flat immersions

As is wel l -known, an arbitrary 4-d imensional Riemannian or Lorentzian manifold M can be immersed locally

in a ~at | 0 -d imens iona l space E ~° ( o f the corresponding s ignature) . It is thus interest ing to consu'uct an extr insic

vers ion o f E ins te in ' s equations, formulated in terms o f restrictions on a 4-d imensional curved surface in E I° ra ther than us ing the intr insic Riemannian geometry o f space-time.

Such an immers ion f : M ~ E ~° determines a class o f coframes {ei, e o} ( i , j , k = I . . . . . 4, a , b = 5 . . . . . 10)

for E t°, cal led Darboux coframes, by requir ing that the e a vanish when pulled back,

p e a = 0 . ( 7 )

The structure equations for E I° ~n'mlediately yield

D. Hartley/Computer Physics Communication¢ i00 (1997) 177-194 189

f * (coai A e i) = 0, (g)

where {¢o0 = _o~i,¢a ib = -¢obi,~a ab = -¢o ~} are the connection l-forms. It follows tha~ an exteri~ system for such immersions can best be written on the orthonormal frame bundle, where the connection l-forms are coframe elements. The problem is thus lifted to a search for immersions f : M -~ OE m, where tbe taJget bundle is isomorphic to the inhomogeneous rotation group ISO(10). Using the Gauss ¢qua~bons for submanifolds, Estabrook and Wahiquist [4] construct an exterior system fer Eins!¢in's vacuum ~uafions,

e~, .

o.,"i ,x e' , (9) OJib A ~ b A * ( e i A e j A e k) ,

with independence condition {el}, where the Hedge map ( , ) refers to the i , j , k indices only. ~ i s system is nonlinear, as signalled by the factor ¢aib A ofl'j. In fact, calculating the. prolongation prod~,.-'es

an iz~act,.'ble system of 16 quadrics in 60 variahl~. Nonetheless, it is still r.ossibl¢ using EDS to cak:uia~ the Caftan characters of this system, and verify (at least locally) that the Cauchy problem is w c l l - ~ . Th.¢ REDUCE session starts with a series of EXCALC declarations to set up the variables and indices for the problem. The input o ( i , j ) is used for the connection i-forms ~dJ. The signature of the space does not affect this calculation, so Euclidean signature is taken.

indexrange {p ,q , s ,Z}={1 ,2 ,3 ,4 ,5 ,6 ,7 ,8 ,9 ,10} , { i , j , k . 1 } = { 1 , 2 , 3 , 4 } , { a , b , c , d } = { 5 , 6 , 7 , 8 , 9 , t O } ;

p fo r=g (p ,q )=O; g (p ,q ) := g ( - p , - q ) := 05 g ( -p , - p ) := g(p,p) := t$ p f o z ' e e ( p ) = t , o ( p , q ) = l ; index_symleerSes o ( p , q ) : a n t i s ~ e ~ r i c ; korder index_expand{e(p)};

The c o f r a = i n g operator produces a (coframiag) object holding the information about ISO(10) needed for the exterior system: the coframe and the stn~,~ture equations. Note that the problem uses moving fra,,r~.s, so the[e are no coordinates at all in use. The o~tput is ,mppressed using REDUCE's $ terminator, since a¢ this stage, the sums and lists are expanded out over the index ranges, so the output becomes too profuse to be informative.

ISO := cof ras ingC{e(p) ,o(p ,q)} , {d e(p) => -oCp,s) 'eCq)*gC-s,-q) ,

d o(p,q) => -oCp , s ) ' oCt ,q )*g~- t , - s )} )$

Next an abbreviation for the nonlinear factor (which represents the 4-dimcusional curvalure 2-form) is made. and the exterior system is formed. The Hedge map in the syste~ (9) is implemented using a .4-dimension~! Levi-Civita tensor. The (somewhat lengthy) output shows the "advantage of entering the system using indexed variables.

p f o r m e p s i l o n ( i , j , k , l ) - O , R ( i , j ) = 2 ; index_symmetries e p s i l o n ( i , j , k , l ) , R ( i , j ) : ~ - ~ t i s y m = e t ~ i c ; e p s i l o n ( - l , - 2 , - 3 , - 4 ) := I$ R ( i , j ) := - o ( i , b ) ' o ( a , j ) * g ( - b , - a ) $ GRO := e d s ( { e C a ) , R ( j , k ) ' e p s i l o n ( - i , - j , - k , - 1 ) ' e ( 1 ) } ,

~eCi)}, ISO),

D. Hartley/ Computer Physics Cor~muni6,ations I00 11997) 177-194

a R o : = EDS ( { e 5, e 6 , e 7, e 8 . e 9 , e I 0

015 A 025 A 6,3 _ o l 5 / \ 035 A e 2 -k 025 A o 35 /', e I h- ,,1,'i ?, ,;;6 A e 3 -- 016 A . . t 6 / , e2 -.I- 026 A cy ,36 A e I --i- ~;17 /, (p27 A e 3 --017 A 037 A e 2 4- 027 A o ~ A e I -k 018 A 028 A e 3 -- 018 A 038 A e 2 4- 028 A 038 .& e I -k 019/'~, 029 A e 3 -- 019 A 039 A e 2

--~o 29 A 0 "~9 A ¢1 @ o11O A 0210 A e 3 -- o I10 A 03111 A e 2 + 0210 A 03111 A e l ,

015 A 0 2 5 A 6, 4 _olSAo45Ae2+o25AoaSAe I +o16Ao26Ae 4 _o16Ao46Ae2.ko26A046Ae I .t_017 A o 2 7 A e 4

_ 0 1 7 A 047 A ~,2..1_ 027 A 0 4 7 A e I -~-olSAo21~Aea_o18Ao48Ae2q-02gAo48t~e I q-olgAo29Aea_olgAo49Ae 2 + 0 ~ A 0 ¢9 A e I + 011° A o 21° A e 4 - v ml° A o 41° A e 2 + 0 "1° A o 41° A e ~,

015 A 035 A ¢4 _ 015 A t ,45 A e 3 -4- 035 A 045 A e I -.k 016 A 036 A e 4 -- 016 A 046 A e 3 -[.- 036 A t ,46 A e I ~k 017 A 037 A ~74

- - t ) l j A 047 A e ~t -~ 037 ,", o 47 .", e I 4.- 018 A 038 A e 4 -- 018 A 048 A e 3 -I 0 )8 A 048 A E I "k i;19 ta 039 ,^ e4 _ 019 A 049 /k 6 ,3

-~O -~/9 A t; '19 A e I q" O Illl A O TM A 6,4 -- OII0 A O diit A e 3 4- o TM A 0410 A 6,1 ,

~25 A~.~5 Ae~-~2~ A ~ 5 /``e3 +~2. A~5 Ae2-k~6 A~36 Ae~.-~2~ A0 .̀~6 A~3 + ~ A~';~ Ae2 +~2~ A~37 Ae 4 ~--r7~ A047 Ae3 q-~37 A~47 A6,2~28 A~38 Ae4-~2~ A~4~ Ae3-~-~38 A048 A¢2-F~29 A~39 Ae4-~29 A~49 Ae ~ + o 39 A 049 A e 2 + 021° A 0310 A e 4 -- 0210 A 0411) A e 3 -t- 0310 A 0411) A e 2 } , e I A • 2 A e 3 A e 4 !

As already mentioned, a straightforward attack on the prolongation will become computatior.,all),, inefii,:'ient very qaickly, so a inore selective approach, borrowing ideas from Wahiquist [ 16], is used. On a:.q~' eonr,,':'~'e~ compom~..nt in the varlet,, decomposition, the Caftan characters are constant. Therefore it suffic,:~s t:) check the characters at one siag!e point on each component. A single prolongation point (ie an integral e le~ent) ca:a be generated at randon, ':cry quickly. If desired, the exterior differential system can be linearised abcm this single point, since the linearised and original systems share the same Cartan characters there. By generating random soEutions repeatedly, it turns out that the Cartan characters are the same on all the main components in this problem.

The variety dimension and the small sum in Ihe comment line constitute C~a.an's test (which EDS can also do automatically, of course), and show that the exterior differential system (9) is involutive, so that local solutions indeed exist. The fiaal Caftan character, 2 I, indicates that there are 2 i free functions of 4 variables in the general solution, which seems odd at first sight. However, the final command in the script shows that there are 55 - 3 4 = 2i Cauchy vectors in the problem (they correspond to the freedom to spin the ta,lgent frames (6 vectors) and normal frames ( 15 vectors) without changing the immersion). Thus the "Cauchy-corrected" Cartan c,haracters read 16, i0, 8, 0}. so there are no free functions of 4 variables in the solution. This demonstrates very neatly the uniqueness of the solution given the boundary data.

CRZ := !ine~rise(GRO)integra1_element GRO)$ ch~urac~ers GRZ;

16.10.s.20

dim_grassmann_var iet y GRZ;

~,34

Z 6+2"I0+3"8+4~21=134 ==> involution!

l e ~ h cauchy_system GRZ;

34

4.2. Twisting D,pe-N solutions

The .search for exact solutions of Einstein's equations of type-N with nonvanishing twist has been rather barren. So far, there is only one known solution, due to Hauser [ 11 ] over 20 years ago, despite considerable effort on the part of many people (see [6] for a historical review). In Hauser's original (and successful!) api~-oach, null : e ~ s ,.,,.ere used in 4 real dimensions, making a series of gauge choices and an ansatz to simplify !be fe,.,~ of Ein~zin's equat'o=~.-: t~traduci~g some coordinates led to a s:Jstem of p.-,a.;al differential equations

D. Hartley/Compu,et Pb.ysics Communications !00 (1997) 177-194 191

which were no' compatible, but pos~_=~ui integrability conditions. Adding these integrability conditions and their consequences finally led (after some further coordinate choices and some algebra) to the who~ system collapsing to a single ODE, solvable in teans of hypergeometric functions. The solution was shown to have a Killing vector (and later a hcmothetic Killing vector), although this information was not explicitly fed into the derivation.

Subsequent attempts to find an alternative solution of the same type have invariably led to a morass of unmanageable equations, with little hope of determining solutions or even telling whether these exist or t~ow general they might be. One recent approach which has succeeded in yielding some information is due to Fink'), and Pleba~ski [5], and aims at solutions ha~ing no Killing symmetries. It involves moving first to comph~x general relativity (doubling the real dimension of the working space), and using so-ca!led hyper-heavenly spaces. These have the useful property that the geometry is described by ~, single potential function satisfying a single partial differential equation. Inserting an ansatz designed to ensure real N-type solutions leads to an implicit relation between the four "independent" variables, yielding ,:ome rather untidy partial differential eqvations. A neat choice of anholonomic coordinates for the independent variables improves things considerably, lea,~,ing a quasilinear set of partial differential equations of mixed second and fourth order,

F33 - TF = 0 ,

~ 2 x l + 2F~.xl2 + x j 2 2 f - x l F A = O,

2F23xlX23 -- F! (x23) 2 -b F2XlX233 + F3x| ~223 q" l x l x2233F = O,

A 3 ~-0, "Y2 ----0, A I =~ O, Yl :~ 0 (10)

for the unknowns ~F, x, y, A}. The indices {1,2, 3} are derivatives wi~b respect to the anholonomic basis with; for e~.ample,

F21 - F I 2 = 0 ,

F31 -- El,_ = - - F , x 1 3 / x l ,

F32 - F23 = - F I x 2 3 / x l . ( ! 1 )

Analysing this ~.ystem for involution poses severe c~:mputational problems (the final calculation took the original authors at least ,J months by hand!). First, there is the anhok, r.amlc coframe to be estal~lished. This is accomplished in EDS by a series of coordinate and coframe changes on an ordinary jet bundt~. N,,,,i, :h-_ f_ac*.or_ x2233 in the third equation in (10) would suggest working on the fourth jet bundle for three independent and four independent vadable~. 11~is space has dimension 143. More importantly, there are 60 principal (i.e. fo',a'th order) coordinates, which in effect means potentially having to invert 60 x 60 matrices to check involution. It is more efficient to work on a hybrid jet bundle of mixed orders, j421 (as was done by hand), reflecting the different orders with which the unknowns appear in the equations. This space has dimension 56, and the principal component has dimension 23, which is much more satisfactory.

The REDUCE script showing the EDS calculation follows the d,~scription given in [6]. After dzclaring the variables and setting some switches, the first jet bundle contact system for the unknown x is generated using the command c o n t a c t . The aim is to do the coframe transformation on this space, and then prolong to fourth order, adding the remaining unknowns at the appropriate stages, to achieve the final hybrid space.

on eva l lhseqp ,eds$1oppy; o f f arbvars ; pform { F , x , D e l t a , g a ~ a , v , y , u , v ( i ) } = O , o m e g a ( i ) = l ; indexrange ~i,j,k,1}={l,2,3}; 21 := ¢ontact(1,{v,y,tl},{x});

192 D. Hartley /Computer Physics Communications !00 (1997) 177-194

"Ga: anholonomic cofi'anae is achieved by a combination of coordinate and coframe transformations. Note that tim coordinate transformation is valid only if xt 4: 0, so this restriction is added with a r e s t r i c t commat~d. The (suppressoJ) ( c o f r a m i n g ) object is automaticMly updated with each tiansformation of the ( E D S ) .

korder x ( -1 ) ,x ( -2 ) ,v ( -3 ) ; col := {x(-v) = x ( - t ) ,

x(-y) = x(-2) , xC-u) = - x C - t ) * v ( - 3 ) } ;

CCI:= {Xv= Xl ,Xy=X2, Xu= --XIV3}

J l := r e s t r i c t ( tml lback( J i , cc l ) , {x ( -1 ) neq 0});

Jt :=

b c l := {omega( l ) = d v - v ( - 3 ) * d u, omega(2) = d y, o m e g a ( 3 ) = d u } ;

bet:= {wl= - v 3 d u + d v , w 2 =dr ~3 =d,}

J1 := t r ~ n ~ f o r ~ ( J l , b c l ) ;

J, := E D S ( { d x - ±,wl _ x2w2}.w, A ~2 A w 3)

Next follows a sequence of prolongations to create the contact system for j421, each increasing the order of the unknowns present by !. The c~'oa~s commands extend the space by the named coordinates. After this, the partial differential equations and restrictions are applied to the system using the p u l l b a c k command.

J2 := p r o l o n g 3 1 5 J20 := J2 c ross {F}$ 331 : = p r o l o n g J205 J310 := J31 cross { D e l t a , g a ~ a } $ J4~l := p ro long J3105

DE1 := {De l taC-3 ) - O, g ~ ( - 2 ) = O, D e l t a & l ) neq O, g ~ a L - 1 ) neq O, Delta. neq O, G a ~ neq O, F neq O} ;

DEt := {A3 = 0,y2 = 0,At ~0, Yt ~t0, A ~0,T ~0, F ~0}

DE2 := {F(-3,-3)=gam~a*F, FC-2,-2)=C(x(-1)*Delta- x( -1 , -2 -2))*F

- 2*x ( - I , -2 )*F( -2) ) /X( - I ) , F(-2,-3)=(2*x(-2,-3)**2*F(-1)

- 2*x (-1)*x(-2,-3,-3)*F(-2) - 2*x (- I )*x(-2, -2 , -3)*F(-3) - z( - l )*x( -2 , -2 , -3 , -3 )*F ) / (4ex ( - i ) * x ( - 2 , - 3 ) ) } ;

{ F3; = Fy, DE2 :.-..

r ~ F - 2~x:2 - x:22F F22 =

xl

D. Hortley /Computer Physic ¢ Communication,~ I00 (1997) 177-194

F2~ = (-2.rtF2x233 - 2x; F3x223

F + 2F! (x23)2)/(4xzx~_~) } -g2233

Y421 : = pul lback(J421 ,append(DEl ,DE,?.) )$

|93

Finally, the resulting system is checked using Cartan's test and found to he noninvolutiv,,. One prolongation is sufficient to bring it into ;nvolution. The on ranpos statement is necessary to give the co~,-ect Cartan characters, since the given anholo~omic frame is characteristic. With raapos on, a linear t r a n s f ~ with random constant coefficients of the base coframe is made to avoid the characteristic directio~,s. The t o r s i o n command checks for integrability conditions~

on ~anpGs; c h a r a c t e r s Y421;

{15.7.0}

dim_grassmann_variegy ¥421;

28

15+2.7 - 29 > 28:Y421 not i n v o l u t i v e , so pro long

¥532 := p ro long Y4215 c h a r a c t e r s Y532;

{22,6,0}

dim_gras~mann_vartety Y532;

34

Z 22+2*6 " 34: check i n t e g r a b i l i t y condi t ion~

t o r s i o n ¥532;

{}

Running on a 50 Mhz Sparc server under SunOS5, the calculation took about 40 seconds using around 8 MB of storage. This calculation shows the potential of such computer algebra iools. It is possible to repeat this calculation over and over with alternative parameters and ans~ze, whereas the thought of the calculation ahead would make most people think twice before repeating she analysis by hand. The features of EDS which were important here in achieving this result are the flexibility in setting up the woblem on the hybrid :;pace, and the ability provided by the (coframing) object to work with anholonomic coframes.

Acknowledgements

The work described here was carried out with the support of the Graduate College on Scientific Computing, University of Cologne and GMD St Augustin, funded by the Deutsche Forschungsgemeinschafi (DFG). I would like to express my thanks to R.W. Tucker, E. Schnifer, P.A. Tucke~, EW. Hehl and R.B. Gardner for helpful and encouraging discussions.

Appendix A

Tables 1-8 summagise the commands available in EDS. More detailed descriptions of the syntax ~ function of each command are to be found in the manual [8]. In each table, examples of the command age given,

194. D. Hartle y l Computer Physics Communications 100 (1997) 177-194

w h e r e b y t h e a r g u m e n t va r i ab les have the fo | l owin~ .~_nes ( . tee Sec t ion 2 ) :

E, E' ( ~ S ) S (system) M, N (coframing), o r a (~stem) s rm:: ifying a (coframing) r (integer) l'l (p-form) f (~,v) rsx (list of inequalities) cob (list o f 1-form variables) crd, dep, ind (list of O-form variables) dry (list of ru!es for exterior derivatives) pde (list of expressions or equations) x (tra_~orm) T (tableau) P (integral element)

References

[ l ] E.A. Arais, V.P. Shapeev and N.N. Yanenko, Computer realization of Cartan's exterior calculus, Soviet Math. Dokl. 15 ~1974) 203-205.

[2] R.L. Bytaat, S.S. Chem, R.B. Gardner, H.L. Goldschmidt and P.A. Griffiths, Exterior Differential Systems (Springer, New York, 19'01).

[3] E. Schrfifer, EXCALC, a system for doing calculations in the calculus of modem differential geometry, User's manual (Rand Corporation, Santa Mort|ca, 1986).

|46 EB. ~ and H.D. Wahlquist, Immersion ideals and the causal structure of Ric~.i-flat geometries, Class. Quantum Gray. 10 (1993) 1851-1858.

65] .I.D. Haley ill and J.E P!ebadski, Equations for twisting, type-N, vac,'um Einstein spaces without a need for Kitling vectors, J. Geom. Phys. 8 (1992) 173-193.

666 i.D. Finley !11 and A. Price, The invo|utive prolongation of the (complex) ~wisting, type-N vacuum field equations, Proc. Aspects of Genet~ Re~rivity ~ Mathematical Physics, eds. N. Bret6n, R. ~Capovilla and T. Matos (1993) pp. 1-18.

67] V.G. Ganzha, S.V. Meleshko, F.A. Murzin, V.P. Shapeev and N.N. Y~i.~oko, Computer realization of an algorithm for inv©stigating c o m ~ l i t y of ~ysterns of partial differential eqtmtions, Soviet Math. Dokl. 24 ( 1981 ) 638-640.

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