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\chapter{EDS: Exterior differential systems}
\label{EDS}
\typeout{{EDS: Exterior differential systems}}

{\footnotesize
\begin{center}
David Hartley \\
Physics and Mathematical Physics \\
University of Adelaide SA 5005, Australia \\
e-mail: DHartley@physics.adelaide.edu.au
\end{center}
}
\ttindex{EDS: Exterior differential dystems}
\ttindex{EDS}

\section{Introduction}

Exterior differential systems give a geometrical framework for partial
differential equations and more general differential geometric problems.
The geometrical formulation has several advantages stemming from its
coordinate-independence, including superior treatment of nonlinear and
global problems. {\tt EDS} provides a number of tools for setting up and 
manipulating exterior differential systems and implements many features of 
the theory. Its main strengths are the ability to use anholonomic or moving 
frames and the care taken with nonlinear problems.

The package is loaded 
%\footnote{The package {\tt EXCALC}
%(Chap. \ref{EXCALC} p. \pageref{EXCALC}) and the package {\tt XIDEAL} 
%(Chap. \ref{XIDEAL} p. \pageref{XIDEAL}) are loaded automatically with
%this package.} 
by typing \quad {\tt load eds;} \par
Reading the full documentation, which comes with this 
package, is strongly recommended.
The test file eds.tst, which is also in the package,  provides
three inspiring examples on the subject. \\
EDS uses E.~Schr{\"u}fer's EXCALC package for the underlying
exterior calculus operations.

\section{Data Structures and Concepts}
\subsection{EDS}
A simple \meta{EDS}, or exterior differential system, is a triple
{\tt (S,$\Omega$,M)}, where {\it M} is a {\it coframing}, {\it S} is a
system on {\it M}, and {\it $\Omega$} is an independence condition. 
Exterior differential equations without independence condition are not
treated by {\tt EDS}. {\it $\Omega$} should be either a decomposable
\meta{p-form} or a \meta{system} of 1-forms on {\it M}. \\
More generally an \meta{EDS} is a list of simple \meta{EDS} objects
where the various coframings are all disjoint. \\
The solutions of {\it (S,$\Omega$,M)} are integral manifolds, or immersions
on which {\it S} vanishes and the rank of $\Omega$ is preserved. Solutions
at a single point are described by integral elements.

\subsection{Coframing}
Within the context of {\tt EDS}, a {\it coframing} means a real 
finite-dimensional differentiable manifold with a given global cobasis. 
The information about a coframing required by {\tt EDS} is kept in a
\meta{coframing} object. The cobasis is the identifying element of
an {\tt EDS}. In addition to the cobasis, there can be given {\it coordinates,
structure equations} and {\it restrictions}. 
In addition to the cobasis, {\it coordinates, structure equations} and
{\it restrictions} can be given.
The coordinates may be an incomplete or 
overcomplete set. The structure equations express the exterior derivative of the
coordinates and cobasis elements as needed. All coordinate differentials must 
be expressed in terms of the given cobasis, but not all cobasis derivatives
need be known. 
The restrictions are a set of inequalities describing point sets
not in the manifold. \\
Please note that the \meta{coframing} object is by no means a full description
of a differentiable manifold. However, the \meta{coframing} object carries 
sufficient information about the underlying manifold to allow a range of exterior
systems calculations to be carried out.

\subsection{Systems and background coframing}
The label \meta{system} refers to a list $\{<${\it p-form expr}$>,\ldots\}$ of
differential forms. If an {\tt EDS} operator also accepts a \meta{system} as 
argument, then any extra information which is required is taken from the
background coframing. \\
It is possible to activate the rules and orderings of a \f{COFRAMING} operator
globally, by making it the {\it background coframing}. All subsequent \f{EXCALC}
\ttindex{EXCALC} operations will be governed by those rules. Operations on 
\meta{EDS} objects are unaffected, since their coframings are still activated
locally.

\subsection{Integral elements}
An \meta{integral element} of an exterior system $(S,\Omega,M)$ is a subspace
$P \subset T_pM$ of the tangent space at some point $p \in M$. This integral
element can be represented by its annihilator $P^\perp \subset T^*_pM$, comprising
those 1-forms at $p$ which annihilate every vector in $P$. This can also be understood
as a maximal set of 1-forms at $p$ such that $S \simeq 0 \pmod{P^\perp}$ and the
rank of $\Omega$ is preserved modulo $P^\perp$. \\
An \meta{integral element} in EDS is a distribution of 1-forms on $M$,
specified as a \meta{system} of 1-forms.

\subsection{Properties and normal form}
For large problems, it can require a great deal of computation to establish
whether, for example, a system is closed or not. In order to save
recomputing such properties, an \meta{EDS} object carries a list of
\meta{properties} of the form 
\begin{list}{}
\item {\tt \{\meta{keyword} = \meta{value},$\cdots$\}}
\end{list}
where \meta{keyword} is one of \f{closed}, \f{quasilinear}, \f{pfaffian} or
\f{involutive}, and \meta{value} is either \f{0} (false) or \f{1}
(true). These properties are suppressed when an \meta{EDS} is printed,
unless the \f{nat} switch is \f{off}. They can be examined using the
\f{PROPERTIES} operator. \\
Parts of the theory of exterior differential systems apply only at points
on the underlying manifold where the system is in some sense
non-singular. To ensure the theory applies, EDS automatically works all
exterior systems $(S,\Omega,M)$ into a {\em normal form}. This means that
the Pfaffian component of $S$ and the independence condition $\Omega$ are 
in {\it solved} forms, distinguished terms from the 1-forms in $S$ have
been eliminated from the rest of $S$ and from $\Omega$ and any 1-forms in
$S$ which vanish modulo the independence condition are removed from the
system and their coefficients are appended as 0-forms.

\section{The EDS Package}

In the descriptions of the various operators we define the following
abbreviations for function parameters:
\vspace{0.25cm}

\begin{tabular}{ll}
$E$, $E'$   & \meta{EDS}\\
$S$         & \meta{system}\\
$M$, $N$    & \meta{coframing}, or a \meta{system} specifying a \meta{coframing}\\
$r$         & \meta{integer}\\
$\Omega$    & \meta{p-form}\\
$f$         & \meta{map}\\
$rsx$       & \meta{list of inequalities}\\
$cob$       & \meta{list of 1-form variables}\\
$crd$, $dep$, $ind$     
            & \meta{list of 0-form variables}\\
$drv$       & \meta{list of rules for exterior derivatives}\\
$pde$       & \meta{list of expressions or equations}\\
$X$         & \meta{transform}\\
$T$         & \meta{tableau}\\
$P$         & \meta{integral element}\\
\end{tabular}


\subsection{Constructing EDS objects}

An EDS \meta{coframing} is constructed using the \f{COFRAMING} operator.
In one form it examines the argument for 0-form and 1-form variables. The more
basic syntax takes the \meta{cobasis} as a list of 1-forms, \meta{coordinates}
as a list of 0-forms, \meta{restrictions} as a list of inequalities and 
\meta{structure equations} as a list giving the exterior derivatives of the
coordinates and cobasis elements. All arguments except the cobasis are optional. \\
A simple \meta{EDS} is constructed using the \f{EDS} operator where the
\meta{indep. condition} can be either a decomposable \meta{p-form} or a
\meta{system} of 1-forms. The \meta{coframing} and the \meta{properties}
arguments can be omitted. The {\it EDS} is put into normal form before being 
returned. With \f{SET\_COFRAMING} the background coframing is set. \\
The operator \f{PDS2EDS} encodes a PDE system into an \meta{EDS} object. \\

\begin{tabular}{lll}
\f{COFRAMING}(cob,crd,rsx,drv)\ttindex{COFRAMING} &
\f{COFRAMING}(S)\ttindex{COFRAMING} &
\f{EDS}(S,$\Omega$,M)\ttindex{EDS} \\
\f{CONTACT}(r,M,N)\ttindex{CONTACT} &
\f{PDE2EDS}(pde,dep,ind)\ttindex{PDE2EDS} &
\f{SET\_COFRAMING}(M)\ttindex{SET\_COFRAMING} \\
\f{SET\_COFRAMING}(E)\ttindex{SET\_COFRAMING} &
\f{SET\_COFRAMING}()\ttindex{SET\_COFRAMING} 
\end{tabular}
\vspace{0.5cm}

{\bf Example:}
\begin{verbatim}
1: load eds;

2: pform {x,y,z,p,q}=0,{e(i),w(i,j)}=1;

3: indexrange {i,j,k}={1,2},{a,b,c}={3};

4: eds({d z - p*d x - q*d y, d p^d q},{d x,d y});

EDS({d z - p*d x - q*d y,d p^d q},d x^d y) 

5: OMrules:=index_expand {d e(i)=>-w(i,-j)^e(j),w(i,-j)+w(j,-i)=>0}$

6: eds({e(a)},{e(i)}) where OMrules;

      3    1  2
EDS({e },{e ,e })

7: coframing ws;
            3  2    1  2         1        2  2 
coframing({e ,w   ,e ,e },{},{d e  =>  - e ^w   ,
                 1                             1 
               2     1  2 
            d e  => e ^w   },{})
                          1  
\end{verbatim}


\subsection{Inspecting EDS objects}

Using these operators you can get parts of your \meta{EDS} object. The
\f{PROPERTIES}(E) operator for example returns a list of properties which are
normally not printed out, unless the \f{NAT}\ttindex{NAT} switch is off. \\

\begin{tabular}{lll}
\f{COFRAMING}(E)\ttindex{COFRAMING} &
\f{COFRAMING}()\ttindex{COFRAMING} &
\f{COBASIS}(M)\ttindex{COBASIS} \\
\f{COBASIS}(E)\ttindex{COBASIS} &
\f{COORDINATES}(M)\ttindex{COORDINATES} &
\f{COORDINATES}(E)\ttindex{COORDINATES} \\
\f{STRUCTURE\_EQUATIONS}(M)\ttindex{STRUCTURE\_EQUATIONS} &
\f{STRUCTURE\_EQUATIONS}(E)\ttindex{STRUCTURE\_EQUATIONS} &
\f{RESTRICTIONS}(M)\ttindex{RESTRICTIONS} \\
\f{RESTRICTIONS}(E)\ttindex{RESTRICTIONS} &
\f{SYSTEM}(E)\ttindex{SYSTEM} &
\f{INDEPENDENCE}(E)\ttindex{INDEPENDENCE} \\
\f{PROPERTIES}(E)\ttindex{PROPERTIES} &
\f{ONE\_FORMS}(E)\ttindex{ONE\_FORMS} &
\f{ONE\_FORMS}(S)\ttindex{ONE\_FORMS} \\
\f{ZERO\_FORMS}(E)\ttindex{ZERO\_FORMS} &
\f{ZERO\_FORMS}(S)\ttindex{ZERO\_FORMS} &
\end{tabular}
\vspace{0.5cm}

{\bf Example:}
\begin{verbatim}
8: depend u,x,y; depend v,x,y;

9: pde2eds({df(u,y,y)=df(v,x),df(v,y)=y*df(v,x)});

EDS({d u - u *d x - u *d y, d u  - u   *d x - u   *d y,
            x        y         x    x x        y x

     d u  - u   *d x - v *d y, d v - v *d x - v *y*d y},d x^d y)
        y    y x        x             x        x

10: dependencies;

{{u,y,x},{v,y,x}}

11: coordinates contact(3,{x},{u});

{x,u,u ,u   ,u     }
      x  x x  x x x 

12: fdomain u=u(x);

13: coordinates {d u+d y};

{x,y} 

\end{verbatim}

\subsection{Manipulating EDS objects}

These operators allow you to manipulate your \meta{EDS} objects. The
\f{AUGMENT}(E,S) operator, see example below, appends the extra forms in the second
argument to the system part of the first. The original \meta{EDS} remains
unchanged. As another example by using the \f{TRANSFORM} operator 
a change of the cobasis is made, where the argument \meta{transform} is a list of
substitutions. \\

\begin{tabular}{llll}
\f{AUGMENT}(E,S)\ttindex{AUGMENT} &
$M$ \f{CROSS} $N$\ttindex{CROSS} &
$E$ \f{CROSS} $N$\ttindex{CROSS} &
\f{PULLBACK(E,f)}\ttindex{PULLBACK} \\
\f{PULLBACK}(S,f)\ttindex{PULLBACK} &
\f{PULLBACK}($\Omega$,f)\ttindex{PULLBACK} &
\f{PULLBACK}(M,f)\ttindex{PULLBACK} &
\f{RESTRICT}(E,f)\ttindex{RESTRICT} \\
\f{RESTRICT}(S,f)\ttindex{RESTRICT} &
\f{RESTRICT}($\Omega$,f)\ttindex{RESTRICT} &
\f{RESTRICT}(M,f)\ttindex{RESTRICT} &
\f{TRANSFORM}(M,X)\ttindex{TRANSFORM} \\
\f{TRANSFORM}(E,X)\ttindex{TRANSFORM} &
\f{TRANSFORM}(S,X)\ttindex{TRANSFORM} &
\f{TRANSFORM}($\Omega$,X)\ttindex{TRANSFORM} &
\f{LIFT(E)}\ttindex{LIFT} \\
\end{tabular}
\vspace{0.5cm}

{\bf Example:}
\begin{verbatim}
% Non-Pfaffian system for a Monge-Ampere equation

14: PFORM {x,y,z}=0$

15: S := CONTACT(1,{x,y},{z});

s := EDS({d z - z *d x - z *d y},d x^d y)
                 x        y

16: S:= AUGMENT(S,{d z(-x)^d z(-y)});

s := EDS({d z - z *d x - z *d y,
                 x        y

          d z ^d z },d x^d y)
             x    y
\end{verbatim}


\subsection{Analysing and Testing exterior systems}

{\bf Analysing exterior systems} \par
This section introduces higher level operators for extracting information about
exterior systems. Many of them require a \meta{EDS} in normal form generated
in positive degree as input, but some can also analyse a \meta{system} or a
single \meta{p-form}. \\ 

\begin{tabular}{lll}
\f{CARTAN\_SYSTEM}(E)\ttindex{CARTAN\_SYSTEM} &
\f{CARTAN\_SYSTEM}(S)\ttindex{CARTAN\_SYSTEM} &
\f{CARTAN\_SYSTEM}($\Omega$)\ttindex{CARTAN\_SYSTEM} \\
\f{CAUCHY\_SYSTEM}(E)\ttindex{CAUCHY\_SYSTEM} &
\f{CAUCHY\_SYSTEM}(S)\ttindex{CAUCHY\_SYSTEM} &
\f{CAUCHY\_SYSTEM}($\Omega$)\ttindex{CAUCHY\_SYSTEM} \\
\f{CHARACTERS}(E)\ttindex{CHARACTERS} &
\f{CHARACTERS}(T)\ttindex{CHARACTERS} &
\f{CHARACTERS}(E,P)\ttindex{CHARACTERS} \\
\f{CLOSURE}(E)\ttindex{CLOSURE} &
\f{DERIVED\_SYSTEM}(E)\ttindex{DERIVED\_SYSTEMS} &
\f{DERIVED\_SYSTEM}(S)\ttindex{DERIVED\_SYSTEMS} \\
\f{DIM\_GRASSMANN\_VARIETY}(E)\ttindex{DIM\_GRASSMANN\_VARIETY} &
\f{DIM\_GRASSMANN\_VARIETY}(E,P)\ttindex{DIM\_GRASSMANN\_VARIETY} &
\f{DIM}(M)\ttindex{DIM} \\
\f{DIM}(E)\ttindex{DIM} &
\f{INVOLUTION}(E)\ttindex{INVOLUTION} &
\f{LINEARISE}(E,P)\ttindex{LINEARISE} \\
\f{INTEGRAL\_ELEMENT}(E)\ttindex{INTEGRAL\_ELEMENT} &
\f{PROLONG}(E)\ttindex{PROLONG} &
\f{TABLEAU}(E)\ttindex{TABLEAU} \\
\f{TORSION}(E)\ttindex{TORSION} &
\f{GRASSMANN\_VARIETY}(E)\ttindex{GRASSMANN\_VARIETY} &
\end{tabular}
\par
\ \\
{\bf Testing exterior systems} \par
The following operators allow various properties of an \meta{EDS} to be checked.
The result is either a {\bf 1} or a {\bf 0}, so these operators can be used in 
boolean expressions. Since checking these properties is very time-consuming, the
result of the first test is stored on the \meta{properties} record of an
\meta{EDS} to avoid re-checking. This memory can be cleared using the
\f{CLEANUP}\ttindex{CLEANUP} opearator. \\

\begin{tabular}{llll}
\f{CLOSED}(E)\ttindex{CLOSED} &
\f{CLOSED}(S)\ttindex{CLOSED} &
\f{CLOSED}($\Omega$)\ttindex{CLOSED} &
\f{INVOLUTIVE}(E)\ttindex{INVOLUTIVE} \\
\f{PFAFFIAN}(E)\ttindex{PFAFFIAN} &
\f{QUASILINEAR}(E)\ttindex{QUASILINEAR} &
\f{SEMILINEAR}(E)\ttindex{SEMILINEAR} &
$E$ \f{EQUIV} $E'$\ttindex{EQUIV} \\
\end{tabular}
\vspace{0.5cm}

\subsection{Switches}

EDS provides several switches to govern the display of information and enhance 
the speed or reliability of the calculations. For example the switch \f{EDSVERBOSE} 
if {\tt ON} will display additional information as the calculation progresses, 
which might generate too much output for larger problems. \\
All switches are {\tt OFF} by default.

\begin{tabular}{llllll}
\f{EDSVERBOSE}\ttindex{EDSVERBOSE} &
\f{EDSDEBUG}\ttindex{EDSDEBUG} &
\f{EDSSLOPPY}\ttindex{EDSSLOPPY} &
\f{EDSDISJOINT}\ttindex{EDSDISJOINT} &
\f{RANPOS}\ttindex{RANPOS} &
\f{GENPOS}\ttindex{GENPOS} \\
\end{tabular}

\subsection{Auxilliary functions}

The operators of this section are designed to ease working with exterior forms
and exterior systems in {\REDUCE}\ . \\

\begin{tabular}{lll}
\f{COORDINATES}(S)\ttindex{COORDINATES} &
\f{INVERT}(X)\ttindex{INVERT} &
\f{STRUCTURE\_EQUATIONS}(X)\ttindex{STRUCTURE\_EQUATIONS} \\
\f{STRUCTURE\_EQUATIONS}(X,$X^{-1}$)\ttindex{STRUCTURE\_EQUATIONS} &
\f{LINEAR\_DIVISORS}($\Omega$)\ttindex{LINEAR\_DIVISORS} &
\f{EXFACTORS}($\Omega$)\ttindex{EXFACTORS} \\
\f{INDEX\_EXPAND}(ANY)\ttindex{INDEX\_EXPAND} &
\f{PDE2JET}(pde,dep,ind)\ttindex{PDE2JET} &
\f{MKDEPEND}(list)\ttindex{MKDEPEND} \\
\f{DISJOIN}(f,g,...)\ttindex{DISJOIN} &
\f{CLEANUP}(E)\ttindex{CLEANUP} &
\f{CLEANUP}(M)\ttindex{CLEANUP} \\
\f{REORDER}(E)\ttindex{REORDER} &
\f{REORDER}(M)\ttindex{REORDER} &
\end{tabular}

\subsection{Experimental Functions}

The following operators are experimental facilities since, they are
either algorithmically not well-founded, or their implementation is
very unstable, or they have known bugs. \\

\begin{tabular}{lll}
\f{POINCARE}($\Omega$)\ttindex{POINCARE} &
\f{INVARIANTS}(E,crd)\ttindex{INVARIANTS} &
\f{INVARIANTS}(S,crd)\ttindex{INVARIANTS} \\
\f{SYMBOL\_RELATIONS}(E,$\pi$)\ttindex{SYMBOL\_RELATIONS} &
\f{SYMBOL\_MATRIX}(E,$\xi$)\ttindex{SYMBOL\_MATRIX} &
\f{CHARACTERISTIC\_VARIETY}(E,$\xi$)\ttindex{CHARACTERISTIC\_VARIETY} \\
\end{tabular}
\vspace{0.5cm}

{\bf Example:}
\begin{verbatim}
17: % Riemann invariants for Euler-Poisson-Darboux equation.
17: % Set up the EDS for the equation, and examine tableau.
17: depend u,x,y; EPD :=PDE2EDS{DF(u,x,y)=-(df(u,x)+df(u,y))/(x+y)}$

19: tableau EPD;

[d u       0   ]
[   x x        ]
[              ]
[  0     d u   ]
[           y y]

20: % 1-form dx is characteristic: construct characteristic EDS.
20: xvars {}; C := cartan_system select(~f^d x=0,system closure epd)$

22: S := augment(eds(system EPD,d y),C)$

23: % Compute derived flag
23: while not equiv(S,S1 := derived_system S) do S := S1;

24: % Stabilised. Find the Riemann invariants.
24: invariants(S,reverse coordinates S);

{x,

 u *x + u *y + u,
  x      x

  - u   *x - u   *y - 2*u }
     x x      x x        x
\end{verbatim}
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