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\chapter[CAMAL: Celestial Mechanics]{CAMAL: Calculations in Celestial Mechanics} 
\label{CAMAL}
\typeout{{CAMAL: Calculations in Celestial Mechanics}}

{\footnotesize
\begin{center}
J. P. Fitch \\
School of Mathematical Sciences, University of Bath\\
BATH BA2 7AY, England \\[0.05in]
e--mail: jpff@cs.bath.ac.uk
\end{center}
}

\ttindex{CAMAL}
The CAMAL package provides facilities for calculations in Fourier
series similar to those in the specialist Celestial Mechanics system
of the 1970s, and the Cambridge Algebra system in
particular.\index{Fourier Series}\index{CAMAL}\index{Celestial
Mechanics} 

\section{Operators for Fourier Series}

\subsection*{\f{HARMONIC}}\ttindex{HARMONIC}

The celestial mechanics system distinguish between polynomial
variables and angular variables.  All angles must be declared before
use with the \f{HARMONIC} function. 
\begin{verbatim}
        harmonic theta, phi;
\end{verbatim}


\subsection*{\f{FOURIER}}\ttindex{FOURIER}

The \f{FOURIER} function coerces its argument into the domain of a
Fourier Series.   The expression may contain {\em sine} and {\em
cosine} terms of linear sums of harmonic variables.
\begin{verbatim}
        fourier sin(theta)
\end{verbatim}

Fourier series expressions may be added, subtracted multiplies and
differentiated in the usual \REDUCE\ fashion.  Multiplications involve
the automatic linearisation of products of angular functions.

There are three other functions which correspond to the usual
restrictive harmonic differentiation and integration, and harmonic
substitution.

\subsection*{\f{HDIFF} and \f{HINT}}\ttindex{HDIFF}\ttindex{HINT{}}

Differentiate or integrate a Fourier expression with respect to an angular
variable.  Any secular terms in the integration are disregarded without
comment.
\begin{verbatim}
    load_package camal;
    harmonic u;
    bige := fourier (sin(u) + cos(2*u));
    aa := fourier 1+hdiff(bige,u); 
    ff := hint(aa*aa*fourier cc,u);
\end{verbatim}

\subsection*{\f{HSUB}}\ttindex{HSUB}

The operation of substituting an angle plus a Fourier expression for
an angles and expanding to some degree is called harmonic substitution.
The function takes 5 arguments; the basic expression, the angle being
replaced, the angular part of the replacement, the fourier part of the
replacement and a degree to which to expand.
\begin{verbatim}
    harmonic u,v,w,x,y,z;
    xx:=hsub(fourier((1-d*d)*cos(u)),u,u-v+w-x-y+z,yy,n);
\end{verbatim}

\section{A Short Example}

The following program solves Kepler's Equation as a Fourier series to
the degree $n$.

\begin{verbatim}
        bige := fourier 0;
        for k:=1:n do <<
          wtlevel k;
          bige:=fourier e * hsub(fourier(sin u), u, u, bige, k);
        >>;
        write "Kepler Eqn solution:", bige$
\end{verbatim}