# PlanetPhysics/Maple for Planet Physics

\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle [{\it poincare},{\it generate_ic},{\it zoom},{\it hamilton_eqs}\\ \mbox{}]$ } \end{maplelatex}\begin{maplelatex}\begin{Maple Heading 1}{The Toda Hamiltonian}\end{Maple Heading 1} \end{maplelatex} \begin{maplelatex}\begin{Maple Normal}{Reference: A.J. Lichtenberg and M.A. Lieberman, "regular and Stochastic motion", Applied Mathematical Sciences 38 (New York: Springer Verlag, 1994).}\end{Maple Normal} H := 1/2*(p1\symbol{94 2 + p2\symbol{94}2) + 1/24*(exp(2*q2+2*sqrt(3)*q1) + exp(2*q2-2*sqrt(3)*q1) + exp(-4*q2))-1/8;}\end{maplelatex} \mapleresult \begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle H\, := \,1/2\,{{\it p1}}^{2}+1/2\,{{\it p2}}^{2}+1/24\,{e^{2\,{\it q2}+2\,\sqrt {3}{\it q1}}}\\ \mbox{}+1/24\,{e^{2\,{\it q2}-2\,\sqrt {3}{\it q1}}}+1/24\,{e^{-4\,{\it q2}}}-1/8$ } \end{maplelatex}\begin{mapleinput} \mapleinline{active}{1d}{H, t=-150..150, \{[0,.1,1.4,.1,0]\ :}}{} \end{mapleinput} \begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}{poincare( }{} \end{mapleinput}

\mapleresult \begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle {\it _____________________________________________________}$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt H = .99005020}},\,\mbox {{\tt  Initial conditions:}},\,t=0,\,{\it p1}= 0.1,\,{\it p2}= 1.4,\,{\it q1}= 0.1,\,{\it q2}=0$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt Number of points found crossing the (p1,q1) plane: 127}}$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt Maximum H deviation : .5740000000e-5 \ \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ [itex]{\it _____________________________________________________}$ } \end{maplelatex}\begin{maplelatex}\mapleinline{inert}{2d}{ $\displaystyle \mbox {{\tt Time consumed: 12 seconds`}}$ } \end{maplelatex}

\medskip \begin{figure} \includegraphics{poly.eps} \end{figure} \medskip

\begin{maplelatex}\begin{Maple Normal}{Figure 1.a. shows a 2-D surface-of-section (2PS) over the q2=0 plane, with 127 intersection points lying on smooth curves. }\end{Maple Normal} \end{maplelatex}\begin{mapleinput} \mapleinline{active}{1d}{\begin{Maple Normal}{poincare( }{} \end{mapleinput}