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periodic orbits, chaos, anisotropic Manev problem, symmetry
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\documentclass[12pt]{article}
\begin{document}
\parindent 0 pt
\markright{Periodic Orbits of the Anisotropic Manev Problem}
\pagestyle{myheadings}
%\renewcommand{\baselinestretch}{1.5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def \ep{\epsilon}
\def \intR {\int_{-\infty}^{+\infty}}
\def \grad {\nabla}
\def \ov{\over}
\def \q {\quad}
\def \qq {\qquad}
\def \bar{\overline}
\def \beq{ \begin{equation} }
\def \eeq{\end{equation}}
\def \r#1{$^{#1}$}
\newtheorem{proposition}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{theorem}{Theorem}
\renewcommand{\labelenumi}{(\roman{enumi})}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Symmetric Periodic Solutions of the Anisotropic Manev Problem}
\author{{Manuele Santoprete}\renewcommand{\thefootnote}{\alph{footnote})}\footnotemark[1]
\\ Department of Mathematics and Statistics \\ University of
Victoria, P.O. Box 3045 Victoria B.C., \\ Canada, V8W 3P4}
\renewcommand{\thefootnote}{\alph{footnote})}
\date{}
\maketitle
\footnotetext[1]{Electronic mail: msantopr@math.uvic.ca}
\begin{abstract}
\noindent We consider the Manev potential in an anisotropic space, i.e., such
that the force acts differently in each direction. Using a
generalization of the Poincar\'e continuation method we study the
existence of periodic solutions for weak anisotropy. In particular we
find that the symmetric periodic orbits of the Manev system are
perturbed to periodic orbits in the anisotropic problem.
\end{abstract}
\vskip 0.5truecm
\hspace{25pt} PACS(2001): 05.45.-a, 45.50.Jf, 45.50.Pk
\vfill\eject
~
\vskip 2truecm
\small\normalsize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{\large\bf I. Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this paper we consider the Anisotropic Manev Problem (AMP) that was
introduced by Diacu\r{1} in the early 1990s. The work on the
AMP was inspired by the Anisotropic Kepler Problem introduced by
Gutzwiller in the early 1970s. Gutzwiller aimed to find connections
between classical and quantum mechanics. His interest was stimulated
by an old unsolved quantum mechanical problem formulated in a paper
written by Einstein:\r{2} even if the
Born-Sommerfeld-Einstein condition (e.g. see Ref. 2) were
appropriate to describe the semi-classical limit of quantum theory it
was unclear how to find a classical approximation for nonintegrable
systems.
Similarly the main reason for considering the AMP is to further
analyze similarities between classical mechanics and quantum
theory. Moreover, as it was remarked in Ref. 1, the AMP also
brings general relativity into the game, since, the Manev's potential,
explains the perihelion advance of the inner planets with the same
accuracy as general relativity.\r{3} It should be remarked
that bringing general relativity into the game is of particular
importance since a satisfactory quantum theory of gravitation does not
exist.
Some of the qualitative features of the Anisotropic Manev Problem have
already been studied. In Ref. 1, a large class of
capture-collision and ejection-escape solutions is studied by means of
the collision and infinity manifold techniques. In particular that
paper also brought arguments favouring the chaoticity and
nonintegrability of the system by showing the existence of
heteroclinic orbits within the zero energy manifold. In Ref. 4
the occurrence of chaos on the zero energy manifold and the
nonintegrability are finally proved, putting into evidence that the
AMP is a very complex problem.
In this work, to gain a better understanding of the complicated
dynamics of the AMP, we find the symmetric periodic orbits.
Analyzing those orbits is especially important since,
by now, it is well known that studying periodic orbits is a valuable
general approach to tackle complex problems in classical mechanics.
The existence of periodic orbits for small
values of the anisotropy is proved using generalizations
of the Poincar\'e continuation method developed in Refs. 5-7.
The (planar) anisotropic Manev problem is described by the Hamiltonian
\beq H={1 \over 2}{\bf p}^2 - {1 \over \sqrt{x^2+ \mu y^2}}-{b \over
x^2 +\mu y^2}.
\label{H}
\eeq where $\mu>1$ is a constant and ${\bf q}=(x,y)$ is the position
of one body with respect to the other considered fixed at the origin
of the coordinate system, and ${\bf p}=(p_x,p_y)$ is the momentum of
the moving particle. The constant $\mu$ measures the strength of the
anisotropy and for $\mu=1$ we recover the classical Manev problem.
Furthermore the equation of motion can be expressed as \beq \left\{
\begin{array}{l}
\dot {\bf q} = {\bf p} \\ \dot{\bf p} = -{\partial H \over
\partial{\bf q}}
\end{array}
\right. .
\label{eqmotion}
\eeq
Now consider weak anisotropies, i.e choose the parameter $\mu>1$ close
to 1. Introducing polar coordinates $x=
r\cos \theta$, $y= r\sin \theta$ and the notation $\ep=\mu-1$ with
$\ep \ll 1$ we can expand the Hamiltonian (\ref{H}) in powers of $\ep$
and obtain
\beq H={1\over 2}{\bf p}^2-{1\over r}-{b \over r^2} +\ep \left ({1
\over 2r} +{b \over r^2}\right )\cos^2\theta \equiv H_0+\ep
W(r,\theta).
\label{Hep}
\eeq
It should be pointed out that the term $W(r,\theta)$ becomes
unbounded as $r \rightarrow 0$ so that a perturbation analysis is not
correct on the ejection-collision orbits. This means that the global
dynamics of the AMP cannot be completely described by perturbations to
the Manev problem even at the limit $\ep \rightarrow 0$. However many
interesting results concerning the Hamiltonian (\ref{H}) for weak
anisotropies (i.e. $\ep \ll 1)$ can be found studying the Hamiltonian
(\ref{Hep}), some of which are presented in this paper.
In the next section we describe the symmetries of the AMP and we find
some properties that will be useful to find symmetric periodic orbits.
In Section III we prove a continuation theorem for the symmetric
periodic orbits of ``second kind'', i.e. the non-circular ones. In
Section IV we prove a continuation theorem for the orbits of ``first
kind'', i.e. the circular ones, following the method developed in Ref. 8.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{\large\bf II. Symmetries of the Anisotropic Manev Problem }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To find periodic orbits in the anisotropic problem it is peculiarly
important to know the symmetries of the system, as it was, for example
observed in Refs. 5, 6. The symmetries of the problem under
discussion have been examined in Ref. 1 and they are the same
as the ones found in Ref. 7 for the anisotropic Kepler
problem:
\beq \begin {array}{l} \
\hspace{3pt} E~:~(x,y,p_x,p_y,t)\longrightarrow(x,y,p_x,p_y,t)\\\
S_0~:~(x,y,p_x,p_y,t)\longrightarrow(x,y,-p_x,-p_y,-t)
\\\
S_1~:~(x,y,p_x,p_y,t)\longrightarrow(x,-y,-p_x,p_y,-t)\\\
S_2~:~(x,y,p_x,p_y,t)\longrightarrow(-x,y,p_x,-p_y,-t)\\\
S_3~:~(x,y,p_x,p_y,t)\longrightarrow(-x,-y,-p_x,-p_y,t)\\\
S_4~:~(x,y,p_x,p_y,t)\longrightarrow(-x,y,-p_x,p_y,t)\\\
S_5~:~(x,y,p_x,p_y,t)\longrightarrow(x,-y,p_x,-p_y,t)\\\
S_6~:~(x,y,p_x,p_y,t)\longrightarrow(-x,-y,p_x,p_y,-t)
\label{symmetries}
\end {array}
\eeq where $E$ is the identity.
The symmetries above can be interpreted in the following way: let
$\gamma(t)$ be a solution of (\ref{eqmotion}), then
$S_i(\gamma(t))$is another
solution for $i \in \{0,1,2,3,4,5,6\}$. For $i\in\{0,1,2,3,4,5,6 \}$ the orbit $\gamma(t)$ will be
called symmetric if and only if $S_i(\gamma(t))=\gamma(t)$.
Let us remark that the symmetries in (\ref{symmetries}), together
with the composition of functions, denoted by $\circ$, form an abelian
group in which the operation acts according to the table below.
$$
\begin{array}{c|cccccccc}
\circ & E & S_0 & S_1 & S_2 & S_3 & S_4 & S_5 & S_6 \\
\hline
E & E & S_0 & S_1 & S_2 & S_3 & S_4 & S_5 & S_6\\
S_0 & S_0 & E & S_5 & S_4 & S_6 & S_2 & S_1 & S_3\\
S_1 & S_1 & S_5 & E & S_3 & S_2 & S_6 & S_0 & S_4\\
S_2 & S_2 & S_4 & S_3 & E & S_1 & S_0 & S_6 & S_5\\
S_3 & S_3 & S_6 & S_2 & S_1 & E & S_5 & S_4 & S_0\\
S_4 & S_4 & S_2 & S_6 & S_0 & S_5 & E & S_3 & S_1\\
S_5 & S_5 & S_1 & S_0 & S_6 & S_4 & S_3 & E & S_2\\
S_6 & S_6 & S_3 & S_4 & S_5 & S_0 & S_1 & S_2 & E \\
\end{array}
$$
From the table above %and the Fundamental Theorem of Finite Abelian Groups
it is easy to deduce the following
\begin{proposition}
The symmetries of the Anisotropic Manev Problem form an elementary abelian
group of order eight, i.e. a group isomorphic to ${\bf Z}_2 \times {\bf Z}_2 \times
{\bf Z}_2$.
\end{proposition}
The symmetries in (\ref{symmetries}), (except $E$ and $S_6$) are very
useful to find symmetric periodic orbits, especially by means of the
continuation method, as we show in the next two sections. Some
important properties of the symmetric orbit, summarized in Ref. 7
, are expressed in the following lemma:
\begin{lemma}
\begin{enumerate}
\item For $i=1$ (resp. $i=2$) we have that an orbit $\gamma(t)$ is $S_i$-symmetric
if and only if it crosses the $x$ axis (resp. $y$ axis) orthogonally.
\item An orbit $\gamma (t)$ is $S_0$-symmetric if and only if it has a
point on the zero velocity curve. \item For $i=4,5$ an orbit
$\gamma(t)$ is $S_i$-symmetric if and only if it is $S_0$-symmetric.
\item All the $S_3$-symmetric orbits are periodic.
\end{enumerate}
\end{lemma}
The properties of the $S_i$-symmetric orbits were first studied by
Birkhoff\r{9} for the restricted three body problem and
later by many other authors. In particular Casasayas and Llibre\r{7}
state a proposition that gives a technique useful to
obtain symmetric periodic orbits with respect to $S_0$, $S_1$,$S_2$
for the anisotropic Kepler problem that are verified also for the
problem under discussion in this paper:
\begin{proposition}
\begin{enumerate}
\item For $i=1$ (resp. $i=2$) we have that an orbit $\gamma(t)$ is a
$S_i$-symmetric periodic orbit if and only if it crosses the
$x$ axis (resp. $y$ axis) orthogonally at two distinct points.
\item An orbit $\gamma(t)$ is a $S_0$-symmetric periodic orbit if and
only if it meets the zero velocity curves at two distinct points.
\item An orbit $\gamma(t)$ is a $S_1$ and $S_2$ symmetric periodic
orbit if and only if it crosses the x axis and the y axis
orthogonally.
\item For $i=1,2$ an orbit $\gamma(t)$ is a $S_0$ and $S_i$-symmetric
periodic orbit if and only if it meets the zero velocity curve and
crosses the x, respectively y axis orthogonally.
\item For $i=4,5$, if an orbit $\gamma(t)$ is $S_i$-symmetric then it
is $S_0$-symmetric and periodic.
\end{enumerate}
\end{proposition}
Now we want to find the symmetric periodic orbit for the unperturbed
problem ($\ep=0$ or $\mu=1$) and continue them to periodic solutions
of the anisotropic system (for $\ep \ll 1$). Firstly we observe that,
by Proposition 2, the $S_i$ symmetric orbits with $i=0,4,5$ must meet
the zero velocity curve at two points, i.e. there must be a point
where the angular momentum $K=xp_y-yp_x$ is zero, but since $K$ is a constant
of motion it must be zero
along the orbit. Therefore such orbits are ejection-collision orbits,
are not periodic and cannot be studied by means of the continuation
method. Hence we are going to consider the symmetric periodic orbit
with $i=1,2$, and also the ones with $i=3$ that are the circular
orbits of the unperturbed problem.
To exploit those properties of the symmetric periodic orbits it is
convenient to write the equation of motion in different coordinates.
For the $S_i$ symmetric orbits with $i=1,2$, as it was noted in Ref. 5
, it is convenient to write the canonical equations of
the restricted three body problem using the Delaunay variables in the
rotating frame.\r{5} Also the Poincar\'e synodic variables
can be used to find symmetric periodic orbits of the restricted three
body problem.\r{6} The anisotropic Manev problem is
different since the Hamiltonian that describes it is time independent,
hence the idea of using rotating coordinates in the present case
cannot be applied. Moreover our problem is nondegenerate, however even
in our case it is advantageous to perform a change of variables and apply a
variation of the action angle variables used in Refs. 4, 12.
Here the nondegeneracy of the problem plays a role similar
to the rotating coordinate system in the restricted three body
problem.
For the $S_3$ symmetric orbits we can instead consider the equations
in the rotating frame, and prove a theorem similar to the one proved
in Ref. 8 for the anisotropic Kepler problem (in Ref. 8
the author remarks that the analysis of the Kepler problem can be
redone in the Manev case, but he doesn't provide a proof) .
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{\large\bf III. The $\bf{S_i}$ Symmetric Orbits with $\bf{i=1,2}$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We recall that the action variables introduced in Refs. 4, 12 are given by
\beq \left \{
\begin{array}{l}
I={{1 \over 2 \pi} \oint p_r dr= - \sqrt{ K^2-2b} +{1 \over 2}{\sqrt
{2 \over |h|}}}\\ K= xp_y-yp_x
\end{array}
\right. \eeq
where $h$ is the energy constant and $K$ is the angular
momentum. These variables are defined for $h<0$ and $K^2>2b$, $I>0$,
to avoid collision orbits as well as circular orbits. The related
frequencies are
$$ \left \{
\begin{array}{l} \medskip
\omega_I= {1 \over (I+ \sqrt {K^2-2b})^3}\\ \medskip \omega_K={K \over
\sqrt {K^2-2b}(I+\sqrt {K^2-2b})^3},
\end{array}
\right.$$
and $\theta$ and $\phi$ are the angle variables associated to $K$ and $I$
respectively.
The unperturbed Hamiltonian in the new variables can
be written as
$$ H_0=-{1 \over 2(I+ \sqrt { K^2-2b})^2}. $$
Now we can consider new variables that are linear combination of the
previous ones. They are defined by the following canonical
transformation
\beq \left \{
\begin{array}{l}
L=K+I\\ G=-I\\ l=\theta\\ g=\theta - \phi
\end{array}
\right .
\label{newvar}
\eeq
Where $l$ is the mean anomaly (where $l(t)= \omega_L(t-t_0)$ and $t_0$
is the time of pericenter passage), $g$ is the longitude of
pericenter as they are defined for the Manev problem in Ref. 13.
Moreover also the action variables can be written in terms of the
orbital elements of the Manev problem. If we set
$$ a={1 \over 2|h|} \qquad \mbox{and} \qquad e=\sqrt{1-2(K^2-2b)|h|}
$$
as in Refs. 4, 13 then
$$ G=-a^{1/2} \left[
1-(1-e^2)^{1/2} \right ] \quad \mbox{and} \quad L=-G \pm
\sqrt{a(1-e^2)+2b} $$
where $a$ is the pseudo-semimajor axis, $e$ is
the pseudo-eccentricity, and the sign + (resp. -) holds for $K>0$
(resp. $<0$). The conditions to avoid collision orbits and circular
orbits, on which $g$ becomes meaningless, can be written in terms of
the orbital elements as $a>0$ and $00$ . To compute the
determinant we can by analyticity substitute (\ref{periodicity}) into
(\ref{periodicity1}) to find out at the time $t=\tau/2$ that \beq
D={6b(\tau/2)^2 \over (-G+\sqrt{(G+L)^2-2b})^7((G+L)^2-2b)^{3/2}} \neq
0
\label{det}
\eeq Thus the existence of $S_1$ symmetric periodic orbits of period
$\tau$ obtained from the $\tau$ periodic $S_1$ symmetric solutions of
the unperturbed problem, that at $t=0$ have the pericenter on the
positive $x$ axis, is readily established.
On the other hand, if at $t=0$, $\ep=0$, the apocenter is on the
positive $x$ axis, and it is crossing the $x$ axis perpendicularly, we
have \beq g(0)=\pi/ \lambda \quad \mbox{and} \quad l(0)=-\pi/\lambda
\eeq where $\lambda= (\omega_L-\omega_G)/ \omega_L$. By Proposition
1, at the half period we have \beq g(\tau/2)=(m+1/\lambda)\pi \qquad
l(\tau/2)=(-1/\lambda+k)\pi.
\label{halfp}
\eeq Instead of computing the functional determinant directly, in this
case, it is easier to consider the new variables given by the
relations, \beq \left\{
\begin{array}{l}
\tilde L=L\\ \tilde G=G\\ \tilde l= l+\pi/\lambda_0\\ \tilde g=
g-\pi/\lambda_0
\end{array}
\right.
\label{canon}
\eeq that define a family of canonical transformations parametrized by
$\lambda_0(L_0,G_0)$. For each orbit choose a different transformation
from the family (\ref{canon}), where $\lambda_0=\lambda$ is a fixed
quantity defined by the value of the action variables along the
periodic orbit under consideration.
The equations (\ref{halfp}), expressed in the new variables, are of
the same form as in (\ref{periodicity}). Thus the functional
determinant, in the new variables, is exactly $D$, and the existence
of the remaining $S_1$-symmetric $\tau$-periodic orbits follows.
Now the proof for the $S_2$-symmetric orbits can be done along the
same lines. Consider an
$S_2$ symmetric periodic orbit of period $\tau=2\pi m/k$. If at $t=0$,
$\ep=0$ the pericenter of the orbit is on the positive $y$ axis and it
is crossing the $y$ axis perpendicularly, we have \beq g(0)=\pi/2
\quad \mbox{and} \quad l(0)=0 \eeq Since the periodic orbit is $S_2$
symmetric one has, at the half period \beq g(\tau/2)=m\pi +\pi/2
\qquad \qquad l(\tau/2)=k \pi.
\label{periodicity1a}
\eeq Now we consider only $S_2$ symmetric solutions of (\ref{pe}) for
$\ep \neq 0$ again it follows from the implicit function theorem that
if the determinant $D$ computed at $t=\tau/2$ for $\ep=0$ is non zero
then (\ref{periodicity1a}) would be satisfied for $\ep>0$. It is
trivial to see from (\ref{det}) that $D \neq 0$, and hence we found
$S_2$ symmetric periodic orbits for the perturbed problem.
For the $S_2$-symmetric orbits having the apocenter on the positive
$x$ axis at $t=0$ the canonical transformation (\ref{canon}) can be used.
Again we find the same expression for the functional determinant and
hence, by the implicit function theorem, the existence of the
remaining $S_2$-symmetric periodic orbits is proved.
It is interesting to remark that Theorem 1 and its proof can be
easily extended to consider any $S_i$-symmetric perturbation with $i=1,2$
and a very general class of nondegenerate integrable Hamiltonians,
however such a generalization is trivial and not strictly related to
the problem under consideration and hence it will not be discussed
any further.
We can also observe that for $b=0$,
i.e. for the Kepler problem, the determinant in (\ref{det}) is
zero. Thus in the case of the Anisotropic Kepler Problem, the
continuation theorem proved above cannot be applied, and the existence
of symmetric periodic orbits of ``second kind'' (for weak
anisotropies) remains unclear. On the other hand the continuation
theorem that we prove in the next section (for the circular orbits)
can be applied to the Anisotropic Kepler Problem\r{8} and
hence at least the existence of symmetric periodic orbits of the first
kind is a well established fact.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{\large\bf IV. The $\bf{S_3}$ Symmetric Orbits}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Again we can consider the
Anisotropic Manev Problem taking the
parameter $\mu$ close to 1. Let the flow $\Phi(t,({\bf r},\dot{\bf
r}),\mu)$ of the equation of motion (1). In this section we prove the
following theorem:
\begin{theorem}
Let ${\bf r}^0(t)$ be a $S_3$-symmetric periodic orbit of the Manev
problem, i.e. a circular one. Set $\ep=\mu -1$, and let $\tau$ be the
period of ${\bf r}^0(t)$. Then there exists a $\tau$-periodic solution
$ \Phi (t,({\bf r}(\ep),\dot{\bf r}(\ep)),\ep)$ of the Anisotropic
Manev problem such that $\Phi(t,({\bf r}(0),\dot{\bf r}(0)),0)=({\bf
r}^0(t),\dot{\bf r}^0(t))$.
\end{theorem}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\normalsize\bf A. The equation of motion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now using the same notation as in Ref. 8 let $ \bf{r}^0(t)$ be a
circular solution of the Manev problem which correspond to $\mu=1$ in
the $xy$-plane, $\omega$ its angular speed and $a$ its radius. For
$\ep=\mu-1\neq 0$ we set, \beq {\bf r}(t,\ep)={\bf r}^0(t)+\ep~{\bf s}
(t,\ep)
\label{circular}
\eeq Expanding $\nabla H$ in powers of $\mu -1$ sufficiently small,
after substituting the expression for $\bf r$ given above, considering
the notation ${\bf r}^0(t)=x_0(t)+iy_0(t)$ and ${\bf s} =u+iv$ we have
that ${\bf r}(t,\ep)$ is a solution of equation of motion defined by
(\ref{H}) if, and only if, ${\bf s}(t,\ep)$ is a solution of the
equations
\beq
\begin{array}{l} \ddot u=- \left({1 \over a^3}-{3x_0^2 \over a^5}-{8bx_0^2 \over a^6}+{2b \over a^4} \right)u +\left({3x_0y_0 \over
a^5}+{8bx_0y_0 \over a^6}\right)v +\eta(t)+O(\ep) \\\noalign{\medskip}
\ddot v=\left({3x_0y_0 \over a^5}+{8bx_0y_0 \ov a^6}\right)u-\left({1
\ov a^3}-{8by_0^2 \ov a^6}-{3y_0^2 \ov a^5}+{2b \ov
a^4}\right)v+\xi(t)+ O(\ep)
\end{array}
\label{eqofmotion}
\eeq
where
$$
\begin{array}{l}
{\eta(t)}={3x_0y_0^2 \over a^5} +{4bx_0y_0^2 \over a^6} \medskip\\
{\xi(t)}={3y_0^2\over 2a^5}-{y_0 \over a^3}+{4by_0^3 \over a^6}-{2by_0
\over a^4}
\end{array}
$$
Consider the orthonormal frame in ${\bf R}^2$,$~{\bf e}_1(t)$ and
${\bf e}_2(t)$ defined by
$${\bf e}_1={{\bf r}^0 \over |{\bf r}^0|}=e^{i \omega t}=\cos \omega t
+i \sin \omega t, \qq {\bf e}_2= i{\bf e}_1 $$ and using the same
notation as in Ref. 8 where
$$ {\bf s}= x_1 {\bf e}_1 +x_2
{\bf e}_2, \qquad \dot{\bf s}=y_1{\bf e}_1+y_2{\bf e}_2 $$ equation
(\ref{eqofmotion}) can be written in an equivalent form as: \beq {\bf
\dot z}=A_0(t)+ A{\bf z} + O(\ep),
\label{rotsystem1}
\eeq
where $ {\bf z}=(x_1,x_2,y_1,y_2)^T$, and
$$ A_0=\left (\
\begin{array}{c}
0\\ 0\\ \alpha(t)\\ \beta(t)
\end{array}\right)
\qquad A=\left (\begin {array}{cccc}
0&\omega&1&0\\\noalign{\medskip}-\omega&0&0&1
\\\noalign{\medskip}2\,{\omega}^{2}+2\,{\frac {b}{{a}^{4}}}&0&0&\omega
\\\noalign{\medskip}0&-{\omega}^{2}&-\omega&0\end {array}\right ) $$
where
$$
\begin{array}{l}
\alpha(t)\cos\omega t-\beta(t)\sin\omega t=\eta(t)\\
\alpha(t)\sin\omega t +\beta(t)\cos\omega t=\xi(t)
\end{array}
$$ or equivalently, \beq
\begin{array}{l}
\alpha(t)=\sin^2\omega t\left ({1\over 2a^2}+{2b \over
a^3}\right)\medskip\\ \beta(t)=-\sin\omega t~ \cos\omega t \left({1
\over a^2}+{2b \over a^3}\right)
\label{beta}
\end{array}
\eeq The eigenvalues of $A$ are $0$, with multiplicity two, $ {i \over
a^{3/2}}$ and $-{i \over a^{3\ 2}}$. One of the two eigenvalues
vanishes because the system is autonomous, and the second due to the
presence of the first integral $H$.
Now consider the real Jordan form $J$ of $A$. The matrix $J$ is
defined by the relation $ J= {\cal T} ^{-1} A {\cal T} $ where ${\cal
T}$ is
$$\scriptstyle{\mathrm{
{\cal T}= \left (\begin{array}{cccc} \scriptstyle{2\,{\omega}^{2}{a}^{3}}&0&{\frac
{{\omega}^{2}{a}^{4} +2\,b}{a}}&0\\\noalign{\medskip}0&-{\frac
{\omega\left (3 \,{\omega}^{2}{a}^{4}+2\,b\right )}{a}}&0&-2\,{\frac
{\omega{a }^{2}\left ({\omega}^{2}{a}^{4}+2\,b\right )}{\left( a
\right )^{3/2}}}\\\noalign{\medskip}0&{1 \over 2}\,{\frac {4\, a\left
({\omega}^{2}{a}^{4}+b\right )+2\,\left ({\omega}^{2}{a}
^{4}+2\,b\right )^{2}}{{a}^{5}}}&0&{ \frac {\left
({\omega}^{2}{a}^{4}+2\,b\right
)^{2}}{{a}^{7/2}}}\\\noalign{\medskip}-{\frac {\omega\left
({\omega}^{2} {a}^{4}+2\,b\right )}{a}}&0&-{\frac {\omega\left
({\omega}^{2} {a}^{4}+2\,b\right )}{a}}&0\end{array}\right )}}$$
and the columns of ${\cal T}$ are the generalized eigenvectors of
$A$.
The vector $J_0={\cal T}^{-1}A_0$ and the matrix $J$ are: $$
J_0=\left( \begin{array}{c} j_1(t) \\ j_2(t) \\ j_3(t) \\ j_4(t)
\end{array} \right ), \qquad
J=\left (\begin {array}{cccc} 0&0&0&0\\\noalign{\medskip}1&0&0&0
\\\noalign{\medskip}0&0&0&{\frac {\sqrt {a}}{{a}^{2}}}
\\\noalign{\medskip}0&0&-{\frac {\sqrt {a}}{{a}^{2}}}&0\end {array}
\right ) $$
where the fact that $j_1(t)=(2\omega^3a^2-{\omega(\omega^2a^4+2b)\over
a^2})^{-1}\beta(t)$ is the only information about $J_0$ that we need
to retain. Furthermore we remark that $\omega^2a^4-a-2b=0$ gives the
relation between $a$ and $\omega$ and solving this equations gives
only one positive solution (for $b>0$).
Letting ${\bf z}={\cal T}\zeta$, the equation of motion becomes
\beq \dot\zeta=J_0(t)+J\zeta +O(\ep),
\label{rotsystem2}
\eeq
and its flow is given by
\beq \psi(t,\zeta, \ep)=\gamma(t)+ e^{Jt}+O(\ep)
\label{jacobflow}
\eeq
where by the variation of constants \beq \gamma(t)= e^{Jt} \int_0^t
e^{-Js} J_0(s) ds
\label{variation}
\eeq
Therefore we have
$$ e^{Jt}=\left(\begin {array}{cccc}
1&0&0&0\\\noalign{\medskip}t&1&0&0 \\\noalign{\medskip}0&0&\cos
{\frac {\sqrt {a}}{{a}^{2}}}t&\sin {\frac {\sqrt {a}}{{a}^{2}}}t
\\\noalign{\medskip}0&0&-\sin {\frac {\sqrt {a}}{{a}^{2}}}t &\cos
{\frac {\sqrt {a}}{{a}^{2}}}t \end {array} \right ) $$
and from (\ref{variation}) we obtain
\beq \gamma(t)=\left (
\begin{array}{c}
\gamma_1(t)\medskip\\ \gamma_2(t)\medskip\\ \gamma_3(t)\medskip\\
\gamma_4(t)
\end{array} \right )
\eeq
where we retain only the information that
\beq
\gamma_1(t)=(2\omega^3a^2-{\omega(\omega^2a^4+2b)\over
a^2})^{-1}\int_0^t\beta(s)ds.
\label{gamma1}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\normalsize\bf B. The periodicity equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since the Hamiltonian $H$ of the anisotropic Manev problem is
$S_3$-symmetric, as we have shown, we can write the periodicity
equation as in\r{8}, \beq \Phi \left({\tau \over 2},({\bf r},
{\bf \dot r}),\ep \right)=-({\bf r}, {\bf \dot r}).
\label{periodicity3}
\eeq Then it easy to check that $\Phi(t,({\bf r}, {\bf \dot r}),\ep)$
is a periodic solution of the equation of motion with period $\tau$.
To find periodic solutions we have to verify that (\ref{periodicity3})
is satisfied for a family of initial conditions. Equation
(\ref{periodicity3}) in $\zeta$ coordinates is \beq \psi \left( {\tau
\over 2}, \zeta , \ep \right) -\zeta=0 \eeq where $\psi(t,\zeta, \ep)$
is the flow of (\ref{rotsystem2}). Let us denote by ${\cal
P}(\zeta,\ep)$ the left hand side of the periodicity equation
(\ref{periodicity3}), that is, let \beq {\cal P}(\zeta,\ep)=\psi
\left({\tau/2}, \zeta,\ep \right)
-\zeta=\gamma(\tau/2)+\left(e^{J{\tau\over 2}}-I\right)= 0.
\label{p}
\eeq
Using (\ref{jacobflow}) we notice that the requirement \beq {\cal P}
(\zeta^\ast,0)=\gamma(\tau/2)+\left( e^{J{\tau \over 2}}-I
\right)\zeta^\ast=0, \eeq imposes the restrictions \beq
\gamma_1(\tau/2)=0 \quad \zeta_1^\ast=-{2\over \tau}\gamma_2(\tau/2)
\quad \zeta_2^\ast=\mbox{arbitrary}\quad \eeq and \beq
\begin{array}{l}
\zeta_3^\ast={1 \over
2(1-\cos\alpha^\ast)}\left(-\gamma_3(\tau/2)(\cos\alpha^\ast
-1)+\gamma_4(\tau/2)\sin\alpha^\ast \right)\medskip\\
\zeta_4^\ast={-1 \over
2(1-\cos\alpha^\ast)}(\gamma_3(\tau/2)\sin\alpha^\ast+\gamma_4(\tau/2)(\cos\alpha^\ast-1))
\end{array}
\eeq where $\alpha^\ast=\pi (1+2b/a)^{-1/2}$. It easy to see from
(\ref{beta}) and (\ref{gamma1}) that $\gamma_1(\tau/2)=0$, therefore,
we take
\beq\zeta^\ast=(\zeta_1^\ast,\zeta_2^\ast,\zeta_3^\ast,\zeta_4^\ast)^T,\eeq
with $\zeta_2^\ast$ arbitrary, for the moment. Now using the flow
(\ref{jacobflow}) , we determine that the Jacobian matrix of $\cal P$ with
respect to the variables $\zeta$ evaluated at the point
$(\zeta^\ast,0)$ is given by \beq \left ( \begin{array}{cccc}\ 0&0
&0&0\\\ \tau/2&0&0&0 \\\ 0&0& \cos \alpha^\ast -1& \sin \alpha^\ast
\\\ 0&0&-\sin \alpha^\ast & \cos \alpha^\ast -1\end{array} \right ).
\eeq
Consider the system of three equations formed by those in (\ref{p})
corresponding to the indices i=2,3,4 and fix the variable
$\zeta_2=\zeta_2^\ast$. Its Jacobian matrix has determinant $\tau (
1-\cos\alpha^\ast)$, that is always positive since $0< \pi
(1+2b/a)^{-1/2}\leq \pi $. Therefore the implicit function theorem
guarantees the existence of analytic functions $\zeta_i=\zeta_i(\ep),
~i=1,3,4$ in a neighborhood of $\ep=0$, satisfying the equations
\beq{\cal P}_i(\zeta, \ep)=0, \q (i=2,3,4)\eeq where
\beq\zeta(\ep)=(\zeta_1(\ep),
\zeta^\ast_2,\zeta_3(\ep),\zeta_4(\ep))\eeq and such that \beq
\zeta_i(0)=\zeta_i^\ast \q (i=1,2,3,4). \eeq It remains to show, in
order to have periodicity, that also the remaining equation \beq {\cal
P}_1(\zeta(\ep),\nu(\ep),\ep)=0, \eeq is satisfied in a, possibly
smaller neighborhood of $\ep=0$. That will be done employing a first
integral of the system under discussion, i.e. the Hamiltonian.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{\normalsize\bf C. Integral of motion}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Since the Hamiltonian is a integral of motion of the problem under
discussion we can apply the same analysis as in Refs. 8, 14. In
particular using the same notations as in Ref. 8 we can define
$$ H_\ep({\bf z},t)=H({\bf r},\dot{\bf r},\ep),$$
where $H_\ep({\bf z},t)$ is a time-dependent, $\tau$-periodic first integral
for system (\ref{rotsystem1}). The above integral satisfies the following
relation
\beq
H_\ep({\bf z},t+\tau/2)=H_\ep({\bf z},t)
\label{int1}
\eeq
for all $t$, since $H(-{\bf r},-\dot{\bf r})=H({\bf r},\dot{\bf r})$,
$~{\bf r}(t)={\bf r}^0(t)+\ep~{\bf s}(t)$ and
$${\bf r}^0(t+\tau/2)=-{\bf r}^0(t)\q,\q {\bf s}({\bf z},t+\tau/2)=-{\bf s}({\bf z},t).$$
Performing a change of coordinates we can define ${\cal H}_\ep(\zeta,t)=H_\ep({\cal T}\zeta,t)$,
hence (\ref{int1}) can be written as
\beq
{\cal H}_\ep(\zeta, t+\tau/2)={\cal H}_\ep(\zeta,t).
\label{int2}
\eeq
Moreover since ${\cal H}_\ep$ is an integral of motion it verifies that
\beq
{\cal H}_\ep(\phi(\zeta, \ep,t))=H_\ep(\zeta,0).
\label{int3}
\eeq
Thus applying equations (\ref{int2}-\ref{int3}) it follows that
$${\cal H}_\ep(\psi(\tau/2,\zeta,\ep),0)={\cal H}_\ep(\zeta,0)$$
and by means of the Mean Value Theorem we obtain
\beq \nabla _\zeta {\cal H}_\ep (\tilde{\zeta},0) \cdot {\cal
P}(\zeta,\ep)=0,
\label{meanval}
\eeq
where$\nabla_\zeta{\cal H}_\ep$ is the gradient of ${\cal H}_\ep$ with
respect to $\zeta$, and $\tilde{\zeta}$ is a point on the segment
joining $\zeta$ to $\psi(\tau/2,\zeta,\ep)$.
Expanding $\Psi(\ep)=\psi(\tau/2,\zeta,\ep)$ in power of $\ep$ sufficiently small
it is easy to show (see Ref. 8) that $\Psi(\ep)=\zeta^\ast+O(\ep)$ and consequently
$$\tilde{\zeta}=s\zeta(\ep)+(1-s)\Psi(\ep)=\zeta^\ast +O(\ep)$$
for some $s\in (0,1)$.
Moreover if we also expand the Hamiltonian $H_\ep({\bf z},0)$ in powers of $\ep$ we get
$$H_\ep({\bf z},0)=H_0+\ep(H_1+H_2\cdot {\bf z} )+O(\ep^2)$$
or, in $\zeta$ coordinates
\beq {\cal H}_\ep(\zeta,0)={\cal H}_0 + \ep({\cal H}_1 +{\cal
H}_2\cdot \zeta) +O(\ep^2),
\eeq
where ${\cal H}_0=H_0=\left( {1\over 2}\omega^2a^2-{1\over a} -{b
\over a^2 }\right)$, ${\cal H}_1=H_1$ and ${\cal H}_2={\cal T}^T
H_2={\cal T}^T(a^{-2}+2ba^{-3},0,0,a\omega)=( a \omega^2 \zeta_1,0,0,0)$.
Hence we obtain
\beq {1 \over \ep} \nabla_\zeta {\cal
H}_\ep(\tilde{\zeta},0)={\cal H}_2 +O(\ep).
\eeq
With these preparations equation (\ref{meanval}) reduces to the equation
in the unknown ${\cal P}_1$
\beq [a \omega^2+O(\ep)]{\cal P}_1=0. \eeq
since, for small $\ep$, we already found in Sec. IV B that ${\cal P}_i=0$ for $i=2,3,4$.
It is easy to see that for $\ep=0$ the equation above has solution ${\cal
P}_1=0$. Thus, by continuity, $[a \omega^2+O(\ep)]$ is
different from zero for $\ep$ sufficiently small. Therefore for such values of
$\ep$ this equation has a unique solution that is the trivial
one. Consequently the remaining equation $${\cal
P}_1(\zeta(\ep),\ep)=0,$$ is also satisfied in a possibly smaller neighborhood of $\ep=0$.
Hence all the equations of the periodicity system (\ref{p}) are satisfied when
$\zeta=\zeta(\ep)$, as long as $\ep$ is sufficiently small. This completes
the proof of Theorem 2.
\qq\qq
\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{\large\bf Acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The author is grateful to Professor Florin Diacu for
his enlightening comments and suggestions. This work was supported
by an University of Victoria Fellowship and a Howard E. Petch Research
Scholarship.
\qq\qq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{\large\bf References}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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