You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'We study the orbits in the MOND theory within a dwarf galaxy of mass $M_d\sim 10^8M_\odot$ at a distance of $\sim 100$kpc from a neighboring galaxy of mass $M_g=5\times 10^{11} M_\odot$, such as ours. It is assumed that a second mass $m<<M_d$ is gravitationally bound to $M_d$ by a previously calculated potential for the MOND theory. This potential is obtained for a free falling mass $M_d$ in a constant external gravitational acceleration field $\nabla\phi_g$. The numerical technique of surfaces of section is used to study the stability of the phase-space orbits in the dwarf galaxy. Equatorial orbits with sufficiently small eccentricities $e<0.65$ are found to be stable with respect to small changes in the initial conditions. (The equatorial plane is perpendicular to the direction of $\nabla\phi_g$, which is along the line joining $M_d$ and $M_g$.) For decreasing values of the conserved component of the angular momentum, in the direction of $\nabla\phi_g$, equatorial stability is lost.'
author:
- Daniel Müller and Reuven Opher
title: Orbits in a Neighboring Dwarf Galaxy According to MOND
---

Introduction
============

The inconsistency between luminous and dynamical mass measurements is well known. An alternative way to explain this inconsistency without dark matter, is by Modified Nonrelativistic Dynamics (MOND) for galaxy and galactic systems [@M1] [@M2], [@M3]. This theory was later developed into an alternative theory of gravitation by [@BM] (1984), which we hereafter refer to as the Bekenstein-Milgrom (BM) theory. MOND agrees with the [@TF] and [@FJ] laws, and explains the dynamics in elliptical and spiral galaxies, without the need for dark matter. It satisfies conservation of energy, momentum, angular momentum, as well as the weak equivalence principle ([@BM] 1984).

Recent studies of rotation disks of Low Surface Brightness (LSB) Galaxies by [@SG], [@Block] and [@McG] are in agreement with MOND.

Departure of MOND from Newtonian theory is not connected with a distance scale. In MOND, deviations from Newtonian theory become significant for very small accelerations. The main argument presented is that Newtonian gravitation has not been tested for very weak fields and it could be that the theory is not valid in this regime. Other theories modifying Newtonian theory were reviewed by [@B1] (1987).

Stability of galactic disks in BM theory was first studied in a WKB approximation by [@MM]. The WKB scheme deals with perturbation wavelengths $\lambda=2\pi/k$, much smaller than the involved distances $\rho$ on the disk ($|k\rho|>>1$). In Newtonian gravitation, the presence of a dark halo is important in stabilizing the disk against violent bar formation [@OP]. Recent N-body simulations for the BM theory, indicate that disks are more stable in MOND than in Newtonian dynamics with dark halos ([@BrdM] 1999).

We obtain the conservative Hamiltonian that describes the motion of a particle in a previous calculated potential for the BM theory in Section \[2\]. This Section also includes a summary of the stability theory for phase-space orbits. Orbits in a dwarf galaxy are discussed in Section \[3\]. Our conclusions are presented in Section \[4\].

Theory \[2\]
============

In the BM theory, the Poisson equation for determining the gravitational potential is modified to $$\nabla .\left( \mu \left( \frac{\mid \nabla\varphi \mid }{a_0}\right) 
\nabla\varphi \right) =4\pi G\rho ,  \label{mond}$$ where $a_0=2\times 10^{-8}\mbox{cm}/\mbox{s}^2$ is a constant with the dimension of acceleration, set so as to agree with the Tully-Fisher law (Milgrom [@M2]), the function $\mu(x)$ (where $x=|\nabla\varphi|/a_0$) obeys $0<\mu (x)<1,$ with $\lim_{x\rightarrow 0}\mu(x)=x$ and $G$ is the Newtonian gravitational constant. With this particular value of $a_0$, non-Newtonian effects due to the Solar gravitational field are expected to begin only beyond the Oort cloud [@Oo].

Let us assume the existence of a constant external gravitational acceleration $\nabla\phi_g$ due to some source (e.g., a neighboring massive galaxy, such as ours) and a free falling sphere of mass $M_d$ (e.g., the center of a dwarf galaxy). Sufficiently close to $M_d$ so that $\mu(x)\approx 1$, the solution of equation (\[mond\]) yields the well known Newtonian potential $$\varphi\approx \frac{GM_d}{r}.$$

When the solution $\varphi$ of equation (\[mond\]) implies gravitational accelerations much bigger than the external gravitational acceleration $(|\nabla\varphi|>>|\nabla\phi_g|)$ but $|\nabla\varphi|< a_0$, spherical symmetry can be assumed and the potential is obtained from equation (\[mond\]) in the limit $\mu(x)\approx x$, using Gauss’s theorem $$\varphi\approx \sqrt{GM_da_0}\log r .\label{sesf}$$ In the opposite limit, when $|\nabla\varphi|<<|\nabla\phi_g|$, [@BM] (1984) have calculated the approximate potential of equation (\[mond\]) in cylindrical coordinates $$\varphi=-\frac{(\mu _g\sqrt{1+L_g})^{-1}M_dG}{\sqrt{z^2(1-\alpha _g)+\rho ^2}}, 
\label{pot}$$ where $z$ is the direction of $\nabla\phi_g$, $\mu_g=\mu(|\nabla\phi_g|/a_0)$, $L_g=d\ln (\mu )/d\ln (x)\mid _{x=\mid \nabla \phi _g\mid /a_0}$ and $\alpha _g=L_g/(1+L_g)$. In the asymptotic Newtonian limit $|\nabla\phi_g|>>a_0,$ $\mu_g\rightarrow1$, $L_g \rightarrow0$, $\alpha_g\rightarrow0$. In the non-Newtonian limit, which we call the MOND limit, $|\nabla\phi_g|<<a_0$, $$\begin{aligned}
\mu_g(x)&\rightarrow&x\label{ml1}\\ 
L_g(x)&\rightarrow& 1\label{ml2}\\ 
\alpha_g&\rightarrow& 1/2\label{ml3}. \end{aligned}$$ It is to be noted that equation (\[pot\]) is not a consequence of tidal effects, since the external field is constant ([@BM] 1984).

We study the orbit of a very small mass $m<<M_d$ such that it does not modify the potential in equation (\[pot\]) and consider the motion of $m$, bounded by this potential. This is a central force type problem, so that conservation of linear momentum implies that the coordinate dependence of the Hamiltonian is in the relative distance between $M_d$ and $m.$ According to equation (\[pot\]), and considering the azimuthal symmetry of the potential, the dynamics of the $M_d-m$ system are governed by the conservative Hamiltonian $$\begin{aligned}
H&=&\frac{1}{2\mu_r}\left(p_\rho^2+p_z^2+\frac{l_z^2}{\rho^2} 
\right)-\frac{\lambda}{\sqrt{z^2(1-\alpha _g)+\rho^2}}\label{H},\\ 
\lambda&=&(\mu _g\sqrt{1+L_g})^{-1}GM_dm,\label{lbd} \end{aligned}$$ where $\mu_r=M_dm/(M_d+m)\approx m$ is the reduced mass and $l_z$ is the angular momentum component in the direction of $\nabla\phi_g$ (the direction joining the neighboring massive galaxy and the dwarf galaxy). Thus, the three dimensional motion in a axisymmetric potential is reduced to the motion in a plane. In equations (\[H\]) and (\[lbd\]), cartesian coordinates $(\rho,z)$ are used to describe this (non-uniformly) rotating plane, which is often called the meridional plane.

In the MOND limit, when $|\nabla\phi_g|<<a_0$, equation (\[pot\]) with the condition $|\nabla\varphi|<<|\nabla\phi_g|$, yields a minimum distance between $M_d$ and $m$: $$r_{min}=\sqrt{\frac{GM_d}{\sqrt{2}a_0}}\frac{a_0}{|\nabla\phi_g|}\label{cdv},$$ where the values for $L_g$ and $\mu_g$ are obtained from equations (\[ml1\])- (\[ml3\]). We note that the forces in the $\rho$ and $z$ directions are not equal.

The Hamiltonian (\[H\]) has an elliptic equilibrium position at $z=0$, $\rho=l_z^2/(\mu_r\lambda)$, which is called the guiding center. Expanding the Hamiltonian in a power series about the guiding center, we have $$\begin{aligned}
& &H=H_0(I)+H_1(I,\theta)\nonumber\\ 
& &H_0(I)=\omega_zI_z+\omega_\rho I_\rho\label{H0}\\ 
& &\omega_\rho=\frac{\lambda^2\mu_r}{l_z^3},\;\;\;\omega _z= 
\frac{\lambda^2\mu_r\sqrt{1-\alpha _g}}{l_z^3}. 
\label{freq} \end{aligned}$$ This expansion is called the epicycle approximation and is appropriate in the neighborhood of the equilibrium. $H_0$ is the unperturbed motion and describes the epicycles around the guiding center; the two frequencies $\omega_\rho$ and $\omega_z$ given in equation (\[freq\]), are called the epicycle frequency and the vertical frequency, respectively. In phase space, the conservation integrals define a $2$-dimensional torus. When $j_z\omega_z+j_\rho\omega_\rho=0,\;j=|j_z|+|j_\rho|$ for integers $j_{z,\rho}$, the linearized frequencies are said to satisfy a resonance of order $j$. The corresponding torus is called a resonant torus. From equations (\[ml3\]) and (\[freq\]), the allowed linearized resonances are restricted to $\alpha_g<1/2$.

For a resonance of order $j$, it is possible to put the Hamiltonian into a Birkhoff normal form of degree $j$ by a canonical transformation $$H=K_0(J)+K_1(J,\Phi),$$ where $K_0$ is a polynomial of degree $j$ in the new actions $J$ and $\Phi$ are the new angles [@MCA]. $K_0$ is a good approximation when the perturbation $K_1$ is sufficiently small. The new frequencies are, then, $$\nu_i=\frac{\partial K_0(J)}{\partial J_i}$$ and are functions of the new amplitudes $J_i$, a phenomenon first discovered by [@LI] in connection with nonlinear oscillators.

Nondegenerescence is defined as the nonvanishing of the Hessian determinant $$\det\left|\frac{\partial\nu_i}{\partial J_k}\right|\neq 0\label{nd}.$$

There exists a theorem of [@K] on the behavior of a nonresonant torus under a small perturbation $K_1$ of a nondegenerate Hamiltonian $K_0$ [@MCA], which was subsequently proved by [@A1] for Hamiltonian systems and by [@M] for area-preserving maps (see also [@A2] and [@Mo]). The theorem, known as KAM in recognition of their work, states the existence of invariant tori, densely filled with phase space curves winding around them, which are conditionally-periodic with the number of independent frequencies equal to the number of degrees of freedom. This theorem is valid if the following condition holds $$|(\nu.j)|>C|j|^\tau\label{nr}.$$ In equation (\[nr\]) $C$ is dependent on the magnitude of the perturbation and $\tau$, on the number of degrees of freedom. The existence of an invariant, or KAM, torus surrounding a periodic orbit, characterizes the stability. Orbits whose frequencies satisfy equation (\[nr\]), occupy most of the volume in phase space for sufficiently small $C$ ([@Mo]).

The conserved $l_z$ is an isolating integral for the full three degrees of freedom system; the two degrees of freedom system described by equation (\[H\]), is a subspace of constant $l_z$ of the original three degrees of freedom system. A surface of section can be constructed in the spirit of [@HH] (1964). In a two degrees of freedom system, for instance equation (\[H\]), the dimension of the phase space is four. Since the energy is an isolating integral, motion is constrained to a three dimensional constant energy surface, for example, $\rho$, $z$ and $p_z$. Successive intersections of the trajectory with the plane $\rho=\mbox{const.}$ and $p_\rho>0$, described by the set of points $z$ and $p_z$ (or $\dot{z}=dz/dt$), is called a surface of section. Each intersection of the orbit is a point in this plane and the passage of one point to the next can be considered as a mapping. After an infinite time interval, the points corresponding to a unique orbit fill up a whole area of the surface of section, in the absence of further isolating integrals.

Motion is constrained to the intersection of the surfaces of constant $H$ and $I$ in the presence of a third isolating integral. Consider an orbit whose initial conditions lie in the phase space region of conserved $I$. Its points in the surface of section form a smooth curve.

According to the Poincare-Birkhoff theorem for resonant tori, after the inclusion of the perturbation $K_1$, we have an even number of periodic orbits in the vicinity of a stable periodic orbit. Periodic orbits are exact resonances of the nonlinear problem. The corresponding surface of section is a fixed point, surrounded by an even number of other fixed points. Half of them are stable, while the other half are unstable. Stable points are surrounded by closed invariant curves. Unstable points are connected by separatrices, forming a structure called a chain of islands. Chaos is present in the vicinity of the separatrices. (For further details, see [@LL] and [@DN].)

Orbits in a Dwarf Galaxy \[3\]
==============================

In the following, we describe the motion of a particle $m$ (e.g. a globular cluster) around a spheroid of mass $M_d$ (the nucleus of a dwarf galaxy) with $M_d>>m$, as shown in Figure \[esq\]. We take $$\begin{aligned}
M_d&=&10^8M_\odot,\nonumber\\ 
m&=&10^5M_\odot. \label{massas} \end{aligned}$$

We assume that the $M_d-m$ system is gravitationally bounded to a nearby massive galaxy $M_g=5 \times 10^{11}M_\odot$ (such as the Milky Way), separated by a distance $d=100kpc$. From equation (\[sesf\]), the nearby massive galaxy generates a gravitational acceleration on the order of $$|\nabla\phi_g|\sim\sqrt{GM_ga_0}/d\sim 5.36^{-1}a_0,\label{condi1}$$ within the $M_d-m$ system. The period of the test particle is appreciably less than that of the dwarf galaxy’s orbit. Possible resonances between the dwarf galaxy’s orbital period and that of the test particle are neglected. According to equation (\[cdv\]), equation (\[pot\]) is valid for distances greater than $r_{min}=1189$pc and we choose $$r\approx 2378\mbox{pc} \label{condi2}.$$ Tidal accelerations for distances such as $r$ are neglected, so that $\nabla\phi_g$ is considered to be a constant external gravitational acceleration within the $M_d-m$ system. The geometry is shown in Figure \[esq\].

If the initial conditions are chosen as $z=p_z=0$, according to equation (\[H\]), Kepler’s Theory is recovered with an effective gravitational constant (eq. (\[lbd\])). These initial conditions correspond to orbits constrained to the equator of $M_d$ and are known as equatorial orbits. (The equatorial plane is defined as the plane containing $M_d$, perpendicular to the $z$ direction (see Fig. \[esq\]).) It then follows that $$\begin{aligned}
l_z^2&=&\mu_r\lambda a(1-e^2)\label{exc}\\ 
H&=&E=-\frac{\lambda}{2a},\label{sem}\\ 
T&=&2\pi a^{3/2}\sqrt{\frac{\mu_r}{\lambda}}, \label{Kpl} \end{aligned}$$ where $a$ is the semi-major axis, $e$ the eccentricity, $T$ the period and $E$, the energy. In Newtonian theory, the relation between the period and the semi-major axis is Kepler’s third law. The equatorial orbits are uniquely defined by equations (\[exc\]) and (\[sem\]).

In the MOND limit, the period predicted by equation (\[pot\]), for equatorial orbits, follows from equations (\[ml1\])-(\[ml3\]), (\[lbd\]) and (\[Kpl\]). $$T\approx2\pi r^{3/2}\sqrt{\frac{\sqrt{2}|\nabla\phi_g|/a_0}{G(M_d+m)}},$$ which, according to equations (\[massas\]), (\[condi1\]) and (\[condi2\]), results in $$T\approx 5.6\times 10^{8}\mbox{yr}\label{periodo}.$$

The physical configuration defined by equations (\[massas\]) and (\[condi1\]) is described in the following. We assume the MOND limit in equations (\[ml1\]) and (\[ml2\]) and choose $\alpha_g=7/16$, corresponding to the linearized lower order resonance of $3/4$ in Table \[tb1\]. Under these conditions, the surfaces of section for the phase orbits of equation (\[H\]) (Figs. \[f4\] - \[fg2\]) are obtained for an energy $E=-6.817\times 10^{50}\mbox{erg}$, which corresponds to distances of the order of that in equation (\[condi2\]). As a numerical check, the energy was found to be conserved to one part in $10^{10}$.

Regions of instability increase for smaller values of $l_z$. Equatorial orbits are defined as $z=p_z=0$ in equation (\[H\]). For equatorial orbits, $l_z$ is connected to the eccentricity by equation (\[exc\]). According to equation (\[nr\]), for equatorial orbits with sufficiently small eccentricities, almost the entire phase space is occupied by KAM tori surrounding the stable orbit, as shown in Fig. \[f4\]. The surface of section in Fig. \[f4\] is obtained for $l_z/\mu_r=57.09\mbox{kpc}\:\mbox{km}/\mbox{s}$, which corresponds to an equatorial orbit with eccentricity $e=0.4$. It can be seen in Fig. \[f4\] that this particular equatorial orbit is stable.

The MOND potential given in equation (\[pot\]) is valid for distances greater than $r_{min}$, defined in equation (\[cdv\]). This potential predicts equatorial orbits which belong to a plane perpendicular to the external gravitational acceleration produced by $M_g$ (see Fig. \[esq\]). We numerically found that the equatorial orbits are stable for eccentricities $e<0.65$.

With decreasing values of $l_z$, for example, $l_z/\mu_r=41.20\mbox{kpc}\:\mbox{km}/\mbox{s}$, the equatorial orbit becomes unstable. There exist other stable periodic orbits, which are not in the plane $z=0$ (see Fig. \[fg2\]).

Associated with the upper and lower stable islands in Fig. \[fg2\], there is an exact $2:1$ resonance, which is a periodic orbit in the meridional plane, defined by $\rho,z$ coordinates. In three dimensional space $x,y,z$, this periodic orbit corresponds to a quasi-periodic orbit in a spatial region, which is a rounded cylindrical shell surrounding $M_d$, as shown in Figure \[orb\]. A closed orbit in three dimensional space requires an exact resonance between the azimuthal, vertical and epicycle frequencies.

Conclusions \[4\]
=================

There exists a previously calculated potential $\varphi$ in the MOND theory, given by equation (\[pot\]), for a free falling sphere of mass $M_d$ in a constant external gravitational acceleration $\nabla\phi_g$. $\varphi$ is valid when $|\nabla\phi_g|>>|\nabla\varphi|$. We assume that the existence of a second mass $m<<M_d$ does not modify the potential.

The system $M_d-m$, bounded by the potential $\varphi$, is described by the Hamiltonian given in equation (\[H\]). Since the potential is axially symmetric, only the component of the angular momentum in the direction of $\nabla\phi_g$, $l_z$, is conserved. As a consequence, when $z=p_z=0$ and $l_z$ is the total angular momentum, motion is periodic and occurs in a plane perpendicular to $\nabla\phi_g$, which we call the equatorial plane (Fig. \[esq\]). Kepler’s theory is then recovered with an effective gravitational constant (eq. (\[lbd\])), and $l_z$ is related to the eccentricity by equation (\[exc\]).

We analyze the phase space of $H$, using the numerical technique of surfaces of section. It is found that for eccentricities $\epsilon<0.65$, the equatorial orbits are stable with respect to small variations in the initial conditions (Fig. \[f4\]). For smaller values of $l_z$, the stability of the equatorial orbit is lost and there is an increase in the size of the chaotic regions (Fig. \[fg2\]). There is a disk perpendicular to $\nabla\phi_g$, as well as some regions not in the disk, which are stable against variations of initial conditions.

D. M. would like to thank the Brazilian agency CAPES for support and R. O. would like to thank the Brazilian agency CNPq and the project PRONEX/FINEP (no. 41.96.0908.00) for partial support. D. M. would also like to thank the Dynamics Group at IAG-USP for enlightening conversations.

[cc]{} $7/16$ & $3/4$\
$9/25$ & $4/5$\
$11/36$ & $5/6$\
$24/49$ & $5/7$\
$...$ & $...$

[99]{} Arnold, V.I. 1989, Mathematical Methods of Classical Mechanics, (New York: Springer-Verlag) Arnold, V.I. 1961, Soviet Math. Dokl., 2, 501 Arnold, V.I. 1963, Russian Math. Surv., 18:6,85 Bekenstein, J. D., Milgrom, M. 1984, , 286, 7 Bekenstein, J. D. 1987, in: 2nd Canadian Conference on General Relativity and Relativistic Astrophysics, ed. C. Dyer, (Singapore: World Scientific) de Block, W.J., McGaugh, S.S. 1998, , 508, 132 Brada, R., Milgrom, M. 1999, , 519, 590 Danby J. M. A. 1971, Qualitative Methods in Celestial Mechanics, (São Paulo: Publicações do Instituto de Matemática e Estatística da Universidade de São Paulo) Faber, S.M., Jackson, R.E. 1976, , 204, 668 Hénon M., Heiles C. 1964, , 69, 73 Kolmogorov, A.N. 1954, Dokl. Akad. Nauk. SSSR, 98, 527 Lichtenberg A.J. & Lieberman M.A. 1983, Regular and Stochastic Motion Second Edition, (New York: Springer-Verlag) Lindstedt, M. 1882, Astrom. Nach., 103, 211 McGaugh, S.S., de Block, W.J.G. 1998, , 499, 66 Milgrom, M. 1983a, , 270, 365 Milgrom, M. 1983b, , 270, 371 Milgrom, M. 1983c, , 270, 384 Milgrom, M. 1989, , 338, 121 Moser, J. 1962, On Invariants Curves of Area-Preserving Mappings on an Annulus, Nachr. Akad. Wiss. Götingen, Math. Phys. K1, p.1 Moser, J., Siegel, C. 1971, Lectures in Celestial Mechanics, (Berlin: Springer-Verlag) Oort, J.H. 1963, The Moon, Meteorites, and Comets, ed. B. M. Middlehurst and G. P. Kuiper (Chicago: University of Chicago Press) Ostriker, J.P., Peebles, P.J.E. 1973, , 186, 467 Salcedo, F.J.S., Gámez, A.M.H. 1999, , 345, 36 Tully, R.B., Fisher, J.R. 1977, , 54, 661