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Geometric Mechanics and the Dynamics of Asteroid Pairs

Wang-Sang Koon, Jerrold E. Marsden and Shane RossControl and Dynamical Systems 107-81

Caltech, Pasadena, CA 91125

Martin LoNavigation and Mission Design

Jet Propulsion LaboratoryCalifornia Institute of Technology, M/S 301-140L

4800 Oak Grove Drive, Pasadena, CA 91109

Dan J. ScheeresDepartment of Aerospace Engineering

University of MichiganAnn Arbor, MI 48109-2140

Conference on New Methods in Astrodynamics, January, 2003

Abstract

The purpose of this paper is to describe the general setting for the applicationof techniques from geometric mechanics and dynamical systems to the problem ofasteroid pairs. It also gives some preliminary results on transport calculations andthe associated problem of calculating binary asteroid escape rates. The dynamics ofan asteroid pair, consisting of two irregularly shaped asteroids interacting throughtheir gravitational potential is an example of a full body problem or FBP in which twoor more extended bodies interact.

One of the interesting features of the binary asteroid problem is that there is cou-pling between their translational and rotational degrees of freedom. General FBPshave a wide range of other interesting aspects as well, including the 6-DOF guidance,control, and dynamics of vehicles, the dynamics of interacting or ionizing molecules,the evolution of small body, planetary, or stellar systems, and almost any other prob-lem where distributed bodies interact with each other or with an external field. Thispaper focuses on the specific case of asteroid pairs using techniques that are generallyapplicable to many other FBPs. This particular full 2-body problem (F2BP) concernsthe dynamical evolution of two rigid bodies mutually interacting via a gravitationalfield. Motivation comes from planetary science, where these interactions play a keyrole in the evolution of asteroid rotation states and binary asteroid systems.

The techniques that are applied to this problem fall into two main categories.

The first is the use of geometric mechanics to obtain a description of the reduced

phase space, which opens the door to a number of powerful techniques such as the

energy-momentum method for determining the stability of equilibria and the use

of variational integrators for greater accuracy in simulation. Secondly, techniques

from computational dynamical systems are used to determine phase space structures

important for transport phenomena and dynamical evolution.

1

CONTENTS 2

Contents

1 Introduction 2

2 The F2BP and Asteroid Binaries 5

3 Phase Space Structures 13

4 A Simplified Model for Binary Asteroids 14

5 Conclusions and Future Directions 19

1 Introduction

Full body problems (FBPs) are concerned with the dynamical interaction betweentwo or more distributed bodies. This fascinating class of problems has many in-teresting open questions and touches on numerous important issues in science andengineering. Examples include binary asteroids, the evolution and dynamics ofthe Earth-Moon system, the dynamics and control of a high-performance aircraft,reaction and ionization of molecules, interactions and collisions between galaxies,stability and control of underwater vehicles, rendezvous and docking of space ve-hicles, fine pointing control of a space-based telescope, etc. Geometric mechanicsand dynamical systems theory along with appropriate computational and geomet-ric control techniques, provide a unified approach to the analysis and simulation ofthese problems.

There are many examples in which geometric mechanics methods have been usedfor this general class of problems. Two examples are the work of Krishnaprasad andMarsden [1987] concerning the use of reduction and stability methods for rigid bodieswith flexible attachments and also the work of Krupa et al. [2001] on the use of theenergy-momentum method for tethered satellites.

The Binary Asteroid Problem. In this paper we focus on the dynamics ofbinary asteroids in the context of F2BPs. This problem concerns the dynamics oftwo spatially extended bodies that interact via their mutual gravitational fields. Anexample of a motivating goal is the accurate estimation of ejection, collision, andtransport rates, accounting fully for coupling between the rotational and transla-tional states of the bodies. We put this problem into the context of systematicapproaches to determine the stability of relative equilibria as well as the compu-tation of phase space structures, such as periodic orbits, quasi-periodic orbits, andthe division of phase space into regions of regular and chaotic motion. This contextallows one to bring to bear powerful computational transport techniques, such asset oriented methods and lobe dynamics. As we shall review below, there is alreadyquite a bit known about relative equilibria in the binary asteroid problem; we referto Scheeres [2003] for further information. However, rather little has been done onthe energy-momentum method (and its converse) for relative equilibria as well as

1 Introduction 3

the problem of merging geometric mechanics and dynamical systems calculations,including phase space structure. Our goal is to take some first steps in this direction,but much work remains to be done on this problem.

The binary asteroid problem is of considerable astrodynamical interest. Forinstance, the methods are directly relevant to asteroid rotational evolution, variationof planetary obliquities, and the long-term dynamics of Kuiper and asteroid beltbinaries, including an analysis of the Pluto-Charon system.

A Little Biased History. Special cases of the F2BP have been analyzed exten-sively in the literature, with most applications focusing on the interaction of a small,distended body with a much larger body, for example the evolution of irregularlyshaped planetary moons or the dynamics and control of gravity gradient spacecraft.Even for this simplified version of the F2BP it is well-known that chaotic motionexists and can play an essential role in the life of a tidally evolving body (see, forinstance Wisdom [1987] and Touma and Wisdom [1994]). Studies from the astro-dynamics community have discovered many fundamental symmetries that exist inthese ideal problems (see, for example, Beck and Hall [1998], Wang, Krishnaprasad,and Maddocks [1991, 1992], and Wang, Lian, and Chen [1995]), which have in turnbeen used to develop novel methods for the control of these systems (e.g. Rui, Kol-manovsky, and McClamroch [2000]; Shen and McClamroch [2001]; Bloch, Crouch,Marsden, and Ratiu [2002]). The other extreme case of the F2BP has also beeninvestigated, namely the motion of a massless particle in the field of a stronglynon-spherical gravity field, with main application to the motion of spacecraft andejecta relative to an asteroid, comet or moon—see Dobrovolskis and Burns [1980],Chauvineau, Farinella, and Mignard [1993], Scheeres [1994, 1996, 1998], Scheeres,Durda, and Geissler [2003], and Scheeres, Williams, and Miller [2000].

An early paper on the FBP was Duboshin [1958], although his subsequent studiesfocused on the necessary force laws needed for certain solutions and symmetries toexist. In Maciejewski [1995], a modern statement of the F2BP is given, along witha preliminary discussion of relative equilibria possible for these systems, but withno investigation of the dynamical evolution and stability of such problems. Morerecently sharp conditions for the Hill and Lagrange stability of the F2BP have beenfound (Scheeres [2002]) and numerical and analytical investigations of rotational andtranslational coupling in these problems have been studied (Scheeres et al. [2000];Scheeres [2001, 2002a]). These most recent works serve as one starting point for thecurrent paper.

Tools for the FBP. The F2BP is a magnificently symmetric problem with theoverall symmetry group being SE(3), the group of Euclidean motions of three space,together with any symmetries of the bodies themselves. Thus, it is ideal for usingthe tools of geometric mechanics for systems with symmetry, variational integrators,and transport and dynamical systems theory. In this paper we shall take some smallinitial steps in this direction for simplified, but still non-trivial, versions of the F2BP.It is our hope that this effort will help merge work from a number of groups into asingle coherent theory. Some of the salient features of these methods are as follows.

1 Introduction 4

1. Geometric Mechanics and Reductions of the F2BP. As hinted at al-ready and explained further in, for example, Bloch [2003], Marsden and Ratiu[1999] and Marsden [1992], geometric mechanics has had enormous successesin many areas of mechanics. Previous investigations in the FBP, however,have been piecemeal at best and have missed using, for example, the power-ful energy-momentum method and its converse for relative equilibria, as wellas the use of geometric phases. For example, the converse of the energy-momentum method, due to Bloch, Krishnaprasad, Marsden, and Ratiu [1994,1996] allows one to study the destabilizing effect of dissipation and body defor-mations for relative equilibria that are gyroscopically stabilized on the linearlevel (that is, one has a saddle point in the augmented energy, but the systemeigenvalues are on the imaginary axis, so one cannot easily infer stability orinstability in the nonlinear system).

2. Variational Integrators for the F2BP. A novel computational techniquethat is ripe for use on the FBP is that of AVIs, or Asynchronous VariationalIntegrators. These are numerical simulation methods that allow different timesteps at different spatial points and yet have all the advantages of the usualsymplectic integrators used in dynamical astronomy (see, for instance Duncan,Quinn, and Tremaine [1989], Wisdom and Holman [1991] and other referencestherein). For example, in many FBPs there are large differences in time scalesfor the dynamics of rotational and translational motions. AVIs, which al-low for such adaptation, greatly improve computational efficiency as has beendemonstrated in related problems (see, for instance Lew, Marsden, Ortiz, andWest [2003,a]), and hence permit new problems to be directly simulated forlonger times and to higher accuracy than previously achieved.

3. Phase Space Structure and Transport Calculations. A third key toolthat can be brought to bear on this problem concern advances in the computa-tion of transport rates using transition state theory borrowed from chemistryand lobe dynamics from fluid dynamics, where recent computational advanceshave enabled significant capabilities using these approaches (see Jaffe, Ross,Lo, Marsden, Farrelly, and Uzer [2002] and references below). This may becombined with set oriented methods involving concepts from graph theorysuch as the notion of graph partitions, congestion, almost invariant sets andthe Peron-Frobenius operator (see for example, Dellnitz et.al. [2003]). Draw-ing on recent advances in computing phase space objects such as periodicorbits and invariant tori for astrodynamical problems, it would be interestingto investigate the link between periodic orbits, resonances, and restrictions(pinch points) in the phase space of the F2BP, structures important for thecomputation of transport, such as ejection and collision rates.

There are a number of fundamental questions in planetary science that can beaddressed by the techniques that we outline, including: Comet nucleus rotationaland translational evolution due to outgassing; Long-term simulation and evaluationsof the Yarkovsky effect on the rotational and translational motion of asteroids; Tidal


evolution of asteroid and Kuiper belt binaries including dissipation and externalforces; Dynamical evolution of galaxies that undergo a close approach.

2 The F2BP and Asteroid Binaries

While problems involving rotational and translational coupling populate many areasof science and engineering, for tractability in this paper we will draw on examplesthat arise in the field of planetary science and dynamical astronomy, as there aremany FBPs in this field whose complete understanding is still lacking, and whosesimulation depends on gross physical approximations to certain aspects of theirmotion. Examples include the evolution of the obliquities of the terrestrial plan-ets Neron de Surgy and Lasker [1997]; Touma and Wisdom [1994], the rotationaldynamics of Europa Greenberg et al. [2002], the effect of the Yarkovsky force ontranslational and rotational evolution of asteroids Bottke et al. [2000, 2001]; Spitaleand Greenberg [2001]; Rubincam [2000], and the evolution of comet nucleus rota-tion and translation due to outgassing Neishtad et al. [2002]; Yeomans and Chodas[1989], to name a few. In each of these cases interactions between rotation andtranslation are important, but detailed simulation and evaluation of these couplingsover long time spans is beyond current capability. More importantly, the specializedforce environments and physical effects unique to each of these systems has made itdifficult to formulate a general, unifying approach to the analysis and simulation ofthese problems.

Recent studies of the dynamics and evolution of binary asteroids, provides aclass of problems that can serve as fundamental, or “canonical,” models that arekey stepping stones to a unified approach to FBPs. This class of problems are ofinterest in their own right, as recent analysis indicates that up to 20% of Near-EarthAsteroids are binaries (see Margot et al. [2002]), along with the current explosion ofbinary discoveries in the Main Asteroid Belt and the Kuiper Belt Veillet et al. [2002];Merline et al. [2002] (see Figure 2.1). The analysis and modeling of binary asteroids(or similarly binary Kuiper Belt objects) does not allow the usual assumptionsof small gravity coefficients, near equilibrium conditions, or long-term stability ofmotion made in classical approaches to FBPs. Rather, to properly understand thedynamics and evolution of these systems requires a generalized approach to theproblem that incorporates these difficulties into its nominal problem statement.

Recent Advances. The general F2BP has been studied under many different ap-proximations, but its general statement has only been well-posed relatively recently(Maciejewski [1995]), followed by advances in understanding its Hill and Lagrangestability (Scheeres [2002]) and observations on the role of these interactions on theevolution of small body rotation states (Harris [2002]; Scheeres et al. [2000]; Scheeres[2002a]). The coupled motion in this problem can lead to profound dynamics, il-lustrated in Figure 2.2 (Scheeres [2002a]) showing the evolving orbit radius (a)and rotation period (b) of a sphere and tri-axial ellipsoid of equal mass interactingthrough gravity alone. The total energy of this system is slightly negative, mean-ing that the two bodies can never mutually escape (that is, they are Hill stable).


Figure 2.1: Dactyl in orbit about Ida, discovered in 1994 during the Galileo mission (Belton etal. [1996]), and 1999 KW4 radar images of its companion, discovered 2001 (Benner et al. [2001]).

The system is also stable against impact (Lagrange stable), and so will evolve adinfinitum unless some small external perturbation (such as from the sun) boosts itsenergy sufficiently to allow for escape, which would also leave the ellipsoid with anextremely slow rotation rate. The phase space of this apparently chaotic systemcan be tightly constrained using integrals of motion, and separated regions of phasespace can be identified as a function of resonances between the system’s rigid bodyrotation and orbital rotation rates. In Figure 2.2(c) (Scheeres [2002a]) shows anexample of the allowable phase space of this system, with additional restrictions onmotion in the phase space, identified as “pinch points” in the figure, that arise fromthese resonances. We conjecture that such pinch points can be understood via thereduction process from geometric mechanics, and that these will be important inunderstanding transport and ejection rates.

1

10

100

1000

10000

1 10 100 1000 10000 100000 1e+06

Rad

ius

(km

)

Time (hours)

1

10

100

1000

10000

1 10 100 1000 10000 100000 1e+06

Rot

atio

n P

erio

d (h

ours

)

Time (hours)

0

0.2

0.4

0.6

0.8

1

1.2

4 6 8 10 12 14 16 18

Ecc

entr

icity

Periapsis (km)

Arg. of Periapsis = 0Arg. of Periapsis = 90

Pinch PointsSample Run(a) (b) (c)

Figure 2.2: Time history of the orbit radius (a) and rotation period (b) for a gravitationallyinteracting sphere and tri-axial ellipsoid of equal mass. Poincare map (c) showing constraints onthe eccentricity of the evolving orbit, including pinch points that isolate regions of phase space.

Models In Use. The equations of motion for the F2BP model may be explic-itly found and are discussed in a number of references, such as Maciejewski [1995];Scheeres [2002]. A widely used model can be derived when one of the bodies isa massive sphere. The rotational dynamics of the sphere decouples from the sys-


tem, which still has coupling of rotational and translational dynamics between thenon-spherical body and the relative orbit between the two. The normalized andsymmetry reduced equations are

r′′ + 2ω × r′ + ω′ × r + ω × (ω × r) =∂U∂r

I · ω′ + ω × I · ω = −µr × ∂U∂r

, (2.1)

where

K =1µA · I · ω + A · r ×

(r′ + ω × r

)E =

12

(r′ + ω × r

)·(r′ + ω × r

)+

12µ

ω · I · ω − U (2.2)

and where ω is the rotational velocity vector in the body-fixed frame, r is the relativeposition vector in the body-fixed frame, A is the attitude tensor of the non-sphericalbody, I is the specific inertia tensor of the non-spherical body, U is the gravitationalforce potential of the non-spherical body, and K and E are the angular momentumand energy integrals, respectively, of these equations.

The free parameters of the system are the mass distribution of the non-sphericalbody (normalized by its largest dimension) and the mass fraction between the twobodies, µ = M1/(M1 + M2) (the same parameter found in the restricted 3-bodyproblem), where M1 is the mass of the “sphere” and M2 is the mass of the distributedbody. The case when µ → 1 corresponds to the motion of a massless distributedbody about a point-mass, with an application to a satellite in orbit about a planet.It is important to note that the angular momentum and energy integrals still applyto this problem. The case when µ → 0 corresponds to the motion of a material pointin the gravity field of the distributed body, with the main application to particledynamics about an asteroid. In this case, we see that the energy and angularmomentum integrals are dominated by the rotational dynamics of the distributedbody, and that the contribution of the spherical body’s motion decouples from theseintegrals.

In this paper we shall use equation (2.1), often with some further simplifications.These simplifications are made to allow for an abbreviated discussion of results, andwill be relaxed in future papers. The main simplification made is the restricted,uniformly rotating approximation which occurs when µ → 0 and the distributedbody (which now has all the mass of the system) is in principal axis rotation aboutits maximum moment of inertia (assumed to lie along the z-axis). When needed,we will refer to this system as the Restricted Full 2-Body Problem (RF2BP), andwe note its similarity to the celebrated restricted 3-body problem, including theexistence of a Jacobi integral. The RF2BP, while highly idealistic, is importantfor understanding the general and qualitative properties of motion about distendedbodies in uniform rotation (Scheeres [1999a]) and approximately models the motionsof some binary asteroids.

Carrying out such detailed calculations over the time scales of interest for theF2BP is challenging and requires methods designed to conserve the symmetries in-


herent in the problem and to handle the multiple time scales in FBPs. Additionally,the existence and prediction of the restricted regions of phase space far from anequilibrium point is an example of a complex, poorly understood dynamical phe-nomenon. The techniques discussed below take some first steps to address thisdeficiency. The F2BP is ideally suited for the application of geometric mechanicswith symmetry, variational integrators, and transport and dynamical systems the-ory. Due to the generality of its statement, the F2BP can be extended to cover othersystems of interest in planetary science, dynamical astronomy, astrodynamics, andchemical dynamics, briefly discussed later in the paper.

Geometric Mechanics and the F2BP. We ultimate want to carry out an anal-ysis of the F2BP by first looking at the problem’s symmetry. The F2BP has thesymmetry group SE(3) accompanied with the corresponding conserved quantities.As the symmetry groups become larger, we expect that one recovers all the specialcases studied in the literature. For example, if one body is an irregular body andthe other is cigar-like, the symmetry group is SE(3) × S1. Relative equilibria andtheir stability can be studied using the powerful energy-momentum method (Simo,Lewis, and Marsden [1991]; Marsden [1992]; Bloch, Krishnaprasad, Marsden, andRatiu [1996]). Relative equilibria are key ingredients in identifying and characteriz-ing the possible final states that an evolving binary asteroid can reach and they playa similar role as the libration points in the 3-body problem. Moreover, a systematicgeometric mechanical approach will enable the use of modern numerical algorithms,such as variational integrators, in the accurate computations of relevant long termstatistics and transport (Marsden and West [2001]; Lew, Marsden, Ortiz, and West[2003,a]).

Symmetry Reduction. The use of geometric and dynamical techniques to studyLagrangian or Hamiltonian mechanics has been enormously successful in a wide va-riety of engineering and astrodynamical problems, such as the use of the energy-momentum method for stability of satellites with internal rotor controls (see, forexample, Bloch, Krishnaprasad, Marsden, and Ratiu [1996]) and in heteroclinic andresonance structures in the 3-body problem (Koon, Lo, Marsden, and Ross [2000]).Geometric mechanics starts with the usual formulation of Lagrangian mechanics us-ing variational principles, and Hamiltonian mechanics using Poisson and symplecticgeometry. Lagrangian reduction by a symmetry group corresponds to finding re-duced variational principles, whereas reduction on the Hamiltonian side correspondsto constructing appropriate reduced Poisson and symplectic structures.

The reduction of a system with configuration manifold Q and Lie symmetrygroup G occurs at two levels, the first of which corresponds to identifying solutionsthat are related by a symmetry group motion, and this corresponds to derivingequations of motion on the quotient space TQ/G (for Lagrangian reduction), andon T ∗Q/G, (for Hamiltonian Reduction). Symmetry gives an associated Noethertheorem, namely conservation of a momentum map, so that the dynamics are con-strained to a momentum surface in the absence of external forcing and dissipation.When appropriate, imposing this constraint explicitly in the equations of motion,


yields a further reduction in the dimensionality of the reduced equations, whichcorresponds to (nonabelian) Routh and symplectic reduction.

Lagrangian ReductionTQ/G ∼= T (Q/G) ⊕ g

Fl Hamiltonian ReductionT ∗Q/G ∼= T ∗(Q/G) ⊕ g∗

Routh ReductionJ−1

L (µ)/Gµ∼= T (Q/G) ×Q/G Q/Gµ

Variational Principles

FRµ

Symplectic ReductionJ−1(µ)/Gµ

∼= T ∗(Q/G) ×Q/G Oµ

Symplectic/Poisson Geometry

On the Lagrangian side, general theorems give the structure of the quotientspace to be

TQ/G ∼= T (Q/G) ⊕ g,

where g is the Lie algebra of G and where g is an associated bundle over shape spaceQ/G, as shown in Cendra, Marsden, and Ratiu [2001]. There is a similar structuretheorem on the Hamiltonian side. When the momentum constraints are imposed,on the Hamiltonian side one gets well known cotangent bundle reduction theorems(see, for example, Marsden [1992]) while on the Lagrangian side one gets a farreaching, and surprisingly recent, generalization of the classical Routh procedure(see Marsden, Ratiu, and Scheurle [2000]). The general scheme of the reductionprocedure is shown in the above figure. For the F2BP, the choice of Q and G areeasily done and are given in the next section.

Reduction for the F2BP. For the F2BP, the configuration space is Q = SE(3)×SE(3). Denote material points in a reference configuration by Xi, and the points inthe current configuration by xi. Given ((A1, r1), (A2, r2)) ∈ SE(3) × SE(3), pointsin the reference and current configurations are related by xi = AiXi + ri, i = 1, 2.Using the body angular velocity notation defined earlier, the Lagrangian has thestandard form of kinetic minus potential energy:

L(A1, r1, A2, r2) =12

∫B1

‖x1‖2dµ1(X1) +12

∫B2

‖x2‖2dµ2(X2)

+∫B1

∫B2

Gdµ1(X1)dµ2(X2)‖x1 − x2‖

=m1

2‖r1‖2 +

12〈Ω1, I1Ω1〉 +

m2

2‖r2‖2 +

12〈Ω2, I2Ω2〉

+∫B1

∫B2

Gdµ1(X1)dµ2(X2)‖A1X1 − A2X2 + r1 − r2‖

.

Reducing by Overall Translations and Rotations. The preceding Lagrangianhas symmetry group SE(3) using the diagonal left action on Q:

(A, r) · (A1, r1, A2, r2) = (AA1, Ar1 + r, AA2, Ar2 + r).


The momentum map corresponding to this symmetry is the total linear momentumand the total angular momentum. As in Maciejewski [1995, 1999] and Gozdziewskiand Maciejewski [1999], the projection from the configuration space to the shapespace π : Q → Q/G, is obtained by transforming to the body frame of the secondrigid body. The reduction is carried out in stages, by first reducing by translations,R

3, followed by reducing by rotations, SO(3). Results from general reduction bystages Cendra, Marsden, and Ratiu [2001]; Leonard and Marsden [1997]; Marsden,Misiolek, Perlmutter, and Ratiu [1998] shows that this is equivalent to directlyreducing by SE(3) in a single step. This is achieved by applying the inverse of(A2, r2), which is given by (AT

2 ,−AT2 r2) ∈ SE(3),

(AT2 ,−AT

2 r2) · (A1, r1, A2, r2) = (AT2 A1, A

T2 r1 − AT

2 r2, AT2 A2, A

T2 r2 − AT

2 r2)

= (AT2 A1, A

T2 (r1 − r2), e, 0).

Shape space Q/G is isomorphic to one copy of SE(3), being coordinatized by therelative attitude AT

2 A1 = AT and relative position AT2 (r1 − r2) = R. The equations

of motion in T (Q/G) (resp. T ∗(Q/G)) involve A, R, and their velocities (resp.conjugate momenta Γ, P ), which correspond to total angular and linear momentain the body fixed frame of the second rigid body. These are coupled to equationsin se(3)∗, which may be identified with equations for the body angular and linearmomenta of the second rigid body, Γ2, P2.

In Maciejewski [1995, 1999] and Gozdziewski and Maciejewski [1999], the equa-tions for Γ and Γ2 are rewritten in terms of Γ1,Γ2, and the linear momentum of thesecond rigid body is ignored. A reconstruction-like equation (that is, recovering thefull attitude of both bodies) is added at the very end. One thing missing from theliterature is the systematic use of shape space to study geometric phases that areimportant for rotational and translational drifts (Marsden, Montgomery, and Ratiu[1990]; Leonard and Marsden [1997]). The general context of geometric mechanicsand the specific setting of the present paper should enable one to fill in this interest-ing gap. The coordinatization of the reduced space by transforming to the referenceframe of the second rigid body corresponds to a particular choice of connection onthe principal bundle Q → Q/G. Different choices of connection will affect how weidentify TQ/G with T (Q/G) ⊕ g.

The reduced Lagrangian is obtained by rewriting the Lagrangian in terms of thereduced variables:

A = AT2 A1, R = AT

2 (r1 − r2), Ω = AT2 A1, V = AT

2 (r1 − r2),

which are coordinates on T (Q/G), as well as Ω2 = AT2 A2, V2 = AT

2 r2, which arecoordinates on se(3). As with the Euler-Poincare theory (Marsden and Ratiu [1999]),Hamilton’s variational principle on T (SE(3) × SE(3)) is equivalent to the reducedvariational principle,

δ

∫ b

al(A, R, Ω, V, Ω2, V2)dt = 0,


on R18 where the variations are of the form,

δA = −Σ2A + Σ, δR = −Σ2R + S, δΩ = ˙Σ − Σ2Ω + Ω2Σ,

δV = S − Σ2V + Ω2S, δΩ2 = Σ2 + Ω2 × Σ2, δV2 = S − Σ2V2 + Ω2S2,

and S, S2,Σ,Σ2 are variations that vanish at the boundary. The symplectic formcan be obtained from the variational principle by considering variations that do notvanish at the boundary. The boundary terms that arise from integration by partscan be interpreted as a Lagrange one-form, and taking the exterior derivative ofthese yield the symplectic two-form.

Already the above approach is more systematic and complete than that given inthe literature. While simple, the variational structure just given is, in fact, new. Infuture studies, we will complete this task and deal with all of the systematic casesfor the F2BP and to tie in this theory with the stability theory of relative equilibriausing the powerful energy-momentum method.

The Energy-Momentum Method. This is a powerful generalization of the clas-sical energy methods for stability (Simo, Lewis, and Marsden [1991]). A study of thestability of rigid body pairs using the energy-momentum method is directly relevantto the evolution of binary asteroids and the dynamical environments they encounterthrough their lifetime. The current literature, however, uses only the Arnold orenergy-Casimir methods (Maciejewski [1999]) which generally give weaker results.It would be very interesting to make use of the full block diagonalization power ofthe energy-momentum, which has proven useful in related problems (Wang, Krish-naprasad, and Maddocks [1991]; Krupa et al. [2001]). In addition, there is a converseto the method which allows one to deduce the development of instabilities due toforcing and dissipation (Bloch, Krishnaprasad, Marsden, and Ratiu [1994, 1996]),which can have important long term effects on the dynamics (Clerc and Marsden[2001]).

Symmetries of the Rigid Bodies. If, in addition, the individual rigid bodiesexhibit configurational symmetries of their own, there are additional symmetriesacting on the right. Since these commute with the symmetry of SO(3) acting onthe left, one can again make use of reduction by stages.

Example: Spherical Symmetry. If the first body is spherical, and the secondbody has no symmetry, the additional symmetry group is SO(3)×e acting on theright. On the Hamiltonian side, the reduced space is then T ∗

R3 × se(3)∗ × so(3)∗.

The T ∗R

3 terms correspond to position and linear momentum of the center of massof the first rigid body, in the body frame of the second rigid body. The se(3)∗

term involves the spatial linear and angular momentum of the second rigid body,and so(3)∗ involves the body angular momentum of the first rigid body. There is asimilar picture on the Lagrangian side.


Example: Cylindrical Symmetry. If the first body is cylindrical, and the sec-ond body has no symmetry, the additional symmetry group is S1 × e acting on theright. The reduced space is then T ∗(S2 ×R

3)× se(3)∗ ×R. The T ∗(S2 ×R3) terms

will correspond to the orientation of the symmetry axis and the position of the firstrigid body, in the body frame of the second rigid body, as well as the conjugatemomenta. The se(3)∗ term involves the spatial linear and angular momentum of thesecond rigid body, and the R term will involve the component of the body angularmomentum of the first rigid body in the direction of its symmetry axis.

The preceding reduction procedures give a global description of the reducedspaces and dynamics. This is important, since ad hoc choices for coordinate systemswhich are only locally valid can introduce significant computational overheads if itis constantly necessary to switch between local charts in the course of a long timesimulation. The variational, Poisson, and symplectic structures on these reducedspaces can be constructed systematically from the general theory of reduction, and,as we have indicated, this is particularly useful in classifying a hierarchy of simplifiedmodels.

Relative equilibria and relative periodic orbits correspond to equilibria and pe-riodic orbits in the shape space Q/G. The bifurcation of a relative equilibria into arelative periodic orbit through a Hamiltonian Hopf bifurcation in the shape spaceis also particularly interesting for the study of real world objects, which may besubject to parametric uncertainty, and the effects of such parameter dependenceneed to be studied in our models. More generally, the study of these phase spaceobjects, their stability, and their bifurcation patterns can be used to characterizemotion in different regions of phase space (Lara and Scheeres [2003]).

The phenomena of pinch points in the dynamics could possibly be related tothe presence of resonances corresponding to discrete symmetries, or possibly to de-generate values of the momentum map corresponding to the non-free action of acontinuous symmetry group. It is well known that pinch points of this sort arerelated to reduction in singular cases (this is well-known; an example is given in,for instance, Alber et al. [2002]). An example of a problem with such a non-freeaction is the double spherical pendulum, and the bifurcation from the degeneratedownward configuration into relative equilibria and relative periodic orbits has beenanalyzed using the machinery of bifurcation theory and singular reduction theory(Hernandez and Marsden [2002]). A better understanding of the pinch points ob-served in dynamical systems will be directly applicable for identifying the regionsof phase space which contribute to statistical transport phenomenon.

Geometric Phases. The issue of geometric phases is not only of interest in the dy-namics of an asteroid pair, but is also relevant to the problem of landing a spacecrafton a spatially extended and irregular asteroid (as in the recent NEAR mission tothe asteroid EROS). An understanding of geometric phases allows one to relate themotion of internal rotors, and their effect on the relative orientation of the spacecraftwith respect to the asteroid. This provides a precise method of controlling the rela-tive orientation, which is in contrast to the use of micro-thrusters, which introducetorque which have to be compensated for once the desired orientation is achieved.

3 Phase Space Structures 13

This same approach can be applied to the study of the tidally locked Pluto-Charonsystem and may help address some aspects of the current controversy surroundingthat body (Stern, Bottke, and Levison [2003]), and can be applied to spacecraftcontrol in close-proximity to an asteroid (Sawai, Scheeres, and Broschart [2002]).In this context of control, the method of Controlled Lagrangian and Hamiltoniansystems may be a useful tool (see Chang et al. [2002] and references therein). This isgeneralization of potential and kinetic shaping techniques which has been applied inthe context of spacecraft dynamics to modify the system’s Poisson structure usingfeedback control laws. A corresponding theory for the reduction of systems withsymmetry also exists, using the method of Lagrangian and Hamiltonian reduction.

3 Phase Space Structures

Global Theory of Chaotic Transport. After systematically categorizing thesystems of interest using geometric mechanics, the next step is to use efficient compu-tations guided by dynamical systems theory to explore statistical questions. Thesequestions, including the probability of binary pair formation and subsequent escapeor impact, and ejecta redistribution around a rotating small body can be cast asphase space transport problems. The systems of interest, being Hamiltonian in na-ture, likely have a global mixed phase space structure of stable and chaotic zones,i.e., islands of KAM tori and a “chaotic sea” between them. A semi-analytical globaltheory of chaotic transport is emerging which combines the theory and numerics oflobe dynamics and tube dynamics, (see Rom-Kedar and Wiggins [1990]; Koon, Lo,Marsden, and Ross [2000]; Lekien and Marsden [2002]). The lobe dynamics tech-niques come partly from fluid dynamics—e.g., those developed in the last two yearsusing the mangen software—providing unprecedented long-term precision calcula-tions (see Lekien and Coulliette [2002]; Lekien [2003]).

Efficient Computation of Statistical Quantities. This global theory allowsone to tackle transport problems by focusing on the objects important for transport.Dynamical systems theory has been used previously to give a qualitative descriptionof the topological features of the phase space in certain reduced F2BP models, suchas periodic orbits, resonance regions, and chaotic zones. One can make this analysismore quantitative by computing statistical measures such as residence times withinregions of phase space, transport rates between various regions, and the overall levelof mixing between regions.

Since the F2BP is such a fundamental model in studying small body orbital androtational dynamics, the present context should shed light on several interestingand important problems. Specifically, it may help address the problem of slowlyrotating asteroids (Harris [2002]), it will be able to quantify the rate at which aster-oid binaries are disrupted due to their mutual interactions following their creation(Scheeres [2002a]), and the important issue of chaotic planetary obliquities and theirprobability of transition to different states (Neron de Surgy and Lasker [1997]).


4 A Simplified Model for Binary Asteroids

The Restricted Full 2-Body Problem (RF2BP). The simplest model thatexhibits the basic ejection and collision dynamics we are interested in studying isgiven by the RF2BP, which is a simplification of Equation (2.1) in the limit µ → 0.The equations of motion for the massless particle in a rotating Cartesian coordinateframe and appropriately normalized are

x − 2y =∂V

∂x

y + 2x =∂V

∂y, (4.1)

where

V (x, y) =1√

x2 + y2+

12(x2 + y2) + U22

U22 =3C22

(x2 − y2

)(x2 + y2)5/2

The system (4.1) has one free parameter, the gravity field coefficient C22, commonlytermed the “ellipticity”, that varies between 0 and 0.05 for physical systems. Thesystem (4.1) may be readily derived from Lagrangian mechanics using the methodof “moving systems” (see Marsden and Ratiu [1999] and therefore has an energyintegral (a Jacobi integral), namely

E =12

(x2 + y2

)− V (4.2)

This system has 4 equilibrium points symmetrically placed along the x and yaxes, each at a radius of R ∼ 1 + . . . , with the higher order terms arising fromthe U22 potential. In general, we do not consider motion at radii much less thanR ∼ 1, since these trajectories will usually impact on the central body. Also, notethat these equations have very bad behavior as R → 0, which is another reason toavoid radii much inside the equilibrium points.

The region of space where R > 1 is very interesting, however. First, there arecontinuous families of near-circular periodic orbits that exist for any given radius.Occasionally, these periodic orbits are resonant with the rotation rate of the system,and at these points we can find additional families of “elliptical” periodic orbits thatbranch off.

Realms and Regions in the RF2BP. In the RF2BP, the value of the energyE indicates the type of global dynamics possible. For example, for energies abovea threshold value, (i.e., E > ES , where ES corresponds to the energy of symmetricsaddle points along the x-axis), movement between the realm near the asteroid(interior realm) and away from the asteroid (exterior realm) is possible, as in Figure4.1(a). This motion between realms is mediated by phase space tubes, to be described


shortly. For energies E ≤ ES , no such movement is possible. Within each realm,the phase space on appropriately chosen Poincare sections is organized further intodifferent regions, connected via lobes.

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

x (nondimensional units, rotating frame)

y (n

ondim

ensi

onal

unit

s, r

ota

ting f

ram

e)

Exterior Realm

InteriorRealm

U2

Poincare Section

InteriorRealm

U1

Poincare Section

Energetically

Forbidden Realm

f1

f2

f12

f2

f1z0

z1z2

z3z4

z5U2

U1

Exit

Entrance

(a) (b)

Figure 4.1: Planar restricted F2BP with uniform rotation: realms and allowable

motion. (a) For energies, E, above a threshold, ES , movement is possible between the realm near

an asteroid and away from the asteroid. In this rotating frame, the x-axis coincides with the elliptical

asteroid’s long axis. The origin is shown as a large black dot at the center of the asteroid. The

value of the “ellipticity”, C22, is 0.05. Varying this parameter changes the shape of the asteroid and

the subsequent potential through the U22 term in Equation (4.1). (b) Poincare sections in different

realms—in this case in the interior and exterior realms, U1 and U2, respectively—are linked by

phase space tubes which live in surfaces of constant energy (Equation (4.2)). Under the Poincare

map f1 on U1, a trajectory reaches an exit, the last Poincare cut of a tube before it enters another

realm. The map f12 takes points in the exit of U1 to the entrance of U2. The trajectory then evolves

under the action of the Poincare map f2 on U2.

Tube Dynamics: Transport between Realms. In Figure 4.1(a) the interiorand exterior realms are linked by tubes in phase space, bounded by the stable andunstable invariant manifolds associated to periodic orbits around the symmetricsaddle points. The role these tubes play in global transport between realms isreferred to as tube dynamics (Koon, Lo, Marsden, and Ross [2000]). One eachthree-dimensional energy surface these two-dimensional tubes partition the surface,acting as separatrices for the flow through the bottleneck regions around the saddlepoints: particles inside the tubes will move between realms, and those outside thetubes will not. For example, an ejecta particle liberated from the surface of theasteroid into the exterior realm with an energy just above the threshold can impactthe asteroid only by passing through one or the other of the pair of exterior branchstable manifold tubes associated to the two periodic orbits around the saddle points


(the right periodic orbit is shown in Figure 4.1(a)).The projection of these tubes onto the configuration space appear as strips and

trajectories on the tubes wind around them. A few trajectories on a couple of tubesare shown in Figure 4.1(a). On the Poincare section, the last Poincare cut of thestable manifold tube is called the exit, because points in there will exit the interiorrealm and go to the exterior realm. The time reverse situation holds for an entrance,and when a particle goes through the exit in one realm, it must enter the entranceof another.

These transition dynamics are of direct interest to the evolution of a dissipatingbinary system, as it may experience several junctures when it can transition betweena state close to its final equilibrium and a state where more dynamic evolution ispossible. Similarly, the long-term variation of planetary obliquities may be modeledusing this approach.

Lobe Dynamics: Transport between Regions. Tubes are only one part of theglobal transport picture. The study of transport between regions within a particularrealm can be reduced to the study of an associated Poincare section in that realm.Following Wiggins [1992], lobe dynamics theory states that the two-dimensionalphase space M of the Poincare map f can be divided into disjoint regions of in-terest, Ri, i = 1, ..., NR, such that M = ∪Ri (Figure 4.2(a)). The boundaries be-tween regions are pieces of stable and unstable manifolds of hyperbolic fixed points,pi, i = 1, ..., N. Moreover, transport between regions of phase space can be com-pletely described by the dynamical evolution of turnstile lobes enclosed by segmentsof the stable and unstable manifolds (Figure 4.2(b)).

To keep track of points as they move between regions, suppose that, at t = 0,region Ri is uniformly covered with points of species Si. The transport problembecomes one of describing the distribution of species Si throughout the regions Rj

for any time t = n > 0. Two quantities of interest—the flux of species Si into regionRj on the n-th iterate, αi,j(n), and the total amount of species Si in region Rj justafter the n-th iterate, Ti,j(n)—can be expressed compactly in terms of intersectionareas of images or pre-images of turnstile lobes. In our application, the Poincaremap of a reduced F2BP model possesses resonance bands consisting of alternatingunstable and stable periodic points. For instance, we can study the transport fromstable configurations to escape configurations or the transport into and out of spin-orbit resonances, and many other transport questions that have implications for thecurrent state of bodies in the solar system.

Computational Tools for Lobe Dynamics. While lobe dynamics has alwaysbeen recognized as an exact transport theory and can theoretically give short andlong term transport rates, computational issues have limited its applications (Rom-Kedar, Leonard, and Wiggins [1990]; Rom-Kedar and Wiggins [1991]). The man-ifolds computed in such problems are typically rather convoluted, as implied inFigure 4.2. Furthermore, the length of these complicated curves grows quickly withthe size of the time window of interest. The number of points needed to describe long


R1

B12 = U [p2 ,q2] U S [p1 ,q2]

R5

R4

R3

R2

q2

q1q4

q5

q6

q3

p2p3

p1

R1

R2

q0

pipj

f -1(q0)q1

L2,1(1)

L1,2(1)

f (L1,2(1))

f (L2,1(1))

(a) (b)

Figure 4.2: Transport between regions in phase space. (a) A point qk is called a primary

intersection point (pip) if S[pi, qk] intersects U [pj , qk] only at the point qk, where U [pj , qk] and

S[pi, qk] are segments of the unstable and stable manifolds, W u(pj) and W s(pi) respectively, of

unstable fixed points of the Poincare map f . The union of segments of the unstable and stable

manifolds form partial barriers, or boundaries U [pj , qk]∪S[pi, qk] between regions of interest Ri, i =

1, ..., NR, in M = ∪Ri. The region on one side of the boundary B12 is labeled R1 and the other

side labeled R2. (b) Let q0, q1 ∈ W s(pi)∩W u(pj) be two adjacent pips, i.e., there are no other pips

on U [q0, q1] and S[q0, q1], the segments of W u(pj) and W s(pi) connecting q0 and q1. The region

interior to U [q0, q1]∪ S[q0, q1] is a lobe. Then S[f−1(q0), q0]∪U [f−1(q0), q0] forms the boundary of

two lobes; one in R1, labeled L1,2(1), and the other in R2, labeled, L2,1(1). Under one iteration of

f , the only points that can move from R1 into R2 by crossing B12 are those in L1,2(1) and the only

points that can move from R2 into R1 by crossing B12 are those in L2,1(1). The union of the two

lobes, L1,2(1) ∪ L2,1(1), is called a turnstile.

segments of manifolds can be prohibitively large if naive computational methods areused.

Recent efforts made to incorporate lobe dynamics into geophysical and chemicaltransport calculations have brought new techniques to compute invariant manifolds(Coulliette and Wiggins [2001]; Lekien and Marsden [2002]; Lekien and Coulliette[2002]; Lekien, Coulliette and Marsden [2003]; Lekien [2003]). We have been ableto compute long segments of stable and unstable manifolds with high accuracyby conditioning the manifolds adaptively, e.g., by inserting more points along themanifold where the curvature is high. As a result, the length and shape of themanifold is not an obstacle, and many more iterates of lobes than hitherto possiblecan be generated accurately. Initial tests show that this method of computing lobesusing mangen is very relevant to the problems we investigate.

Phase Space Structure in the Exterior Region. A Poincare surface-of-section(s-o-s) of the RF2BP at an energy just above the threshold where the bottleneckappears (i.e., E > ES , the case in Figure 4.1(a)) illustrates the relevance of tube andlobe dynamics in this system. In Figs. 4.3(a,b), the s-o-s was taken in the exteriorregion along the positive x-axis. At the energy chosen, there is a bottleneck around


1 2 3 4 5 6 7 8 9 10

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

R : Radial Distance from Asteroid’s Center (Nondim.)

V :

Rad

ial V

elo

city

(N

on

dim

.)

Collision Tube Slices

Initial Condition for Trajectory in (c)

(a) (b)

-1 0 1 2 3 4 5-3

-2

-1

0

1

2

3

X (Nondim., Inertial Frame)

Y (

No

nd

im.,

Iner

tial

Fra

me)

(c)

Figure 4.3: Mixed phase space structure for the restricted model: the dynamics of

particle ejection from, and collision with, a distended asteroid. A Poincare surface-

of-section taken in the exterior region (R > 1). (a) Particles not ejected after 15 iterates are

shown. The coordinates are the radial velocity of the particle versus its radial distance from the

asteroid’s center. The finger-like structures visible here, the lobes of ejection, have been seen in

some chemistry problems Tiyapan and Jaffe [1995]; Lekien and Marsden [2002]. (b) Semi-major

axis vs. Argument of Periapse with respect to the rotating asteroid (the body-fixed frame). (c) An

interesting trajectory corresponding to the labeled point on the Poincare section in (a) is shown.

The trajectory escapes the asteroid only to fall back upon it after one large elliptical orbit. The

units are in terms of the semiaxis length of the asteroid, which is shown schematically at one instant

of time. The asteroid is rotating counterclockwise around its center (shown as a large dot at the

origin). The long axis of the asteroid sweeps out the circular region bounded by the dashed line.

the equilibrium points along the x-axis. Particles beginning in the exterior regionwill reach the interior region and subsequently collide with the asteroid if they lie


within the phase space tubes associated with the unstable periodic orbits abouteither the left or right saddle points. Thus we refer to the tube slices on this s-o-sas tube slices of collision. Furthermore, particles will be ejected from the system ifthey lie within lobes enclosed by the stable and unstable manifolds of a hyperbolicfixed point at (+∞, 0), referred to as lobes of ejection.

Physical insight is gained into the conditions for ejection by transforming toDelaunay variables. In Figure 4.3(b), the semimajor axis is shown versus the argu-ment of periapse with respect to the rotating asteroid (the body-fixed frame). Thealternate fates of collision and ejection are seen to be intimately intermingled in thephase space. Note that the number of particles remaining in the fourth quadrantis smaller than that in the other three quadrants, in agreement with observationsin Scheeres [1996]. Figure 4.3(c) shows a particular trajectory which escapes fromthe asteroid only to fall back upon it, a peculiar phenomenon encountered in tubedynamics (Koon, Lo, Marsden, and Ross [2000]).

In the RF2BP, we are considering the motion of a massless particle, i.e., the limitµ → 0. The next step in our study is to assume that the particle has mass, i.e.,µ > 0 in Eq. 2.1. Due to the gravitational attraction of the particle, the rotationof the asteroid is now non-uniform and must be tracked. While this adds anotherdimension to the phase space, it also adds a new integral, the angular momentum Kin Eq. (2). Thus we can still use the two-dimensional Poincare map analysis, as inFigure 4.3. We are currently in the process of characterizing the global phase spacestructure which gives rise to the complex behavior seen in Figure 2.2 and similarproblems.

5 Conclusions and Future Directions

We have outlined the general setting for the application of techniques from geomet-ric mechanics and dynamical systems transport calculations to full body problems(FBPs). General FBPs have a wide range of other interesting aspects as well, in-cluding the 6-DOF guidance, control, and dynamics of vehicles, the dynamics ofinteracting or ionizing molecules, the evolution of small body, planetary, or stellarsystems, and almost any other problem where distributed bodies interact with eachother or with an external field.

This paper focused on a motivating example of a full 2-body problem fromdynamical astronomy—the problem of asteroid pairs and to the calculation of binaryasteroid escape rates. We have given some preliminary results for a simplified modelof binary asteroid escape rates, describing how lobe and tube dynamics lead to arich phase space even for this simplified system.

Some future directions for this area of research are as follows.

Variational Integrators for FBPs. As is noted in the literature, the Poissonstructures that are obtained in the reduced models are non-canonical, and as suchapplying standard symplectic algorithms will not provide the long-time stability wehave come to associate with such numerical methods, since they preserve the canon-ical symplectic form, which is not consistent with the reduced dynamics. Instead,


we shall to make use of reduction theory for discrete systems, the abelian Routhcase of which is worked out and applied to the J2 problem (satellites in orbit aboutan oblate planet) in Jalnapurkar, Leok, Marsden, and West [2002]. We will studytransport phenomena in the F2BP using numerical schemes that preserve the geo-metric structures inherent in the system. Toward this end, we will redo the abovecomputations for other F2BP cases and carry out long time accurate time inte-grations using variational integrators that capture transport rates and the detailedstructure of chaotic sets.

Variational integrators provide a systematic and powerful extension of symplec-tic integrators that have a proven track record in celestial mechanics for long-termintegrations; see, for instance Duncan, Quinn, and Tremaine [1989]; Wisdom andHolman [1991]; Saha and Tremaine [1992]. The variational setting allows for exten-sions to partial differential equations, for asynchronous methods as well as the useof symmetry methods that are central to this approach.

The idea of the variational integration method is to discretize Hamilton’s prin-ciple directly rather than discretize the associated Euler–Lagrange equations. Thediscrete form of Hamilton’s principle then determines the numerical algorithm forthat system. The accuracy to which the approximation of Hamilton’s principle isdone is reflected in the accuracy of the algorithm itself; however, much more istrue. When integrators are designed this way, they have remarkable respect for thegeometric mechanics aspects of the problem, including excellent energy behavior,conservation of the symplectic structure and exact conservation of a discrete ver-sion of the Noether quantities associated with symmetries. It is perhaps surprisingthat these methods work well even for dissipative and forced systems. The idea inthis case is to discretize the Lagrange–d’Alembert principle rather than Hamilton’sprinciple.

Recent references that develop and document the success of this methodologyare Marsden, Patrick, and Shkoller [1998]; Lew, Marsden, Ortiz, and West [2003];Marsden and West [2001]. Figure 5.1 shows a computation for a particle movingin the plane under a radially symmetric polynomial potential, without and witha small amount of friction. The exact preservation of the conserved quantities, inthis case energy, is a natural consequence of the discrete variational principle. Inaddition, Figure 5.1 illustrates that these methods can handle dissipative systemsand get the energy decay rate accurate as measured against a benchmark calcula-tion (see Kane, Marsden, Ortiz, and West [2000]). In addition, it is shown in Lew,Marsden, Ortiz, and West [2003] and Rowley and Marsden [2002] that many statis-tical quantities, such as temperature and the structure of chaotic invariant sets, areaccurately captured by variational integrators.

A key feature is the development of AVI’s or asynchronous variational integra-tors (Lew, Marsden, Ortiz, and West [2003]) which allow one to take different timesteps at different spatial points and yet retain all the beautiful structure preservingproperties of variational integrators. This is important in FBPs since the differentbodies may have vastly different spatial and temporal scales, and so one must adaptthe time steps according to these different scales. The AVI approach is a naturalway to study multi-scale dynamics without sacrificing structure preservation. In-


0 200 400 600 800 1000 1200 1400 16000

0.05

0.1

0.15

0.2

0.25

0.3

Time

Ene

rgy

Variational

Runge-Kutta

Benchmark

0 100 200 300 400 500 600 700 800 900 10000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Time

Ene

rgy

Midpoint Newmark

Explicit NewmarkVariational

Runge-Kutta

Benchmark

Figure 5.1: Conservative system (left) and dissipative system (right).

terestingly, this is closely related to methods used in molecular dynamics (work ofBarash and Schlick, in progress; cf. Schlick [2002]).

Reduction for Discrete Mechanics. There has been significant work done onreduction theory for discrete variational mechanics. An example, developed in Jal-napurkar, Leok, Marsden, and West [2002] is that of the dynamics of a satellite in thepresence of the bulge of the Earth, in which many interesting links with geometricphases and computation in the reduced and unreduced spaces are noted. However,for the F2BP, this basic theory needs to be generalized, a key point being the ex-tension to the case of nonabelian symmetry groups and to generalize it beyond theEuler-Poincare context Marsden, Pekarsky, and Shkoller [2000]. We will undertakesuch a study and to apply it to the case of the F2BP. There is a cautionary messagein the work that numerical experience shows—it is important to fully understandthe reduction of discrete variational mechanics, since applying standard numericalalgorithms to the reduced equations obtained from continuous reduction theory maynot yield the desired results and long term stability may not be respected. A relatedgoal is to combine variational integrators and adaptive manifold conditioning to givea state of the art package to compute manifolds, lobes and turnstiles in a varietyof systems, including the F2BP. The use of variational integrators is expected toincrease the accuracy of computation of long term statistical quantities, such astransport rates. We also may combine these techniques with those of the Dellnitzgroup using box elimination methods Dellnitz et.al. [2003].

The Full N-Body Problem. A natural extension of the F2BP is the problem ofmultiple distributed bodies interacting with each other, the Full N -Body Problem.For example, it would be interesting to put the work of Duboshin [1958] into a mod-ern context using the same reduction and simulation theory outlined above. Recentsuccess in applying classical N -Body Problem results to the F2BP (see Scheeres[2002,a]) indicates that progress can indeed be made in developing this approach.


Incorporation of Deformation and Dissipation Effects. The incorporationof deformation and dissipation effects into one or both of the bodies provides animportant path towards understanding full body dynamics in a deeper way. Even ifthe simplest deformation and dissipation models are incorporated, based on smalldeformation and linear dissipation assumptions, these effects can have major influ-ence on full body attitude and orbital dynamics. This phenomenon is well knownin planetary dynamics (c.f. Murray and Dermott [1999]) and has also been charac-terized in the field of space structure dynamics and control.

Models that incorporate attitude, translation, and finite deformation degrees offreedom for a distributed body in a central body gravitational field, have been de-veloped, e.g., in Cho, McClamroch and Reyhanoglu [2000]. Even in this relativelysimplified version of a FBP, the presence of deformation and dissipation can pro-vide mechanisms for energy transfer between translation and attitude. They alsoplay important roles in controlled FBPs, where physical or control effects directlyinfluence the deformation and dissipation, and indirectly influence attitude and/ortranslation.

It is also of interest to extend recent work on the stability and control of satel-lites with flexible appendages to describing the natural dynamics of deformablebodies in orbit about a rigid body. As examples, the stability and Hamiltonianstructure of a rigid body with attached flexible rod was analyzed in Krishnaprasadand Marsden [1987]. More recently, Hagerty, Bloch and Weinstein [2003] consid-ered induced instabilities in a satellite with momentum wheels with an attachedstring. This “radiation induced instability” is related to the dissipation inducedinstabilities studied in Bloch, Krishnaprasad, Marsden, and Ratiu [1994] and Bloch,Krishnaprasad, Marsden, and Ratiu [1996]. Such instabilities can occur when thereis a saddle point in the energy-momentum function but all eigenvalues of the lin-earized system lie on the imaginary axis, situations which occur in the relativeequilibria of many FBPs. These results, when coupled with a recent rediscovery ofan elastic solution for tri-axial ellipsoids (Washabaugh and Scheeres [2002]), couldyield significant enhancements in our current understanding of tidal disruptions anddissipation in small body systems (Asphaug and Benz [1996]; Richardson, Bottke,and Love [1998]).

Astronomy and Planetary Science. Generalized models of the F2BP havemany applications in the field of astronomy and planetary science. In the followingwe mention a few specific targets for our research, chosen as good candidates for theapplication of our systematic approach to full body problems.

An important outcome of this phase of analysis could involve the long-term sim-ulation of the effects of the Yarkovsky effect on the translational and rotationalmotion of asteroids. This effect has recently been realized to be an important com-ponent in describing the evolution and current state of the solar system. Whilemuch work has been done on the modeling and prediction of its effects (Bottke etal. [2000, 2001]; Spitale and Greenberg [2001]; Rubincam [2000]), long-term simu-lations of its effects using realistic models is currently not possible. The heart ofthe difficulty is the delicate relation between an asteroid’s rotation state and the


net result of the Yarkovsky force (due to thermal inertia in the body). The suite ofsimulation tools we are developing will be able to directly address this question, andwould unquestionably advance our ability to model and understand this importanteffect.

A related open question in planetary science is an explanation for the large num-ber of slowly rotating asteroids (Harris [1994, 2002]). Current distribution statisticscannot be accounted for by using the traditional theory of asteroid rotation stateevolution. While Yarkovsky forces have been considered as a possible explanation(Rubincam [2000]), a strong case has not been built as of yet. Mutual disruption ofbinaries under their own gravitational interaction is another potential mechanism forthe creation of slowly rotating and tumbling asteroids. To properly address whetherthis can account for the noted excess of slow rotators a deeper study of ejectionprobability in binary asteroids must be made, starting from the work described inScheeres et al. [2000]; Scheeres [2002a]; Harris [2002], coupled with an analysis ofthe likely formation energies of these binaries, drawing on recent work in this area(Bottke and Melosh [1996]; Durda [1996]; Doressoundiram et al. [1997]; Richardson,Bottke, and Love [1998]; Bottke et al. [1999]; Michel et al. [2001]).

Related to this study is a complete analysis of the formation and evolution of abinary asteroid or Kuiper belt object, including the effect of energy dissipation andexternal force perturbations. The simulation tools being developed for the F2BPwill be ideal for the simulation of these systems over long time spans, allowingfor the development and testing of various hypotheses and constraints on the life ofthese objects. Based on current theoretical models and results (Goldreich, Lithwick,and Sari [2002]; Scheeres et al. [2000]; Scheeres [2002a]) a binary will be subject tomutual disruption or impact during its evolution, and only a fraction of binary bodiesshould survive into a long-term stable state. The resolution and understanding ofthis question is completely open, but would be naturally addressed using a FBPapproach. As part of this analysis, we will also examine the Pluto-Charon system,currently subject to some controversy based on the apparent eccentricity of theirmutual orbit Stern, Bottke, and Levison [2003].

Recent re-analysis of the long-term evolution of comet nucleus rotation statesaccounting for realistic models of outgassing torques on the bodies has shown aremarkably rich set of steady-state states for their rotation states (Neishtad et al.[2002]). Direct simulation of these effects over long-periods is currently not possible,but may be made attainable from the outcomes of our work. Furthermore, thenon-trivial dependence on rotation state on a comet’s heliocentric orbit invites amore complete analysis of the coupling of nucleus rotation and its orbital variationdue to outgassing, which is the classical problem of comet nucleus orbit mechanics(Yeomans and Chodas [1989]).

An additional, novel idea is to apply the F2BP, with dissipation and deformationeffects, to describing the interaction of two or more distributed galaxies. As a firststep, this work would link the classical theory of ellipsoidal figures of equilibrium(Chandrasekhar [1969]) with the effect of a non-collision interaction between twogalaxies. Following such an encounter, each of the distributed bodies would feela near-impulsive change in their energy and angular momentum, due to exchanges


between the distributed bodies (Scheeres [2001]). Treating these stellar collections asFBPs could lead to fruitful new insights and approaches to describe their evolutionand dynamics.

Vehicle Dynamics and Control. Control problems arise in FBPs where one ofthe bodies is an artificial body such as a controlled spacecraft or satellite. There is avast literature on attitude control and orbit control of spacecraft, and the success ofmost space missions has depended on this capability. With only few exceptions, thisliterature treats attitude control problems and orbit control problems as independentproblems. This is reflected in the organization chart of most space agencies and spaceindustries; attitude control groups and orbit control groups operate independently.

The thesis of this area of research, based on the fundamental definition of FBPs,is that attitude and orbit dynamics and control problems should be treated in aunified way. A consequence of this thesis is that control problems for FBPs shouldbe formulated, analyzed, and implemented using prior knowledge of the full bodydynamics. Geometric properties such as symmetries and reduction are importantnot only for analysis of full body dynamics, but are crucial to full body controlas well. Most full body control problems are likely to be nonlinear and to requirespecial methods for control design and analysis that are tailored to the full bodydynamics.

Nonlinear control methods should provide excellent frameworks for studyingfull body control problems. These methods include controlled Lagrangian (Bloch,Chang, Leonard and Marsden [2001]), geometric phases (Marsden, Montgomery,and Ratiu [1990]; Rui, Kolmanovsky, and McClamroch [2000]), and differential ge-ometric approaches (Shen, McClamroch, and Block [2003]; Rui, Kolmanovsky, andMcClamroch [2000]). These approaches have been developed and successfully ap-plied to the control of rigid bodies and multi-body systems, taking into account mo-tion constraints, symmetries, control-actuation assumptions, deformation degrees offreedom, and dissipation. They have been used to construct controllers for specificrigid-body and multi-body spacecraft attitude control problems using thrusters, re-action wheels, proof mass actuators, appendages, and tethers. By analyzing anddescribing these systems from a more general vantage, it is often possible to gainadditional insight into the known symmetries of the simpler problem, and poten-tially to discover new symmetries not appreciated before. A similar philosophy canbe applied to a re-analysis of vehicle motion in fluid fields, where the existence ofthe fluid can be idealized as the limiting model of a second body interaction onthe vehicle. As these vehicles have strong coupling between rotational and transla-tional motion, and are considered to be extremely difficult analytical problems, anyadvance in understanding such vehicles would be important.

Chemical Dynamics. The mathematical description of transport phenomena ap-plies to a wide range of physical systems across many scales Meiss [1992]; Wiggins[1992]; Rom-Kedar [1999]. The recent and surprisingly effective application of meth-ods combining dynamical systems ideas with those from chemistry to the transportof Mars ejecta by several of the co-authors in collaboration with chemists, underlines

REFERENCES 25

this point (Jaffe, Ross, Lo, Marsden, Farrelly, and Uzer [2002]). Thus, techniquesdeveloped to study transport in the solar system are truly fundamental and broad-based. The methods may be applied to diverse areas of study, including fluid mixing(Beigie, Leonard, and Wiggins [1991]; Malhotra and Wiggins [1998]; Poje and Haller[1999]; Haller and Yuan [2000]; Haller [2001]; Coulliette and Wiggins [2001]; Lekienand Coulliette [2002]) and N -body problems in physical chemistry (Jaffe, Farrelly,and Uzer [2000]). A basic framework for this theory has already allowed the severalof the co-authors to develop a new low-fuel mission concept to explore Jupiter’smoons (Ross et al. [2003]). Any improvement made in the software in the course ofthe work outlined above can thus be applied to areas beyond dynamical astronomy.

Acknowledgements. This research was partially supported by A Max PlanckResearch award and NSF ITR grant ACI-0204932.

Author Email Addresses : [email protected], [email protected],[email protected], [email protected], [email protected]

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