Dynamics of multibody systems in planar motion in a central

Dynamical Systems, Vol. 19, No. 4, 2004, 303–343

Dynamics of multibody systems in planar motion

in a central gravitational field

AMIT K. SANYAL*, ANTHONY BLOCHy

and N. HARRIS MCCLAMROCH*

*Department of Aerospace Engineering, University of Michigan,Ann Arbor, MI 48109, USAyDepartment of Mathematics, University of Michigan, Ann Arbor,MI 48109, USA

Abstract. Multibody systems in planar motion are modelled as two or more rigidcomponents that are connected and can move relative to each other. The dynamicsof such multibody systems in planar motion in a central gravitational force field isanalysed. The equations of motion of the system include the equations for the orbitalmotion of the bodies, the orientation (attitude) of the assembly, and the relativeorientation (shape) of the bodies with respect to each other. Dynamic couplingbetween these degrees of freedom gives rise to complex dynamical systems thatare usually not integrable. Relative equilibria, corresponding to circular orbits ofthe multibody system, are obtained. The free dynamics has a symmetry due to acyclic coordinate. Routh reduction is carried out to eliminate this coordinate andobtain the reduced dynamics. The stability of the relative equilibria is analysed usingthe Routh stability criterion when it is applicable; an expansion of the Hamiltonianin normal form is used otherwise. We apply the general results to a multibody systemconsisting of two hinged planar bodies, each modelled as a rigid massless link witha point mass at one end with their other ends connected by a hinge joint. We obtainthe relative equilibria of this model, and carry out a stability analysis for the relativeequilibria. Numerical simulations using a symplectic integrator are carried out forperturbations to these relative equilibria, to confirm their stability properties.

Received 12 April 2004; accepted 28 August 2004

1. Introduction

In this paper, we discuss the dynamics and control of multibody systems in planarmotion, in the presence of a central gravitational force field. The equations of motiondescribe the dynamics of the multibody system in orbit, their rotational dynamics,as well as the relative motion of the connected bodies. This extends the developmentin Bloch (2003) and the work of Oh et al. (1988, 1989) to include a potential force

Correspondence to: Amit K. Sanyal, e-mail: [email protected]

Dynamical Systems ISSN 1468–9367 print/ISSN 1468–9375 online # 2004 Taylor & Francis Ltdhttp://www.tandf.co.uk/journals

DOI: 10.1080/14689360412331309160

given by a central gravitational field. This extension results in non-integrable systems

that are highly non-trivial because of dynamic coupling between the various degrees

of freedom. Unlike the Keplerian two-body system, multibody systems in central

gravity are generally not fully integrable. The motion of coupled rigid bodies in three

dimensions in the absence of any force field has been studied in Patrick (1995).

This work is also related to that in Sanyal et al. (2003, 2004a), which considers

the dynamics and control of a simple elastic dumb-bell-shaped body moving in a

planar orbit about a central body.

The assumption of planar motion is not entirely artificial in this context. Although

the motion of such systems in central gravity is three-dimensional in general, the

motion can be restricted to a plane if all initial conditions are restricted to the plane

and there are no perturbing forces other than gravity. This has been shown in Sanyal

(2004) and, for a particular example, in Sanyal et al. (2004c). In order to avoid

difficulties associated with integration over a rigid body to obtain the gravitational

potential, we study a simple model for the multibody system, where we approximate

the gravitational force as acting on the centre of mass of each body. This simplifica-

tion does not take away from the essential feature of coupling between the various

degrees of freedom of the system. We also obtain algebraic equations that describe

relative equilibria corresponding to circular orbits with a fixed orbital rate. The free

dynamics of a planar multibody system in central gravity has two conserved quan-

tities: the total energy and the angular momentum of the system around the centre of

the force. We also look at the reduced dynamics obtained by eliminating the cyclic

variable (true anomaly) which gives rise to the conserved angular momentum. The

relative equilibria of the full dynamics correspond to the equilibria of the reduced

dynamics. The stability of these relative equilibria is analysed using the Routh

stability criterion when it is applicable, and Hamiltonian expansion in normal

form (Arnold et al. 1988) about the equilibria of the reduced dynamics otherwise.

To ensure that total energy and the angular momentum of the system are

conserved during numerical integration, we use a numerical integrator that conserves

momentum and is based on the variational principle. The prior literature on such

integration schemes is quite voluminous, and a sample can be found in Simo et al.

(1992), Kane et al. (1999), Marsden and West (2001), Hairer et al. (2002), and

references therein. We obtain a variational (symplectic) integrator algorithm by

applying discrete Routh reduction (Jalnapurkar et al. 2003, Sanyal et al. 2004b)

to eliminate the cyclic variable. The variational integrator is also found to keep

the error in total energy bounded during numerical integration.

We apply these results to an example of a planar multibody spacecraft, which

consists of two planar bodies each modelled as a rigid massless link with a point mass

at one end. The other ends are connected together by a planar hinge joint, which

allows each body to rotate freely about the hinge point. The relative orientation of

the two bodies describes the shape of the assembly. We obtain the relative equilibria

of this system in a central gravitational field, and analyse their stability using results

developed in this paper. We also carry out numerical simulations for perturbations

about the relative equilibria of the hinged planar bodies. The simulation results

suggest that the change in shape and attitude of the assembly may be advantageously

used in control schemes to affect the overall orbital motion of the system. Such

schemes may be used, for example, to provide low-energy control of the orbital

motion of a spacecraft without using its costly and limited resources of fuel.

304 A. K. Sanyal et al.

The organization of the paper is as follows: in section 2 we obtain the equationsof motion of planar multibody systems in central gravity. We also obtain algebraicconditions that give the relative equilibria of such systems, corresponding to fixedcircular orbits. We look at the reduced dynamics in section 3, using the process ofRouth reduction (Marsden and Ratiu 1999, Marsden et al. 2000), and obtain theequations of motion in the reduced space. In this section, we also present a generaldevelopment for obtaining the stability properties of the relative equilibria ofplanar multibody systems in a central gravitational field. We obtain conditions forstability or instability of relative equilibria, based on linear and nonlinear analysis.In section 4, we present a scheme to integrate the reduced dynamics using discreteequations of motion obtained by applying the variational principle to a discreteRouthian. We then look at the dynamics of an example of a planar multibodysystem which consists of two identical rigid massless links with tip masses atone end and the other ends connected by a rotary or hinge joint, in section 5.The reduction of the free dynamics, and the corresponding reduced equations ofmotion, are presented in section 6. This section also presents the stability propertiesof the relative equilibria of this system, using the analytical results obtained insection 3. In section 7, we carry out numerical simulations of the free dynamicsfor this example problem. We apply the scheme in section 4 to numerically integratethe reduced dynamics for perturbations of the relative equilibria.

2. Dynamics of planar multibody systems in central gravity

We use polar coordinates, with coordinate frame basis {e1, e2}, to describe thedynamics of the multibody system. This coordinate description gives rise to acyclic coordinate and a corresponding conserved momentum, as well as facilitatingdescription of the relative equilibria we obtain later. The radial distance fromthe origin of the inertial frame, fixed at the centre of force, to a fixed point inthe multibody system, called the base point, is denoted by r. The angle the radialdirection vector e1 makes with a fixed inertial axis is denoted �, and measures the trueanomaly of the system in orbit. Thus, the coordinates (r, �) denote the position ofthe system in the orbital plane. A body-fixed coordinate frame, with basis vectors{b1, b2}, is fixed to the base point, and the attitude of the system with respect to theradial direction is denoted by the angle �. The relative positions and orientations ofthe bodies constituting the multibody system are given by the shape of the system,which is denoted by the generalized coordinates s2QS, where QS denotes the shapespace. Thus, the configuration manifold can be expressed in local coordinates byq¼ (r, �,�, s)2Q. Figure 1 gives a graphical representation of the various quantitiesdescribing a multibody system with nb connected bodies.

2.1. Equations of motion of planar multibody systemThe kinetic energy and the gravitational potential energy, and hence the Lagrangianof the multibody system, are independent of the true anomaly �, which is a cycliccoordinate. We express the other coordinates by the vector x ¼ ½r � sT�T. Thecoordinates q¼ (x, �) give the local representation of the system configuration onthe m-dimensional manifold Q. Let the central gravitational potential be Vg(x),

305Dynamics of multibody systems

and the potential due to shape be Vs(s). The Lagrangian can be expressed as

Lðx; _qqÞ ¼1

2_qqTMðxÞ _qq� VðxÞ; ð1Þ

where _qq represents the generalized velocities, M(x) is the inertia matrix, andV(x)¼Vg(x) þ Vs(s) is the total potential. The configuration space Q can bedescribed as a Riemannian manifold with the metric tensor given by the inertiamatrix. We model the planar multibody system as a lumped mass model, with nbinterconnected bodies with the ith body having mass mi concentrated on the body’scentre of mass. The distance of the centre of mass of the ith body from the base pointof the multibody system is denoted �i(s) in the body-fixed frame. The rotation matrixfrom the body frame to the polar coordinate frame is given by R(�). If � denotes thegravitational force constant, then the gravitational potential is given by

Vgðr; �; sÞ ¼Xnbi¼1

��mi

kre1 þ Rð�Þ�iðsÞk; ð2Þ

where e1 ¼ ½1 0�T is the standard unit vector denoting the radial direction in thepolar coordinate frame.

mnb

r

e1

b1

m2

m1

ν

α

b2

e2

Figure 1. Multibody system in planar motion in a central gravitational force field.


The equations of motion obtained from this Lagrangian have the form

d

dt

�MðxÞ _qq

�¼

@

@q

1

2_qqTMðxÞ _qq

� ��@VðxÞ

@q:

We arrange the coordinates such that the first n¼m� 1 coordinates are expressedby the vector x, the components of the top-left n� n square block of the inertiamatrix are denoted by gij(x), the (m,m) component is denoted by g��(x), the othercomponents in the last row are denoted by gi�(x), while the remaining componentsin the last column are denoted g�i(x). The equations of motion on TQ can then bewritten as

gijðxÞ €xxjþ gi�ðxÞ €�� ¼ �

1

2

@gijðxÞ

@xkþ@gikðxÞ

@x j�@gjkðxÞ

@xi

� �_xx j _xxk þ

@gj�ðxÞ

@xi�@gi�ðxÞ

@x j

� �_xx j _��

þ1

2

@g��ðxÞ

@xi_��2 �

@VðxÞ

@xi; ð3Þ

g�iðxÞ €xxiþ g��ðxÞ €�� ¼ �

@g�iðxÞ

@xk_xxi _xxk �

@g��ðxÞ

@xk_�� _xxk; ð4Þ

where i; j; k ¼ 1; . . . ; n and summation over repeated indices is assumed. Theconjugate momentum corresponding to � is

p ¼@Lðx; _qqÞ

@ _��¼ g�iðxÞ _xx

iþ g��ðxÞ _��; ð5Þ

and equation (4), which is equivalent to _pp ¼ 0, shows that this momentum isconserved.

These Euler–Lagrange equations are equivalent to the canonical Hamilton’sequations in the cotangent bundle T?Q of Q. The Hamiltonian is

H ¼1

2gijðxÞpi pj þ gi�ðxÞpipþ

1

2g��ðxÞp2 þ VðxÞ;

and the Lie–Poisson bracket is given by

fF;Gg ¼@F

@xi@G

@piþ@F

@�

@G

@p�

@F

@pi

@G

@xi�

@F

@p

@G

@�; ð6Þ

where pi is the conjugate momentum corresponding to xi. From the above bracket,one can see that any function f ( p) of the conjugate angular momentum corre-sponding to � is conserved along the flow of the system. However, for a generalmultibody system, it is difficult to obtain independent first integrals that are ininvolution with H, other than p and the Hamiltonian H itself. In fact, as we showlater in section 6, the example system of two planar bodies connected by a hinge jointis not integrable.

2.2. Relative equilibria in circular orbitsWe now look at relative equilibria of the planar multibody system in circular orbits,corresponding to ‘steady’ circular motion about the origin or centre of gravitationalforce. Under these relative equilibrium conditions, there are no other external forces


on the system and

_xx ¼ 0; _�� ¼ ! ¼ constant:

Equation (4) is trivially satisfied at relative equilibria, while equation (3) gives

@g��ðxÞ

@xi_��2 ¼ 2

@VðxÞ

@xi; i ¼ 1; . . . ; n: ð7Þ

In terms of the radial, attitude, and shape coordinates, we obtain

@g��ðr; �; sÞ

@r!2

¼ 2@Vðr; �; sÞ

@r; ð8Þ

@g��ðr; �; sÞ

@�!2

¼ 2@Vðr; �; sÞ

@�; ð9Þ

@g��ðr; �; sÞ

@sa!2

¼ 2@Vðr; �; sÞ

@sa; a ¼ 1; . . . ;m� 3: ð10Þ

These algebraic equations (8)–(10) describe the configuration at a relative equi-librium. If the orbital radius at relative equilibrium, re, is known, then equation (8)gives the orbital rate (angular velocity) ! in terms of the (unknown) shape se andattitude �e. The attitude angle(s) at relative equilibria can then be found from (9)in terms of the shape se. This in turn can be used to obtain the shape se at relativeequilibria from equation (10). Thus, if the orbital radius re is known, one can, inprinciple, solve this system of equations to obtain the values of the other coordinates.These may give rise to multiple solutions for any particular value of re , which givesus all the different relative equilibria.

3. Routh reduction of planar multibody systems

Using the technique of Routh reduction, we eliminate the cyclic variable � 2 S1,

which represents the angle, in polar coordinates, that the position vector to theorigin of the body-fixed coordinate frame of the multibody system makes withan inertially fixed direction. We denote the reduced configuration space, describedlocally by the coordinates x¼ (r,�, s), by S ’ Q=S1. The free dynamics in thisreduced space is then analysed.

3.1. The Lagrange–Routh equations of motionThe Lagrangian has the form

Lðx; _xx; _��Þ ¼1

2gijðxÞ _xx

i _xx jþ g�iðxÞ _xx

i _��þ1

2g�� _��

2� VðxÞ;

where V(x)¼Vg(x)+Vs(q), and i, j are summed over 1 to n. Since � is a cyclicvariable, the angular momentum p given by (5) is conserved in the absence ofexternal applied forces in the direction tangential to the orbit. We carry out Routhreduction of the dynamics to eliminate � and _��, and reduce the dynamics to S. Theclassical Routhian is defined by setting p constant and performing a partial Legendretransformation in the variable �:

Rðx; _xxÞ ¼ Lðx; _xx; _��Þ � p _��;


where we can solve for _�� as

_�� ¼ g��ðxÞ p� g�iðxÞ _xxi

� �;

and g�� ¼ 1=g��.Substituting the expression for the Lagrangian and for _�� from above, we obtain

Rðx; _xxÞ ¼1

2gij _xx

i _xx jþ g�i _xx

iðg��p� g��g�i _xx

iÞ þ

1

2g��ðg

��p� g��g�i _xxiÞ2

� pðg��p� g��g�i _xxiÞ � VðxÞ: ð11Þ

The terms linear in _xx can be evaluated to give

g�ig��p _xxi ¼ A�

i p _xxi;

where the A�i ¼ g�ig

�� are referred to as the connection coefficients. Since in this casethere is only one cyclic coordinate, we define the vector of connection coefficientsAi ¼ A�

i . The terms quadratic in _xx are given by

1

2gij � g�ig

��g�j� �

_xxi _xx j¼

1

2hij _xx

i _xx j;

where hij ¼ gij � g�ig��g�j is the modified metric for the reduced dynamics. The

reduced configuration manifold S, together with this metric, is a Riemannianmanifold. The terms depending only on x can all be combined together to give themodified potential

VpðxÞ ¼ VðxÞ þ1

2g��ðxÞp2:

Introducing the modified Routhian (Marsden and Ratiu 1999, Bloch 2003),

eRRðx; _xxÞ ¼1

2hijðxÞ _xx

i _xx j� VpðxÞ; ð12Þ

the equations for the reduced dynamics take the form

d

dt

@eRR@ _xxi

�@eRR@xi

¼ �Bijp _xxj; ð13Þ

where the Bij are curvature coefficients of the connection A, given by

Bij ¼@Ai

@x j�@Aj

@xi:

In this case, when there is only one cyclic coordinate, the curvature coefficients canbe denoted by the coefficients of a matrix Cij¼Bijp. The equations of motion (13) onTS can then be rewritten as

€xxi ¼ ��ijk _xx

j _xxk � Cik _xx

k� hil

@Vp

@xl; ð14Þ

where

�ijk ¼

1

2hil

@hlj

@xkþ@hlk@x j

�@hjk

@xl

� �;

are the Christoffel symbols for the metric hij (Sakai 1996, Marsden and Ratiu 1999)and Ci

k ¼ hilClk.


3.1.1. Hamiltonian formulation for reduced dynamics. We present a reduction of theplanar multibody system also on the Hamiltonian side, since it is useful for stabilityanalysis of relative equilibria, which we carry out later. We introduce the variables

yi ¼@Rðx; _xxÞ

@ _xxi¼ hij _xx

jþ pAi; ð15Þ

using the expression (11) for the Routhian, which is like a Legendre transform for thereduced dynamics. The Hamiltonian for the reduced dynamics can then be obtainedfrom the Routhian by

Hðx; yÞ ¼ yi _xxi�Rðx; _xxÞ; with _xx j

¼ hijðyi � pAiÞ:

Carrying out the substitutions, one obtains the following form for the Hamiltonian

Hðx; yÞ ¼1

2hijðxÞyiyj � phijðxÞAiðxÞyj þ

1

2p2hijðxÞAiðxÞAjðxÞ þ VpðxÞ: ð16Þ

Making further substitutions of the form

zi ¼ yi � pAi;

one can express the Hamiltonian in the compact form

Hðx; zÞ ¼1

2hijðxÞzizj þ VpðxÞ: ð17Þ

The dynamics in the canonical Hamiltonian form on T?S are given by

_xxi ¼@Hðx; zÞ

@yi¼ hijðxÞð yj � AjðxÞÞ ) _xxi ¼ hijðxÞzj; ð18Þ

_yyi ¼ �@Hðx; zÞ

@xi) _zzi þ pBijðxÞh

jkðxÞzk ¼ �

1

2

@h jkðxÞ

@xizjzk �

@VpðxÞ

@xi: ð19Þ

It is obvious that equation (18) is equivalent to (15); one can also verify thatequation (19) is equivalent to the Lagrange–Routh equations of motion (13) or(14). The Poisson bracket structure for the reduced system is given by

fF;Gg ¼@Fðx; yÞ

@xi@Gðx; yÞ

@yi�@Fðx; yÞ

@yi

@Gðx; yÞ

@xi: ð20Þ

3.2. Equilibria of the reduced dynamicsThe equilibria xe of the reduced dynamics are given by substituting ð _xx ¼ 0; €xx ¼ 0Þin the equations of motion (14), which gives us

@VpðxÞ

@x

x¼xe

¼ 0:

Near an equilibrium xe of the reduced dynamics, the linearized equations of motionfor the free dynamics are given by

hli� €xxi¼ �Clk� _xx

k�

@2V

@xk@xl�xk;


or eMM� €xxþ C� _xxþ K�x ¼ 0; ð21Þ

where eMM is the symmetric matrix representation of the metric hij , C is the skew-symmetric matrix with coefficients Cij, and K is a symmetric matrix (the Hessianof Vp) whose coefficients are given by

Kij ¼@2VpðxÞ

@xi@x j;

all evaluated at the equilibrium configuration x¼ xe. Equation (21) has the formof a linear ‘gyroscopic system’ (Marsden and Ratiu 1999), which is Hamiltonian.If these linearized equations are unstable, then the nonlinear equations (14) arealso unstable, and control forces would be required to stabilize the equilibria.Note that the equilibria of the reduced dynamics given by (14) correspond to therelative equilibria of the full dynamics given in the earlier section; this is true in thegeneral sense (Jalnapurkar and Marsden 2000). Hence, assessing the stability ofthe equilibria of the reduced dynamics is equivalent to assessing the stability ofthe relative equilibria of the full dynamics. We present some results which givesufficient and necessary conditions for the stability of these equilibria in thefollowing section.

3.3. Stability of equilibria of reduced systemThe stability of the equilibria of the reduced system, i.e. the relative equilibria of thefull system, is given by the following condition.

Proposition 1. An equilibrium xe of the reduced dynamics is stable if the Hessianmatrix K ¼ d2VpðxeÞ is positive definite, and it is unstable if K ¼ d2VpðxeÞ has negativeeigenvalues.

The sufficiency of the positive definiteness for stability is Routh’s stability criterion(see Marsden and Ratiu 1999, Bloch 2003). A necessary condition for stability (basedon linearization) is given in Marsden and Ratiu (1999: 39), which states that if K hasnegative eigenvalue(s), then the reduced system (21) is unstable.

Routh’s stability criterion is actually a special case of the energy–momentummethod and the Lagrange–Dirichlet criterion (Bloch 2003). The stability of a relativeequilibrium is not easily determined when K is positive semidefinite with some zeroeigenvalues. This corresponds to an elliptic equilibrium with zero eigenvalues of thereduced system. Note that the eigenvalues of the system (21) are given by

S ¼ s 2 C j detðs2 eMM þ sC þ KÞ ¼ 0n o

:

Suppose the matrix V is an orthogonal matrix that diagonalizes K, i.e.

K ¼ V�VT;

where � is the diagonal matrix of real eigenvalues of K. The set of eigenvalues of thereduced system can be expressed as

S ¼ s 2 C j det s2 �MM þ s �CC þ��

¼ 0 �

;


where

�MM ¼ VT eMMV; �CC ¼ VTCV :

From the assumption, � has one zero along its main diagonal. It is easy to verifythat this leads to

det ðs2 �MM þ s �CC þ�Þ ¼ s2pðsÞ;

where p(s) is a polynomial in s. Thus, the set S will have an even number of zeroeigenvalues (for each zero eigenvalue of K), as expected in a Hamiltonian system.Thus, there will be some mode of the linearized system which grows linearlyin time. Hence the linear system (21) is unstable if K is positive semi-definite withzero eigenvalues. However, this does not necessarily imply that the nonlinear systemis unstable if K is positive semi-definite. In this case, a centre manifold analysis couldbe carried out to obtain the stability of such equilibria. We do this in the followingsection.

3.4. Centre manifold analysis near elliptic equilibria with K � 0In this section, we analyse the higher-order stability of elliptic equilibria with positivesemi-definite Hessian K, where K has one or more zero eigenvalues. This is moreconveniently done using the Hamiltonian formulation and canonical Hamilton’sequations. We use the Hamiltonian formulation for the reduced dynamics alreadydeveloped in section 3.1.1. Then we use a low-order series expansion of theHamiltonian in the Birkhoff normal form (Birkhoff 1927) to analyse stability ofsuch equilibria (Arnold et al. 1988, Giorgilli 1988, Giorgilli et al. 1989, Fasso et al.1998). We also use the Birkhoff normalization process to analyse whether thesystems being considered are analytically integrable.

3.4.1. Series expansion of Hamiltonian. To analyse the stability of the dynamicsaround elliptic relative equilibria with K � 0, we use the tool of Hamiltonian expan-sion in normal form, which has a rich literature (see Giorgilli 1988, Giorgilli et al.1989, Fasso et al. 1998, Jorba and Masdemont 1999, and references therein). Thisexpansion is usually carried out using action-angle variables in R

2n, so we first carryout a transformation of the Hamiltonian description of the dynamics in T?S to adescription in R

2n. The Hamiltonian function can be expanded about an equilibriumxe (ze¼ 0) of the reduced dynamics as follows

Hðx; zÞ ¼1

2hijðxeÞzizj þ VpðxeÞ þ

1

2

@2VpðxÞ

@xi@x j

x¼xe

ðxi � xieÞðxj� x j

eÞ

þ1

2

@hijðxÞ

@xk

x¼xe

ðxk � xke Þzizj þ1

6

@3VpðxÞ

@xi@x j@xk

x¼xe

� ðxi � xieÞðxj� x j

eÞðxk� xke Þ þ Oð4Þ; ð22Þ

in terms of a Taylor series in the coordinates ðx; zÞ. We assume that the systemis linearly normalized, i.e. the inertia matrix eMM ¼ ½hij �ðxeÞ and the HessianK ¼ d2VpðxeÞ evaluated at the equilibrium are diagonal. If not, we can carry out asuitable linear canonical transformation to diagonalize them, and we denote the new


coordinates also by ðx; zÞ. Note that in terms of the diagonal matrices eMM and Kdefined earlier, we can express

H2ðx; zÞ ¼1

2zT eMM�1zþ

1

2�xxTK �xx; �hhij ¼ �kkij ¼ 0 if i 6¼ j: ð23Þ

where �xx ¼ x� xe, �kkjj ¼ kjjðxeÞ, and �hh jj¼ h jj

ðxeÞ. This is the Hamiltonian of a systemof n uncoupled linear oscillators, which is an integrable system. Thus, the expansionin (22) can be expressed as

Hðx; zÞ ¼ H0 þH2ðx; zÞ þXs�3

Hsðx; zÞ; ð24Þ

where

H0 ¼ VpðxeÞ; H2ðx; zÞ ¼1

2hijðxeÞzizj þ

1

2

@2VpðxÞ

@xi@x j

x¼xe

ðxi � xieÞðxj� x j

eÞ;

and Hsðx; zÞ 2 Psðx; zÞ, the set of homogeneous polynomials of order s in thevariables (x, z). Hence the expansion of the Hamiltonian expresses the dynamicsaround the elliptic equilibria as perturbations from an integrable system. Wedefine a real linear transformation of coordinates < : T?S ! R

2n, given by

rj ¼

ffiffiffiffiffi�kkjj

qx j; sj ¼

ffiffiffiffiffiffi�hh jj

pzj : ð25Þ

In these coordinates, we can define the action variables

Ijðr; sÞ ¼1

2

�r2j þ s2j

�; H2ðr; sÞ ¼

Xnj¼1

Ijðr; sÞ; ð26Þ

and one can verify that these are conserved along the flow of the quadraticHamiltonian H2 given in (23).

To expand the Hamiltonian H in Birkhoff normal form up to fourth order, weneed the third and fourth degree homogeneous terms in the expansion of H. For thetype of system we are considering, these are evaluated as

H3ðr; sÞ ¼1

2ffiffiffiffiffiffiffiffiffiffi�hhii �hh jj

p @hijðxðrÞÞ

@rk

r¼re

�rrksisj þ1

6

@3VpðxðrÞÞ

@ri@rj@rk

r¼re

�rri �rrj �rrk;

H4ðr; sÞ ¼1

4

"1ffiffiffiffiffiffiffiffiffiffi�hhii �hh jj

p @2hijðxðrÞÞ

@rk@rl

r¼re

�rrk �rrlsisj þ1

6

@4VpðxðrÞÞ

@ri@rj@rk@rl

r¼re

�rri �rrj �rrk �rrl

#; ð27Þ

where

re ¼ffiffiffiffiK

pxe; �rr ¼ r� re:

In the following section, we give the procedure to obtain the expansion of theHamiltonian in normal form.

3.4.2. Expansion of Hamiltonian in normal form. An algorithm for the expansion ofthe Hamiltonian in normal form is given in Giorgilli et al. (1989), which is bestimplemented through software that can carry out both symbolic and numericalcomputations. We use the Birkhoff normal form, which expresses the Hamiltonian


in terms of the action variables Ijðrj; sjÞ. Hence, the Birkhoff normal form containspolynomials of even order in the original canonical variables (x, z). Note that anexact Birkhoff normal form expansion may not always exist about an ellipticequilibrium. For an exact Birkhoff normal form to exist, the characteristicfrequencies (!j ¼

ffiffiffiffiffiffiffiffiffiffi�kkjj �hh

jjq

) of the linearized system about this equilibrium haveto satisfy the non-resonance condition given by equation (37) in the followingdiscussion.

To use the algorithm in Giorgilli et al. (1989), we carry out the complexificationC : R2n

! C2n given by

Cð�rrj; sjÞ ¼ ð�j; �jÞ; �j ¼1ffiffiffi2

p �rrj � {sj� �

; �j ¼�{ffiffiffi2

p �rrj þ {sj� �

; j ¼ 1; . . . ; n: ð28Þ

Note that this transformation and its inverse exist even when the matrix K ispositive semi-definite (singular). However, the composition of the transformationsC � < : T?S ! C

2n does not have an inverse defined if K is positive semi-definite.Hence, in this case, it is helpful to carry out the linear transformation < on theoriginal coordinates. As we see later in this section, the complexification C helps inmaking it easier to solve the equations that transform the Hamiltonian to a normalform. The Lie–Poisson bracket in terms of these complex coordinates is given by

L� ¼ f�; �g ¼Xnj¼1

!j

@�

@�j

@

@�j�

@�

@�j

@

@�j

� �; where !j ¼

ffiffiffiffiffiffiffiffiffiffi�hh jj �kkjj

q: ð29Þ

The inverse transformation,

C�1ð�j; �jÞ ¼ ð�rrj; sjÞ ¼

1ffiffiffi2

p �j þ {�j� �

;{ffiffiffi2

p �j � {�j� ��

; j ¼ 1; . . . ; n; ð30Þ

is well defined, as we remarked earlier. This transformation and its inverse arerequired for software implementation of the algorithm in Giorgilli et al. (1989).In these complex coordinates ð�j; �jÞ, the quadratic part of the series expansion ofthe Hamiltonian can be expressed as

H2ð�; �Þ ¼ {Xnj¼1

�j�j: ð31Þ

We consider the space P of all polynomials in ð�; �Þ 2 C2n and denote the subspace of

all homogeneous polynomials of degree k in ð�; �Þ by Pk; P is thus a graded algebra.We have P0 ¼ C. For � 2 P, we have � ¼

Pk�1 �k, with �k 2 Pk. We use the multi-

index notation to denote �k ¼P

jlþmj¼k �lm�l�m, where l;m 2 Z

nþ, �lm 2 C, and

�l ¼ �l11 �l22 . . . �lnn , l ¼ ðl1; l2; . . . ; lnÞ. Note that � represents a real formal series in the

real variables (r, s) iff �lm ¼ {jlþmj��ml, where

� represents complex conjugation. In thiscase, we refer to the series � as being C-real, where C is the complexification given by(28).

Let � ¼P

j�3 �j 2 P play the role of a generating function for a canonical trans-formation of the HamiltonianH to a normal form Z ¼

Pk�2 Zk. Then, we define the

transformation T� : P ! P in the following way:

T�H ¼Xk�2

Zk; Zk ¼Xk�2

l¼0

Hlþ2;k�l�2; ð32Þ


where

fl;0 ¼ fl; and fl;k ¼Xkm¼1

m

kL�mþ2

fl;k�m; ð33Þ

since H0 is constant and H1¼ 0 at an equilibrium. Note that fl;k 2 Plþk and soZk 2 Pk. Here LXY ¼ fX;Yg represents the Lie–Poisson bracket expressed in �; �coordinates after complexification, as given by (29). Equations (32) and (33) give riseto the following relations from which the normal form can be obtained:

Z2 ¼ H2; LH2�k þ Zk ¼ Fk; for k � 3; ð34Þ

where

F3 ¼ H3; Fk ¼Xk�3

l¼1

l

k� 2L�lþ2

Zk�l þXk�2

l¼1

l

k� 2Hlþ2;k�l�2; for k � 4; ð35Þ

where Zk is the projection of Fk to the normal form. It can be verified that F,and hence � and Z, are C-real, if H is C-real. Note that for the Birkhoff normalform, Zk¼ 0 if k is odd, so we begin algorithm (34)–(35) with the equationLH2

�3 ¼ F3 ¼ H3. Using the multi-index notation and (29), we can also expressthe linear operator LH2

¼ fH2; �g as

LH2¼ {

Xnj¼1

!j �j@

@�j� �j

@

@�j

� �) LH2

� ¼ {Xl;m

ðm� l Þ � !�lm�l�m;

where ! ¼ ð!1; . . . ; !nÞ 2 Rn. Thus, we can express the algorithm (34)–(35) for the

Birkhoff normal form as

Zk ¼ Fll; 2jlj ¼ k; and �lm ¼{

ðl �mÞ � !Flm; l 6¼ m: ð36Þ

One can obtain the unknown generating function � 2 P and the Birkhoff normalform Z 2 P from equations (36). Note that, for the functions � and Z to exist, oneneeds to guarantee that the vector of frequencies ! 2 R

n satisfies a non-resonancecondition of the form,

j � ! 6¼ 0; j 2 Zn: ð37Þ

If this condition is exactly satisfied, then the Hamiltonian system is termed ‘non-resonant’ and an exact Birkhoff normal form exists (Birkhoff ’s theorem, Arnold et al.1988: chap. 4, section 1.3). However, if this non-resonance condition is satisfiedonly up to some order (j jj � k), then one can obtain a normal form expansionwith Birkhoff normalized terms up to that order, and a higher-order remainderthat cannot be rendered into Birkhoff normal form. The series expansion processin this case is called ‘Birkhoff–Gustavson normalization’ (see Verhulst 1996). For thesystems we are considering here, if �kkjj ¼ 0 ¼ rj then !j ¼ 0 for all j ¼ fn1 þ 1; . . . ; ng.We see from (27) that H3ðr; sÞ is a polynomial in the ri , i 2 f1; . . . ; n1g, i.e. H3ð�; �Þis a polynomial in the ð�i þ {�iÞ, i 2 f1; . . . ; n1g. In this case, ðl �mÞ � ! 6¼ 0 unlessl¼m, and the lowest-order terms of the transformation, �3, as well as F4 and thesecond term Z4 in Birkhoff normal form can be found from (36).


3.4.3. Results on instability from expansion of the centre manifold. In general, theBirkhoff normal form expansion of the Hamiltonian near an equilibrium is

divergent; an exception is the case when Hðx; zÞ has n analytic first integrals in

convolution (Arnold et al. 1988). But the satisfaction of the non-resonance condition

is not sufficient to guarantee Lyapunov stability of the elliptic equilibrium, as Arnold

et al. (1988) remark, although Nekhoroshev-type ‘practical stability’ or confinement

for long periods is possible (Giorgilli et al. 1989, Fasso et al. 1998). However, it is

easier to draw conclusions on instability of the Hamiltonian system from its Birkhoff

expansion, even though the expansion may be divergent.

Theorem 1. Let �(o) be an order o series expansion of the generating function, giving

the transformed Hamiltonian expansion ~HH ¼ T�ðoÞH ¼ ZðoÞþ RðoÞ about an equilib-

rium. Here ZðoÞ¼Po

k¼1 Zk is the normal part, and RðoÞ ¼P

k>o Fk is the remainder,

consisting of terms with order l > o. Then the system is unstable about this equilibrium

if _IIj ¼ fIj; eHHg is positive in the interior of a neighbourhood of the equilibrium, for some

j 2 f1; . . . ; ng.

Proof. According to Theorem 5.5 in Giorgilli et al. (1989), a truncated series

expansion of finite order o, giving the transformed Hamiltonian T�ðoÞH ¼

ZðoÞþ RðoÞ, where ZðoÞ

¼Po

k¼1 Zk and RðoÞ ¼P

k>o Fk, is convergent in a compact

polydisk neighbourhood PD(xe) of the equilibrium xe, assuming simple bounds on

the homogeneous terms of the original Hamiltonian, Hk, and satisfaction of the non-

resonance condition up to that order. In this case, the transformed Hamiltonian

gives the local behaviour of the system. Hence, growth of the action variable Ijcorresponds to growth of the states ðx j; zjÞ of the system in some neighbourhood

of the equilibrium. œ

As a corollary of Proposition 2 and the fact that the higher-order remainder terms

are bounded in a compact polydisk neighbourhood of the equilibrium, we state the

following as a stronger sufficient condition for instability.

Corollary 1. Let ~HH ¼ T�ðoÞH ¼ ZðoÞþ RðoÞ be the order o truncated normal form

expansion of the Hamiltonian about an equilibrium. If fIj;ZðoÞg for some j 2 f1; . . . ; ng

is positive in the interior of a subset L T?S containing the equilibrium, then the

equilibrium is unstable.

Proof. Let PD(xe) be a compact polydisk neighbourhood of the equilibrium xe.

The remainder terms RðoÞ are also bounded in PD(xe), according to Theorem 5.5 in

Giorgilli et al. (1989). The rate of change of the action variable Ij is _IIj ¼ fIj;ZðoÞgþ

fIj; RðoÞg. Since fIj; R

ðoÞg consists of terms of order greater than o, we can assume

that the neighbourhood PD(xe) of the equilibrium is small enough, so that we have

fIj;ZðoÞg > jfIj; R

ðoÞgj, with both equal to zero at the equilibrium (if not, then we can

take a smaller compact neighbourhood of the equilibrium where this condition is

satisfied). Hence, _IIj is positive definite in the neighbourhood PDðxeÞ \ L of xe, and we

can apply Proposition 2 to conclude that the equilibrium is unstable. œ

Thus, if the flow along the normal form Hamiltonian expansion ZðoÞ, consisting of

terms in normal form up to a finite order o, about an equilibrium is found to be

unstable, then one can conclude that the equilibrium is unstable. Note that these

results provide conditions for nonlinear instability, which would be weaker than

exponential (linear) instability. For the planar dynamics of multibody systems in


central gravity that we are studying, these results are applicable only when theHessian of the amended potential of the reduced dynamics is positive semi-definite,with zero eigenvalues. We use these results based on the normal form expansionof the Hamiltonian in section 6.3.1, where we apply them to study the nonlinearinstability of degenerate relative equilibria of our example system.

3.4.4. Integrability of the free dynamics of planar multibody systems. The Birkhoffnormal form expansion about an elliptic equilibrium point of the reduced dynamicscan also be used for obtaining the integrability properties of a planar multibodysystem in central gravity. The relations between analytic integrability of Hamiltoniansystems and normal form expansions have been studied extensively before. It hasbeen known that, for non-resonant Hamiltonian systems, a convergent Birkhoffnormal form expansion implies integrability (Birkhoff 1927, Arnold et al. 1988).The inverse is also true for non-resonant systems, though it is much more difficultto prove (Ito 1989). In fact, Arnold et al. (1988) describe as integrable thoseHamiltonian systems for which the Birkhoff normal form transformation abouta non-resonant elliptic equilibrium converges. Note that, a stable relative equilib-rium (where the Hessian of the modified potential is positive definite) is less likelyto be resonant than a degenerate relative equilibrium (where the Hessian is positivesemi-definite). The degree of resonance, which is a measure of resonance of theHamiltonian system about an elliptic equilibrium, is defined below.

Definition 1. Let Z Zn denote the sublattice of Zn consisting of elements j 2 Z

such that j � ! ¼ 0, where ! denotes the vector of characteristic frequencies of theHamiltonian H. The dimension of Z over Zn, denoted by dr, is called the ‘degree ofresonance’ of the Hamiltonian H.

That analytic integrability implies a convergent Birkhoff normal form expansionwas shown by Ito (1989) for the non-resonant (dr¼ 0) case, and later, for the simplyresonant (dr¼ 1) case (Ito 1992). Recent work in this regard has been reportedby Zung (2003) for the case dr � 2, which applies to our example system (seesection 6.3.2).

4. Discrete Routh reduction

Discrete Routh reduction has been studied very recently in Jalnapurkar et al. (2003),and Sanyal et al. (2004b) for systems with cyclic coordinates. Here we followthe development in Sanyal et al. (2004b) of a discrete Routh reduction scheme asapplied to a multibody system moving in a plane under the influence of a centralgravitational field. This scheme provides us with a numerical integration algorithmthat can be used to simulate the free reduced dynamics of the multibody system incentral gravity, and conserve first integrals (total energy and angular momentum)of the system. We use this algorithm to numerically simulate the stability propertiesof the relative equilibria of our example system based on the reduced dynamics,and verify our analytical results. We consider the case when one of the generalizedcoordinates used to represent the dynamics is cyclic, i.e. the Lagrangian does notdepend on this coordinate explicitly. This is in accordance with the Lagrangianstructure of the planar multibody systems we are dealing with. We can then partitionthe coordinate vector as q ¼ ðx; �Þ, where � denotes the cyclic coordinate, andx denotes the remaining n¼m� 1 coordinates.


4.1. Discrete Routhian for a system with one cyclic coordinateThe discrete equations of motion obtained using this procedure approximate theflow of the real system F : TS ! TS with a discrete flow F d : S� S ! S� S. Theapproximation of the state ðx; _xxÞ at time t is given by the discrete state ðxn; xnþ1Þ

at time t ¼ n�t, where �t is a fixed time step, and F d ðxn�1; xnÞ ¼ ðxn; xnþ1Þ. Theconserved momentum corresponding to the cyclic coordinate can be written as

p ¼ g�jðxnÞvjðxn; xnþ1Þ þ g��ðxnÞv

�ðxn; xnþ1Þ:

Thus

v�n ¼ v�ðxn; xnþ1Þ ¼ g��ðxnÞ½ p� gj�ðxnÞvjðxn; xnþ1Þ�: ð38Þ

We define the discrete Lagrangian as

Ldðxn; xnþ1; v�nÞ ¼

1

2gijðxnÞv

iðxn; xnþ1Þv

jðxn; xnþ1Þ þ gi�ðxnÞv

iðxn; xnþ1Þv

�ðxn; xnþ1Þ

þ1

2g��ðxnÞðv

�nÞ

2� VðxnÞ; ð39Þ

where v�n is given by (38) and viðxn; xnþ1Þ is the ith component of the discrete velocityat the nth time step, which depends on the discrete configurations at the nth andðnþ 1Þth time steps. We approximate this velocity component using the trapezoidalrule:

viðxn; xnþ1Þ ¼xinþ1 � xin

�t:

We define the discrete Routhian analogous to the continuous Routhian, as

Rdðxn; xnþ1Þ ¼ Ldðxn; xnþ1Þ � pv�ðxn; xnþ1Þ: ð40Þ

The discrete Routhian indeed satisfies the discrete Euler–Lagrange equations (Sanyalet al. 2004b). We have

D�1Rdðxn; xnþ1Þ þDi

2Rd ðxn�1; xnÞ ¼@Rdðxn; xnþ1Þ

@xinþ@Rdðxn�1; xnÞ

@xin

¼@Ldðxn; xnþ1; v

�nÞ

@xinþ@Ldðxn; xnþ1; v

�nÞ

@v�n

@v�n@xin

þ@Ld ðxn�1; xn; _n�1Þ

@xinþ@Ldðxn�1; xn; v

�n�1Þ

@v�n�1

@v�n�1

@xin

� p� @v�n@xin

þ@v�n�1

@xin

�: ð41Þ

The first and third terms on the right-hand side of the above equation vanish becausethey satisfy the discrete Euler–Lagrange equations (Sanyal et al. 2004b), while theother terms cancel since

@Ldðxn; xnþ1; v�nÞ

@v�n¼

@Ldðxn�1; xn; v�n�1Þ

@v�n�1

¼ p:


4.2. Discrete Lagrange–Routh equationsUsing equations (39) and (40), we obtain the discrete Routhian as

Rd ðxn; xnþ1Þ ¼1

2

�gijðxnÞ � gi�ðxnÞA

�j ðxnÞ

�viðxn; xnþ1Þv

jðxn; xnþ1Þ

þ pA�i ðxnÞv

iðxn; xnþ1Þ �

1

2g��ðxnÞp

2þ VðxnÞ

� �¼

1

2hijðxnÞv

iðxn; xnþ1Þv

jðxn; xnþ1Þ þ pA�

i ðxnÞviðxn; xnþ1Þ � VpðxnÞ; ð42Þ

where

A�i ðxnÞ ¼ g��ðxnÞg�iðxnÞ ¼ AiðxnÞ

are the ‘connection coefficients’ defined previously,

hijðxnÞ ¼ gijðxnÞ � gi�ðxnÞA�j ðxnÞ;

and Vp(xn) is the modified potential as defined for the continuous dynamics. Thediscrete Lagrange–Routh equations then take the form

1

2

@hijðxnÞ

@xknviðxn; xnþ1Þv

jðxn; xnþ1Þ �

hkjðxnÞ

�tv jðxn; xnþ1Þ þ

@AiðxnÞ

@xknpviðxn; xnþ1Þ

�AkðxnÞ

�tp�

@VpðxnÞ

@xknþ

1

�thkjðxn�1Þv

jðxn�1; xnÞ þ

1

�tAkðxn�1Þp ¼ 0: ð43Þ

We use the trapezoidal rule to approximate the velocity coordinates at the nth timestep, which gives us

viðxn; xnþ1Þ ¼xinþ1 � xin

�t)

@viðxn; xnþ1Þ

@xkn¼ �

1

�t�ik;

@viðxn�1; xnÞ

@xkn¼

1

�t�ik; ð44Þ

where �ik denotes the Kronecker delta. From equations (43) and (44), we obtain thefollowing update rule (algorithm) for numerical integration

x jnþ1 ¼ x j

n þ� jilðxnÞ ðx

inþ1 � xinÞ ðx

lnþ1 � xlnÞ þ h jk

ðxnÞnpCikðxnÞ ðx

inþ1 � xinÞ

�p�tAkðxnÞ ��t2@VpðxnÞ

@xknþ hklðxn�1Þ ðx

ln � xln�1Þ þ p�tAkðxn�1Þ

o; ð45Þ

where

� jilðxnÞ ¼

1

2h jk

ðxnÞ@hilðxnÞ

@xkn; CikðxnÞ ¼

@AiðxnÞ

@xkn:

Equation (45) is an implicit relation which gives us a system of m� 1 simultaneousequations in the xinþ1; i ¼ 1; . . . ;m� 1. These equations can be solved by aniteration procedure to obtain the xinþ1 explicitly from the known discrete states xnand xn�1. Sufficient conditions on the structure of the inertia matrix ½hij � of thereduced system, by which one may obtain an explicit algorithm by the aboveprocedure, are given in Sanyal et al. (2004b). However, for the example system wetreat in the following sections, we get only an implicit algorithm based on discreteRouth reduction.


5. Free dynamics of the two connected bodies

We now study a specific example of a planar extended multibody system in centralgravity, which consists of two planar bodies, each modelled as a rigid masslesslink with a point mass at one end. The other ends are connected together by arotary or hinge joint, which allows each body to rotate freely about the hingepoint. The relative orientation of the two bodies describes the shape of the assembly.This model exhibits more complex dynamics compared to the simple dumb-bellspacecraft model idealized as two equal mass particles connected by an elastic mass-less link, which has been studied in Sanyal et al. (2003, 2004a). For the planar bodiesconnected by a rotary joint, the shape of the body is given by the mutual orientationof the two bodies. The polar coordinates ðr; �Þ denote the position of the hinge jointin the orbital plane. We also denote the angles the rigid links make with the radialvector as � and . The description of the configuration of the two coupled bodies interms of these generalized coordinates is depicted in figure 2. The coordinates � and together describe the attitude and shape of the coupled bodies. The configurationat any point of the configuration manifold is given by q ¼ ðr; �; �; Þ, whereðr cos �, r sin �Þ 2 R

2 denotes the position of the system, � 2 S1 denotes its

attitude, and 2 S1 denotes the shape of the system (the roles of � and in this

respect can be interchanged). The configuration manifold, Q, can be expressed as atrivial product Q ¼ R� T

2� S

1, where r 2 R, ð�; Þ 2 T2, and � 2 S

1.

5.1. Equations of motionLet the massless rigid links be of length l each, where l r. The position vectors ofthe two mass particles at the free ends of the rigid links can be expressed as

x1 ¼ ruð�Þ þ luð�� Þ; x2 ¼ ruð�Þ þ luð�þ Þ;

ν

r=|| x ||

1

2

X

Y

αβ

Figure 2. Configuration of two coupled bodies expressed in spherical coordinates.


where

uðÞ ¼cos sin

�:

We have

huðÞ; uð�Þi ¼ cosð � �Þ; _uuðÞ ¼ _JuðÞ; huðÞ; Juð�Þi ¼ sinð � �Þ;

where

J ¼0 �11 0

�is the standard 2� 2 symplectic matrix. The velocities of the two end masses are thengiven by

_xx1 ¼ _rruð�Þ þ r _��Juð�Þ þ lð _�� _��ÞJuð�� Þ;

_xx2 ¼ _rruð�Þ þ r _��Juð�Þ þ lð _��þ _ÞJuð�þ Þ:

We assume that the end masses labelled 1 and 2 have a mass of m each. The kineticenergy of the system is then given by

Tðq; _qqÞ ¼1

2mðk _xx1k

2þ k _xx2k

2Þ ¼ mð_rr2 þ r2 _��2Þ þ

ml2

2ð _�� _��Þ2 þ ð _��þ _Þ2 �

þm lr _��ð _�� _��Þ cos�þ l _rrð _�� _��Þ sin � �

þm lr _��ð _��þ _Þ cos� l _rrð _��þ _Þ sin �

:

The gravitational potential energy of the system is

V ¼ ��m

kx1k�

�m

kx2k¼ �

�m

ðr2 þ l2 þ 2rl cos�Þ1=2�

�m

ðr2 þ l2 þ 2rl cosÞ1=2:

The Lagrangian of the system is then given by

Lðq; _qqÞ ¼ Tðq; _qqÞ � VðqÞ

¼ m

_rr2 þ r2 _��2 þ

l2

2ð _�� _��Þ2 þ ð _��þ _Þ2 �

þ lr _��ð _�� _��Þ cos�þ l _rrð _�� _��Þ sin �þ lr _��ð _��þ _Þ cos� l _rrð _��þ _Þ sin

þ�

ðr2 þ l2 þ 2rl cos�Þ1=2þ

�

ðr2 þ l2 þ 2rl cosÞ1=2

�: ð46Þ

Note that � is a cyclic variable in this system, and the Lagrangian does not dependexplicitly on �. Hence, in the absence of external forces to the system, the conjugatemomentum

p ¼@L

@ _��¼ 2mr2 _��þml2fð _�� _��Þ þ ð _��þ _Þg þmflrð2 _�� _��Þ cos�þ l _rr sin �g

þmflrð2 _��þ _Þ cos � l _rr sin g ð47Þ


is conserved. This is in addition to the total energy

Eðq; _qqÞ ¼ Tðq; _qqÞ þ VðqÞ

¼ m

_rr2 þ r2 _��2 þ

l2

2ð _�� _��Þ2 þ ð _��þ _Þ2 �

þ lr _��ð _�� _��Þ cos�þ l _rrð _�� _��Þ sin �þ lr _��ð _��þ _Þ cos� l _rrð _��þ _Þ sin

��

ðr2 þ l2 þ 2rl cos�Þ1=2�

�


�; ð48Þ

which is also conserved for the free dynamics.The configuration manifold Q is a Riemannian manifold with the metric tensor

given by the inertia matrix of the system

M ¼ ½gij �

¼

2m mlðsin�� sinÞ �ml sin� �ml sin

mlðsin�� sinÞ 2mðr2 þ l2 þ lr cos�þ lr cosÞ �mlðl þ r cos�Þ mlðl þ r cosÞ

�ml sin� �ml ðl þ r cos �Þ 2ml2 0

�ml sin ml ðl þ r cosÞ 0 2ml2

2666437775,

ð49Þ

where gij; i; j ¼ 1; . . . ; 4 is used to denote the (symmetric) metric tensor and thecoordinates are arranged in the order ðq1; q2; q3; q4Þ ¼ ðr; �; �; Þ. The free dynamicsis also influenced by the gradient of the gravitational potential, @VðqÞ=@q 2 �ðQÞ,a vector field on Q given by

@VðqÞ

@q¼

�mðrþ l cos �Þ


�mðrþ l cosÞ


0

��mlr sin �

ðr2 þ l2 þ 2rl cos�Þ3=2

��mlr sin


2666666666664

3777777777775: ð50Þ

For a mechanical system of this type, the equations of motion are given in second-order form by

€qqi ¼ ��ijkðqÞ _qq

j _qqk � gilðqÞ@VðqÞ

@ql; ð51Þ

where gil denotes the components of the inverse of the inertia matrix, and �ijk ¼ �i

kj

are the Christoffel symbols for the associated metric, given by

�ijk ¼

1

2gil

@glj

@xkþ@glk@x j

�@gjk

@xl

� �: ð52Þ

If we label the coordinates in the order ðq1; q2; q3; q4Þ ¼ ðr; �; �; Þ, the non-zeroChristoffel symbols are then given by the expressions in the Appendix.


5.2. Relative equilibriaWe now look at relative equilibria corresponding to fixed circular orbits of thesystem with a constant orbital rate. At such relative equilibria, we have:

€rr ¼ _rr ¼ 0; _�� ¼ !; €�� ¼ 0; _�� ¼ 0; _ ¼ 0;

and there are no external forces except gravity. Denoting the non-cyclic coordinatesof this system by x ¼ ðr; �; Þ, we get the following algebraic equations describing therelative equilibria

�i22ðxÞ!

2¼ �gilðxÞ

@VðxÞ

@xl; i; l ¼ 1; . . . ; 4: ð53Þ

This expression is identical to that in (7), since

�i22ðxÞ ¼ �

1

2gilðxÞ

@g22ðxÞ

@xl:

From (53), we get the following three algebraic equations that are satisfied at relativeequilibria corresponding to circular orbits

f2rþ lðcos�þ cosÞg!2¼

�ðrþ l cos�Þ


�ðrþ l cos Þ

ðr2 þ l2 þ 2rl cosÞ3=2; ð54Þ

ð2lr sin �Þ!2¼

2�lr sin �

ðr2 þ l2 þ 2rl cos�Þ3=2; ð55Þ

ð2lr sin Þ!2¼

2�lr sin

ðr2 þ l2 þ 2rl cosÞ3=2: ð56Þ

Equations (55) and (56) have the following solutions:

sin � ¼ 0; or !2¼

�


sin ¼ 0; or !2¼

�


Equations (54)–(58) give rise to the following sets of conditions that lead torelative equilibria:

(1) sin � ¼ 0 and sin ¼ 0;(2) cos� ¼ cos and !2

¼�

ðr2 þ l2 þ 2rl cos �Þ3=2;

(3) sin � ¼ 0 and !2¼

�

ðr2 þ l2 þ 2rl cosÞ3=2; or

(4) sin ¼ 0 and !2¼

�

ðr2 þ l2 þ 2rl cos�Þ3=2.

The last two possibilities enumerated above are really symmetric, and we need toconsider only one of them. We look at the possible condition (3). Using equation (54),we see that if this is a possible solution, then the angle must satisfy

ðr2 þ l2 þ 2rl cosÞ3=2 ¼ ðr� lÞ3 ) cos ¼ �1:

However, this relation is already satisfied if condition (1) is satisfied. Hence, we canignore (3) and (4) as separate conditions for relative equilibria.


Note 1. Condition (1) gives rise to the following three classes of relative equilibria:

� ¼ 0; ¼ 0; and !2¼

�

ðrþ lÞ3; ð59Þ

� ¼ 0; ¼ �; and !2¼

�

2r

� 1

ðrþ lÞ2þ

1

ðr� lÞ2

�; ð60Þ

� ¼ �; ¼ �; and !2¼

�

ðr� lÞ3: ð61Þ

These relative equilibria, labelled 1, 2, and 3, respectively, are shown graphically infigure 3. The direction of gravity is shown by an arrow. Since the two planar bodiesare identical (equal length massless rigid links with equal end masses), there isanother symmetry which makes the configurations ð�; Þ ¼ ð0; �Þ and ð�; Þ ¼ð�; 0Þ indistinguishable. Hence there are only three unique types of relative equilibriaobtained from condition (1) above. The first type, given by (59) above, correspondsto relative equilibria in which one body is ‘folded’ on top of the other in the plane,and the two bodies are aligned along the radial vector with the hinge joint nearer tothe centre of force. The second type (given by (60)) corresponds to the two bodieslying in an ‘opened up’ position along the radial vector, with the two end massesat either end and the hinge joint midway between them. The third type (given by(61)) corresponds to the configuration in which both bodies are again ‘folded’ ontop of each other and lying along the radial vector, with the end masses nearer to thecentre of force.

31 2

α = 0, β = 0α = 0, β = π

α = π, β = π

Figure 3. First three relative equilibria for the two connected bodies.


Note 2. Condition (2) gives rise to the following two types of relative equilibria:

� ¼ 6¼ f0; �g and !2¼

�


� ¼ 2�� 6¼ f0; �g and !2¼

�

ðr2 þ l2 þ 2rl cos�Þ3=2: ð63Þ

These relative equilibria, labelled 4 and 5, respectively, are shown graphically infigure 4. The direction of gravity is shown by an arrow. The first type given by (62)above corresponds to the two planar bodies ‘opened up’ at equal angles to the radialdirection, with the radial vector bisecting the angle made by the two rigid links.The second type (given by (63)) corresponds to the two planar bodies in a ‘folded’position, with one body on top of another and making a fixed angle with theradial direction. Notes 1 and 2 show that the relative equilibria of this systemform a connected subset of the subspace of the configuration space given bythe coordinates ð�; Þ. This is in contrast to the dumb-bell spacecraft treated inSanyal et al. (2003), where the relative equilibria are discrete in the attitude-shapesubspace.

6. Routh reduction for the two connected bodies

To study the stability properties of the relative equilibria of the system of thetwo coupled rigid bodies in a central gravity field, we look at the reduced systemobtained from Routh reduction. As we remarked in section 3.2, the equilibriaof the reduced system correspond to the relative equilibria of the original system.Since the reduced system is of dimension 3, it is easier to analyse the stability of itsequilibria.

4 5

α β =α α

β = 2π−α

Figure 4. Fourth and fifth relative equilibria for the two connected bodies.


6.1. Equations of motion for the reduced systemIf we rearrange the coordinates of the original system as (q1, q2, q3, q4)¼ (r,�,, �),then the metric tensor (inertia matrix) of the system is

M0¼ ½gij �

¼

2m �ml sin � �ml sin mlðsin �� sinÞ

�ml sin � 2ml2 0 �mlðl þ r cos�Þ

�ml sin 0 2ml2 mlðl þ r cos Þ

mlðsin�� sinÞ �mlðl þ r cos �Þ mlðl þ r cos Þ 2mðr2 þ l2 þ lr cos �þ lr cos Þ

2666437775:

ð64Þ

We denote the blocks of this matrix by

M0¼

M11 M12

M21 M22

�;

where M11 is the 3� 3 upper left block; M12 is the 3� 1 vector consisting of the firstthree elements of the last column; M21 ¼ MT

12, the transpose of M12, and M22 is the(4, 4) element of M0. The metric in the reduced configuration space, given by thecoordinates x¼ (q1, q2, q3)¼ (r,�,), is given by the matrixeMM ¼ M11 �M12ðM22Þ

�1M21: ð65Þ

If we denote the elements of this metric by hij , i; j ¼ 1; 2; 3, then the matrix repre-sentation of this metric is given byeMM¼ ½hij �

¼

m 2�l2ðsin �� sinÞ2

f ðr;�;Þ

( )ml �sin�þ

lðlþ rcos�Þðsin�� sinÞ

f ðr;�;Þ

� �

ml �sin�þlðlþ r cos�Þðsin�� sinÞ

f ðr;�;Þ

� �ml2 2�

ðlþ rcos�Þ2

f ðr;�;Þ

( )

�ml sinþlðlþ rcosÞðsin�� sinÞ

f ðr;�;Þ

� �ml2ðlþ rcos�Þðlþ rcosÞ

f ðr;�;Þ

26666666666664

�ml sinþlðlþ rcosÞðsin�� sinÞ

f ðr;�;Þ

� �ml2ðlþ rcos�Þðlþ rcosÞ

f ðr;�;Þ

ml2 2�ðlþ rcosÞ2

f ðr;�;Þ

( )

377777777775,

ð66Þ

where

f ðr; �; Þ ¼ f ðxÞ ¼ 2ðr2 þ l2 þ lr cos�þ lr cosÞ:


The vector of connection coefficients is given by

A ¼ ðM22Þ�1M12 ¼

l

f ðr; �; Þ

sin �� sin

�ðl þ r cos�Þ

l þ r cos

264375; ð67Þ

while the matrix of curvature coefficients is given by

B ¼

0

2l2ðl � r cos�Þðcos �� cosÞ

�2l2r sin �ðsin �þ sin Þ

8<:9=;

ð f ðr; �; ÞÞ2

�

2l2ðl � r cos�Þðcos�� cosÞ

�2l2r sin �ðsin �þ sin Þ

8<:9=;


�

2l2ðl � r cosÞðcos�� cosÞ

þ2l2r sin ðsin �þ sin Þ

8<:9=;

ð f ðr; �; ÞÞ2�2l2rflðsin �þ sin Þ þ r sinð�þ Þg


2666666666666666666666664

�

2l2ðl � r cosÞðcos�� cosÞ

þ2l2r sin ðsin �þ sin Þ

8<:9=;


2l2rflðsin �þ sin Þ þ r sinð�þ Þg


0

3777777777777775:

ð68Þ

The amended potential is given by

VpðxÞ ¼ VðxÞ þ1

2g��p2

¼ ��m

ðr2 þ l2 þ 2rl cos �Þ1=2�

�m

ðr2 þ l2 þ 2rl cosÞ1=2þ

p2

2mf ðr; �; Þ: ð69Þ

The equations of motion for the reduced system in the absence of external forcesother than gravity are given by

eMMðxÞ €xxþ_eMMeMMðx; _xxÞ _xx�

@

@x

� 12_xxT eMMðxÞ _xx

�þ@VpðxÞ

@xþ Bp _xx ¼ 0: ð70Þ


The gradient of the amended potential is given by

@VpðxÞ

@x¼

�mðrþ l cos �Þ

ðr2 þ l2 þ 2rl cos �Þ3=2þ

�mðrþ l cosÞ

ðr2 þ l2 þ 2rl cos Þ3=2

��mlr sin �

ðr2 þ l2 þ 2rl cos �Þ3=2

��mlr sin

ðr2 þ l2 þ 2rl cos Þ3=2

266666666664

377777777775þ p2

�2rþ l cos�þ l cos

mð f ðr; �; ÞÞ2

lr sin �


lr sin


266666666664

377777777775:

ð71Þ

6.2. Equilibria of the reduced systemThe equilibria of the reduced system are given by the critical points of the amendedpotential, i.e. when the gradient in (71) vanishes. Thus, the algebraic equationssatisfied by an equilibrium point in the reduced configuration space are

�mðrþ l cos�Þ

ðr2 þ l2 þ 2rl cos �Þ3=2þ

�mðrþ l cos Þ

ðr2 þ l2 þ 2rl cosÞ3=2¼

p2ð2rþ l cos�þ l cos Þ

4mðr2 þ l2 þ lr cos�þ lr cosÞ2; ð72Þ

�mlr sin �

ðr2 þ l2 þ 2rl cos �Þ3=2¼

p2rl sin �

4mðr2 þ l2 þ lr cos�þ lr cosÞ2; ð73Þ

�mlr sin

ðr2 þ l2 þ 2rl cosÞ3=2¼

p2rl sin

4mðr2 þ l2 þ lr cos�þ lr cosÞ2: ð74Þ

Equations (73) and (74) have the following solutions:

sin � ¼ 0; or p2 ¼4�m2

ðr2 þ l2 þ lr cos �þ lr cos Þ2


sin ¼ 0; or p2 ¼4�m2

ðr2 þ l2 þ lr cos�þ lr cosÞ2


As with the unreduced dynamics, we obtain two independent conditions forequilibria for the reduced dynamics:

(1) sin � ¼ 0 and sin ¼ 0;(2) cos� ¼ cos and p2 ¼ 4�m2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ l2 þ 2rl cos �

p.

Note 3. Condition (1) gives rise to the following three types of equilibria:

� ¼ 0; ¼ 0; and p2 ¼ 4�m2ðrþ lÞ; ð77Þ

� ¼ 0; ¼ �; and p2 ¼4�m2

ðr2 þ l2Þ3

rðr2 � l2Þ2; ð78Þ

� ¼ �; ¼ �; and p2 ¼ 4�m2ðr� l Þ; ð79Þ

since the configuration ð�; Þ ¼ ð�; 0Þ is indistinguishable from ð�; Þ ¼ ð0; �Þ.


The equilibria in (77)–(79) correspond to the relative equilibria given byequations (59)–(61), respectively, for the unreduced system.

Note 4. Condition (2) gives rise to the following two types of equilibria:

� ¼ ; and p2 ¼ 4�m2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ l2 þ 2rl cos �

p; ð80Þ

� ¼ 2�� ; and p2 ¼ 4�m2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ l2 þ 2rl cos�

p: ð81Þ

The equilibria (80) and (81) for the reduced system correspond to the relativeequilibria (62) and (63), respectively, for the unreduced system. These equilibriaform a connected set in the attitude-shape subspace of the configuration space ofthe reduced system.

6.3. Stability of relative equilibria in circular orbitThe stability of the relative equilibria of the system of coupled planar bodiesis obtained from the stability of the equilibria of the reduced system, given byequations (77)–(81). From section 3.3, we know that this can be obtained bychecking for positive definiteness of the Hessian of the amended potential Vp(x)evaluated at x¼ xe, the equilibrium configuration. Let us denote the equilibria in(77)–(81) by x1 to x5, respectively.

Proposition 2. The equilibrium x1 given by (77) has non-trivial stable and centremanifolds, but no unstable manifold.

Proof. The Hessian of the amended potential at the first equilibrium considered,with � ¼ ¼ 0, is positive semi-definite and has the form

d2Vpðx1Þ ¼

2�m

ðrþ lÞ30 0

0 0 0

0 0 0

2666437775:

Thus, we cannot use Proposition 1 to determine the stability of this equilibrium.œ

This case is further analysed later in this section.

Proposition 3. The equilibrium x2 given by (78) is stable.

Proof. The Hessian of the amended potential at the second equilibrium, with � ¼ 0and ¼ �, has the form

d2Vpðx2Þ ¼

2�mðr4 � 10r2l2 þ l 4Þ

rðr2 � l2Þ30 0

0�mðr2 þ l2Þl

ðr2 � l2Þ2�

�mrl

ðrþ l Þ30

0 0�mðr2 þ l2Þl

ðr2 � l2Þ2�

�mrl

ðrþ l Þ3

2666666664

3777777775:


This Hessian is positive definite, and applying Proposition 1, this equilibrium is stable.This means that the corresponding relative equilibrium of the full system, where thehinged links are opened out and aligned along the radial direction, is stable. œ

The last two (common) eigenvalues of �2Vpðx2Þ are positive but small, since theratio l / r is small.

Proposition 4. The equilibrium x3 given by (79) has non-trivial stable and centremanifolds, but no unstable manifold.

Proof. The Hessian of the amended potential at the third equilibrium considered,with � ¼ ¼ �, is positive semi-definite and has the form

d2Vpðx3Þ ¼

2�m

ðr� l Þ30 0

0 0 0

0 0 0

266664377775:

Thus, we cannot use Proposition 1 to make conclusions about the stability of thisequilibrium. œ

This case is further analysed later in this section.

Proposition 5. The equilibrium x4 given by (80) is unstable.

Proof. The Hessian of the amended potential at the fourth equilibrium, with � ¼ and both these angles not equal to zero or � radians, has the form

d2Vpðx4Þ ¼

2k1 �k2 �k2

�k2 �k3 2k3

�k2 2k3 �k3

264375;

where

k1 ¼�mðrþ l cos�Þ2

ðr2 þ l2 þ 2rl cos�Þ5=2; k2 ¼

�mrlðrþ l cos �Þ sin �

ðr2 þ l2 þ 2rl cos�Þ5=2,

k3 ¼�mr2l2 sin2 �

ðr2 þ l2 þ 2rl cos�Þ5=2:

The eigenvalues of this Hessian matrix (obtained using the fact that k1k3 ¼ k22) areð�3k3; 0; 2k1 þ k3Þ. Hence, the matrix has at least one negative eigenvalue.According to Proposition 1, it is unstable. The corresponding relative equilibriumof the two connected bodies in circular orbit, with the hinge open so that the linksmake an equal angle with the radial direction, is thus unstable. œ

Proposition 6. The equilibrium x5 given by (81) is unstable.

Proof. The Hessian of the amended potential at the fourth equilibrium, with� ¼ 2�� and both these angles not equal to zero or � radians, has the form

d2Vpðx5Þ ¼

2k1 �k2 k2

�k2 �k3 �2k3

k2 �2k3 �k3

26643775;


where k1, k2, and k3 are as given in the proof of Proposition 5. The eigenvalues ofthis Hessian matrix (obtained using the fact that k1k3 ¼ k22) are ð�3k3; 0; 2k1 þ k3Þ,identical to those of d2Vpðx4Þ. Hence, the matrix has at least one negative eigenvalue.According to Proposition 1, it is unstable. The corresponding relative equilibriumof the two connected bodies in circular orbit, with the links folded over eachother and making an arbitrary angle � 6¼ ð0; �Þ to the radial direction, is thusunstable. œ

6.3.1. Stability results for the degenerate relative equilibria x1 and x3. The stabilityproperties of the equilibria x1 and x3 of the reduced dynamics of the planar hingedbodies can be obtained from the stability properties of the lower-order terms inthe normal form expansion of the Hamiltonian, as given in section 3.4.3. Weapply Corollary 1 to the third-order dynamics about these relative equilibria,given by the fourth-order Birkhoff normal form expansion Zð4Þ

¼ Z2 þ Z4. Weobtain the following result based on Corollary 1.

Proposition 7. The equilibrium x1 given by (77) is nonlinearly unstable.

Proof. As mentioned in Proposition 4, this equilibrium has a centre manifold,consisting of the attitude and shape degrees of freedom. The normalized coordinatesof the reduced dynamics about this equilibrium are �xx ¼ ½r� re � �T and�zz ¼ ½ �zz1 �zz2 �zz3�

T¼ VTz, where eMM�1

ðx1Þ ¼ VNVT and N is diagonal. We evaluate thefourth-order Birkhoff normal form expansion Zð4Þ

¼ Z2 þ Z4 and use it to obtain

fI2;Zð4Þg ¼ fI2; I1 þ I2 þ I3g þ fI2;Z4g ¼ fI2;Z4g;

since Z2 ¼ I1 þ I2 þ I3 and I1 ¼ I1ðr� re; �zz1Þ, I2 ¼ I2ð�; �zz2Þ and I3 ¼ I3ð; �zz3Þ areindependent of each other in the normalized coordinates. We also obtain

fI2;Z4g ¼ 2� �zz2 p1 ðr� reÞ2þ

�zz21k1

!þ p2 �2

þ�zz22k2

!þ p3 2

þ�zz23k3

! !;

where pi>0, and ki>0 for i¼ 1, 2, 3. Evaluating this bracket at the closed half-space of the momentum state space given by La ¼fðr;�;; �zz1; �zz2; �zz3Þ j ð�� 0; �zz2 � 0Þ [ð� � 0; �zz2 � 0Þg, we see that fI2;Z

ð4Þg is positive definite in the interior of this

half-space which contains x1. Hence, by Corollary 1, this equilibrium is nonlinearly(weakly) unstable. œ

We obtain the following result for the relative equilibrium x3, based onCorollary 1.

Proposition 8. The equilibrium x3 given by (79) is nonlinearly unstable.

Proof. According to Proposition 6, this equilibrium has a centre manifold,consisting of the attitude and shape degrees of freedom. The normalized coordinatesof the reduced dynamics about this equilibrium are �xx ¼ ½r� re �� T and�zz ¼ ½ �zz1 �zz2 �zz3�

T¼ VTz, where eMM�1

ðx3Þ ¼ VNVT and N is diagonal. We evaluate thefourth-order Birkhoff normal form expansion Zð4Þ

¼ Z2 þ Z4 and use it to obtain

fI3;Zð4Þg ¼ fI3; I1 þ I2 þ I3g þ fI3;Z4g ¼ fI3;Z4g;


since Z2 ¼ I1 þ I2 þ I3 and I1 ¼ I1ðr� re; �zz1Þ, I2 ¼ I2ð�; �zz2Þ and I3 ¼ I3ð; �zz3Þ areindependent of each other in the normalized coordinates. We also obtain

fI3;Z4g ¼ 2ð� �Þ �zz3 p1 ðr� reÞ2þ

�zz21k1

!þ p2 ð�� Þ2 þ

�zz22k2

!

þp3 ð� �Þ2 þ�zz23k3

!!;

where pi>0 and ki>0 for i¼ 1, 2, 3. Evaluating this bracket in the closed half-spaceof the momentum state space given by Lb ¼fðr;�;; �zz1; �zz2; �zz3Þ j ð��; �zz3 � 0Þ [ð � �; �zz3 � 0Þg, we see that fI2;Z

ð4Þg is positive definite in the interior of this half-

space which contains x3. Hence, by Corollary 1, this equilibrium is nonlinearly(weakly) unstable. œ

6.3.2. Non-integrability of the two connected bodies. The normal form expansionsabout the two degenerate relative equilibria of the two connected bodies have degreeof resonance dr¼ 2. The normal form expansions up to fourth order about thedegenerate equilibria of the reduced dynamics were obtained in section 6.3.1.It turns out that an exact Birkhoff normal form expansion is not possible aroundthese equilibria. One can carry out a Birkhoff–Gustavson normalization up to thefourth order, but the higher-order remainder cannot be rendered in normal form.This is because the system is resonant at higher order, and hence the normal formdiverges. This suggests that this system is non-integrable (see, e.g. the recent resultsof Zung 2003). We mention that the motion of three planar coupled rigid bodies inthe absence of gravity is also non-integrable (see Oh et al. 1989).

6.4. Discrete Routh reduction for the two connected bodiesNow we look at the numerical analysis of the free dynamics of the coupled planarbodies in central gravity. The continuous Routhian for the hinged planar bodies incentral gravity is

Rðx; _xxÞ ¼1

2_xxT eMMðxÞ _xxþ

p

mf ðxÞM21ðxÞ _xx�

p2

2mf ðxÞ� VðxÞ:

For ease in numerical computations, we non-dimensionalize the Routhian asfollows. Let R>0 be a constant radial distance from the central body, and let �denote its gravitational force constant. Define the constant ! ¼

ffiffiffiffiffiffiffiffiffiffiffi�=R3

p, which is

a nominal angular rate about the central body. Make the change in the generalizedcoordinate r 7!� given by r ¼ R�, and the change in the time coordinate � ¼ !t. Thecontinuous Routhian is then given by

Rðx; _xxÞ ¼ mR2!2 1

2�0T �MMð�Þ�0 þ

�pp

ð�Þ�MM21ð�Þ�

0�

�pp2

2 ð�Þ� �VVð�Þ

" #¼ mR2!2 �RRð�; �0Þ;

ð82Þ


where

�¼

�

�

264375;

�MMð�Þ¼

2��2ðsin�� sinÞ2

ð�;�;Þ� �sin�þ

�ð�þ�cos�Þðsin�� sinÞ

ð�;�;Þ

� �� sin�þ

�ð�þ�cos�Þðsin�� sinÞ

ð�;�;Þ

� ��2 2�

ð�þ�cos�Þ2

ð�;�;Þ

( )

�� sinþ�ð�þ�cosÞðsin�� sinÞ

ð�;�;Þ

� ��2ð�þ�cos�Þð�þ�cosÞ

ð�;�;Þ

26666666664

�� sinþ�ð�þ�cosÞðsin�� sinÞ

ð�;�;Þ

��2ð�þ�cos�Þð�þ�cosÞ

ð�;�;Þ

�2 2�ð�þ�cosÞ2

ð�;�;Þ

( )

377777777775;

�MM21ð�Þ¼ ½�ðsin�� sinÞ � �ð�þ�cos�Þ �ð�þ�cosÞ�;

� ¼l

R; ð�; �; Þ ¼ 2ð�2 þ �2 þ �� cos�þ �� cosÞ,

�VVð�Þ ¼ �1

ð�2 þ �2 þ 2�� cos�Þ1=2�

1

ð�2 þ �2 þ 2�� cos Þ1=2;

and

�pp ¼ 2�2�0 þ �2ð2�0 � �0þ 0

Þ þ ��ð2�0 � �0Þ cos�

þ ��0 sin �þ ��ð2�0 þ 0Þ cos� ��0 sin :

We use prime ( 0 ) to denote differentiation with respect to �.The discrete scaled Routhian is obtained from the continuous scaled Routhian

by using the trapezoidal rule to approximate first derivatives with respect to �. Thediscrete Routh equations for the hinged planar bodies give

�MMð�nÞð�nþ1 � �nÞ �1

2

@

@�

�ð�nþ1 � �nÞ

TMð�Þð�nþ1 � �nÞ�

�¼�n

þ�pp��

ð�nÞ�MM12ð�nÞ

þ�pp��

ð�nÞ2

d ð�nÞ

d�n�MM21ð�nÞð�nþ1 � �nÞ �

�pp��

ð�nÞ

d �MM21ð�nÞ

d�nð�nþ1 � �nÞ �

ð �pp��Þ2

2 ð�nÞ2

d ð�nÞ

d�n

þd �VVð�nÞ

d�n��2 ¼ �MMð�n�1Þð�n � �n�1Þ þ

�pp��

ð�n�1Þ�MM12ð�n�1Þ: ð83Þ


These equations are used as a numerical integration scheme to obtain theevolution of the free dynamics of the hinged planar bodies when perturbed fromits relative equilibria. The results of the numerical simulation are presented in thenext section.

7. Simulation results for the two connected bodies

The discrete Routh equations (83) give a two-step integration procedure that relatesthe configurations at the (n� 1)th and nth time steps, �n�1 and �n, to the configura-tion at the ðnþ 1Þth time step, �nþ1. However, these equations give an implicitrelation for �nþ1, in terms of the known configurations �n and �n�1. These implicitrelations need to be solved using an implicit equation solver, and a low-ordermethod, like Newton–Raphson, usually works well in solving these implicitequations. However, solving these implicit equations slows down the working ofthis numerical algorithm. Therefore, we numerically simulate over shorter durations(usually the change in the cyclic variable is less 2� radians) to observe the stabilityproperties of the relative equilibria.

We present four sets of simulation results for four different relative equilibria ofthe hinged planar bodies in a central gravity field. The simulations are carried outfor the Routh reduced system, about four of its equilibria. The first set of results areobtained for an initial perturbation to the stable relative equilibrium given by (78).The second set of results are obtained from an initial perturbation to the unstablerelative equilibrium (80). The third and fourth sets of simulation results are obtainedfor initial perturbations to the elliptic relative equilibria given by (77) and (79),respectively.

The first set of simulation results presented here show the evolution of the freehinged planar bodies when a small initial perturbation is given to its stable relativeequilibrium given by (78). The configuration for the reduced system at this equilib-rium is

�s ¼ ½�0 �0 0� ¼ ½1 0 ��; �0s ¼ 1:0000375:

The initial conditions for the simulation are given by

�0 ¼ ½�0 �0 0� ¼ ½1 0 ��; �00 ¼ ½�00 �00 0

0� ¼ ½0 0:01 � 0:01�;

�00 ¼ 1:0000375; �0 ¼ 0:

The evolution of the orbital coordinates obtained from the discrete Routhequations is shown in figure 5. The evolution of the attitude and shape coordinatesobtained from the discrete Routh equations for this initial perturbation is shownin figure 6. The scaled time period of integration (�f) taken for these simulations isa relatively short period (less than half an orbit). Simulations for larger periods oftime are necessary to observe the behaviour of the free dynamics about this stableequilibrium.

Now we carry out a simulation around an unstable hyperbolic relative equilibrium(x4) of the hinged planar bodies, which is given by the reduced configuration

�u ¼ ½�0 �0 0� ¼ 1�

6

�

6

h i; �0u ¼ 0:9935352:


0 0.5 1 1.5 2 2.5 3 3.5 4−0.01

−0.005

0

0.005

0.01

Firs

t ang

le (

α)

0 0.5 1 1.5 2 2.5 3 3.5 43.13

3.135

3.14

3.145

3.15

3.155

Sec

ond

angl

e (β

)

Scaled time (τ)

Figure 6. Evolution of attitude and shape for initial perturbation fromstable relative equilibrium.

0 0.5 1 1.5 2 2.5 3 3.5 4

0.9998

0.9999

0.9999

1

1S

cale

d or

bita

l rad

ius

(ρ)

0 0.5 1 1.5 2 2.5 3 3.5 40.9999

1

1.0001

Orb

ital r

ate

(ν')

Scaled time (τ)

Figure 5. Evolution of orbital coordinates for initial perturbation from stablerelative equilibrium.



�0 ¼ �u; �00 ¼ ½�00 �00 0

0� ¼ ½0 0:01 � 0:01�; �00 ¼ 0:9935352; �0 ¼ 0:

The evolution of the orbital coordinates obtained from the discrete Routhequations is shown in figure 7. The evolution of the attitude and shape coordinatesobtained from the discrete Routh equations for this initial perturbation is shown infigure 8. The scaled radius �, and the angles � and show steady divergence fromtheir equilibrium values over the short duration of time (0.5 units), which agrees withthe unstable nature of this equilibrium.

We now look at the degenerate elliptic relative equilibrium x1, where K(x1)� 0.The reduced configuration of the hinged planar bodies at this equilibrium is

�c1 ¼ ½�0 �0 0� ¼ ½1 0 0�; �0c1 ¼ 0:9925466:


�0 ¼ ½1 0:01 � 0:005�; �00 ¼ ½�00 �00 0

0� ¼ ½0 0 0�; �00 ¼ 0:9925466; �0 ¼ 0:

Note that the initial conditions are in the closed half-space La given in section 6.3.1.According to Proposition 8, the flow should indicate divergence of the angle � awayfrom 0. The evolution of the orbital coordinates obtained from the discrete Routhequations is shown in figure 9. The evolution of the attitude and shape coordinatesobtained from the discrete Routh equations for this initial perturbation is shown infigure 10. The scaled radius � does not change much, and the angles � and showslow divergence from their equilibrium values over the simulation time (1 unit). This

0 0.5 1 1.5

1

1

1.0001

1.0001

1.0001

Sca

led

orbi

tal r

adiu

s (ρ

)

0 0.5 1 1.5

0.9935

0.9935

0.9935

0.9935

0.9935

0.9936

Orb

ital r

ate

(ν')

Scaled time (τ)

Figure 7. Evolution of orbital coordinates for initial perturbation from anunstable relative equilibrium x¼ x4.


0 0.5 1 1.50.52

0.525

0.53

0.535

0.54

0.545

0.55F

irst a

ngle

(α)

0 0.5 1 1.50.505

0.51

0.515

0.52

0.525

0.53

Sec

ond

angl

e (β

)

Scaled time (τ)

Figure 8. Evolution of attitude and shape for initial perturbation from anunstable relative equilibrium x¼ x4.

0.9999

1

1.0001

Sca

led

orbi

tal r

adiu

s (ρ

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0.9925

0.9925

0.9926

0.9926

0.9926

Orb

ital r

ate

(ν')

Scaled time (τ)

Figure 9. Evolution of orbital coordinates for initial perturbation froma relative equilibrium x¼ x1.


agrees with the analytical result obtained in section 6.3.1 about the (weak) instability

of this equilibrium.

We now look at the degenerate elliptic relative equilibrium x3, where K(x3) � 0.

The reduced configuration of the hinged planar bodies at this equilibrium is

�c2 ¼ ½�0 �0 0� ¼ ½1 � ��; �0c2 ¼ 1:00754715:


�0 ¼ ½1 � �þ 0:01 �� 0:005�; �00 ¼ ½�00 �00 0

0� ¼ ½0 0 0�;

�00 ¼ 1:00754715; �0 ¼ 0:

Note that the initial conditions are in the closed half-space Lb given in section 6.3.1.

According to Proposition 9, the flow should indicate divergence of the angle away

from �. The evolution of the orbital coordinates obtained from the discrete Routh

equations is shown in figure 11. The evolution of the attitude and shape coordinates

obtained from the discrete Routh equations for this initial perturbation is shown in

figure 12. The scaled radius � does not change much, and the angles � and show

slow divergence from their equilibrium values over the simulation time (1 unit). This

agrees with the analytical result obtained in section 6.3.1 about the (weak) instability

of this equilibrium. Note that the simulation results for the elliptic unstable equili-

bria (x1 and x3) shown in figures 9–12 show that these are weakly unstable when

compared to the simulation results for the hyperbolic unstable equilibrium x4 shown

in figures 7 and 8.

0.01

0.01

0.01

0.01

0.0101

0.0101

Firs

t ang

le (

α)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−5.08

−5.06

−5.04

−5.02

–5

−4.98x 10 –3

Sec

ond

angl

e (β

)

Scaled time (τ)

Figure 10. Evolution of attitude and shape for initial perturbation froma relative equilibrium x¼ x1.


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−3.1316

−3.1316

−3.1316

−3.1315

−3.1315

−3.1315

−3.1315

Firs

t ang

le (

α)

3.1365

3.1365

3.1365

3.1366

3.1366

3.1366

3.1366

Sec

ond

angl

e (β

)

Scaled time (τ)

Figure 12. Evolution of attitude and shape for initial perturbation froma relative equilibrium x¼ x3.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80.9999

1

1.0001

Sca

led

orbi

tal r

adiu

s (ρ

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

1.0075

1.0075

1.0076

1.0076

1.0076

Orb

ital r

ate

(ν')

Scaled time (τ)

Figure 11. Evolution of orbital coordinates for initial perturbation froma relative equilibrium x¼ x3.


8. Conclusion

This paper provides a general development for obtaining and analysing the dynamicsof multibody systems in planar motion under the influence of a central gravitationalfield. Multibody systems are complex dynamical systems, and the introduction of acentral potential field reduces the symmetry of the system since linear momentumis no longer conserved. There is conservation of total angular momentum and totalenergy, but this does not make the system integrable.

The reduction of the dynamics is carried out on the Lagrangian side usingRouth reduction, to eliminate the cyclic variable associated with conservationof angular momentum. The configuration manifold of the reduced system has atleast three degrees of freedom, corresponding to radial, attitude, and one or moreshape degrees of freedom. Dynamic coupling between these degrees of freedom isnon-trivial and leads to the complexity in the dynamics.

The stability of the relative equilibria of the multibody system in a circular orbitwith fixed attitude and shape about the centre of the gravitational field are analysed.We obtain five relative equilibria, of which two are linearly unstable, one is linearlystable, and the remaining two are degenerate. A centre manifold analysis near thedegenerate elliptic (relative) equilibria, based on series expansion in the Birkhoffnormal form, is presented. Such an analysis applied to the degenerate equilibria oftwo connected bodies in central gravity, shows these equilibria to be nonlinearlyunstable. Simulation results confirm the stability properties of the linearly stableand unstable equilibria, and also confirm the weak (nonlinear) instability of thedegenerate elliptic relative equilibria.

The simulations are carried out using a discrete Routh reduction approach, whichgives a variational integrator that preserves first integral invariants of the system andis implicit for our model. The development of this integration algorithm, and itsapplication to the system of two coupled bodies in central gravity, is also presentedhere. Control schemes for such systems, which utilize the coupling from attitude andshape degrees of freedom to orbital degrees of freedom to change the orbit, are beingdeveloped and will be reported elsewhere.

Acknowledgements

The first author thanks Daniel Scheeres for attracting his attention to theexisting work on analysis of centre manifolds in the three-body problem andrelated problems in celestial mechanics. The authors thank Antonio Giorgilli andFrederic Gabern for their help in clarifying some questions on the use of normalform expansions around elliptic equilibria of Hamiltonian systems. The authorsalso thank Richard Montgomery for helpful comments on analysis ofnonlinear instability of Hamiltonian systems with series expansions, and JerroldMarsden for helpful discussions and comments on discrete Routh reduction.This research has been supported in part by NSF under grants DMS-0103895,DMS-0305837, and ECS-0140053.

Appendix. Christoffel symbols for coupled planar bodies

Label the coordinates in the coupled planar bodies system as (q1, q2, q3, q4)¼(r, �,�,). The non-zero Christoffel symbols are then obtained using (52)


as follows:

�112¼�1

21¼ 2l ð4rþ lcos��rcos2�þcosð3lþrcos�ÞÞsin��

�ð4rþ lcos�rcos2þcos�ð3lþrcosÞÞsin��=f ðr;�;Þ;

�122¼� 8l3ðcos�þcosÞþ l2r 22þ5cos2�þ6cosð��Þð

�þ5cos2þ10cosð�þÞÞþ24lr2ðcos�þcosÞþr3 17�cosð2ð�þÞÞð Þ

�=f ðr;�;Þ;

�123¼�1

32¼ 2l 4l2 cos�þ lrð3þcos2�þcosð��Þþ3cosð�þÞÞ��

þr2ð5cos��cosð�þ2ÞÞ��=f ðr;�;Þ; �1

33¼��123;

�124¼�1

42¼� 2l 4l2cosþ lrð3þcos2þcosð��Þþ3cosð�þÞÞ��

þr2ð5cos�cosð2�þÞÞ��=f ðr;�;Þ; �1

44¼�124; ð84Þ

�212 ¼�2

21 ¼�2�lðcos�þ cosÞþ rð3� cosð�þÞÞ

�ð3þ cosð�þÞÞ

�=f ðr;�;Þ;

�222 ¼

�4l�lðcos�þ cosÞþ rð3� cosð�þÞÞ

�ðsin�� sinÞ

�=f ðr;�;Þ;

�223 ¼�2

32 ¼��2l�lðsinð��Þþ sin2�� sinð�þÞÞþ rð5sin�þ sinð�þ2ÞÞ

��=f ðr;�;Þ;

�224 ¼�2

42 ¼�2l�lðsinð��Þ� sin2þ sinð�þÞÞ� rð5sinþ sinð2�þÞÞ

��=f ðr;�;Þ;

�233 ¼��2

23;

�244 ¼�2

24; ð85Þ

�312 ¼ �3

21 ¼

�4�lðcos � cos�Þ þ rð3þ cosð�þ ÞÞ

�sin2

� �þ

2

��.f ðr; �; Þ;

�322 ¼ �

�2�l2ð1þ cosð�� ÞÞ þ 4lr cos�þ r2ð3� cosð�þ ÞÞ sinð�þ Þ

�=f ðr; �; Þ;

�323 ¼ �3

32 ¼

�2�l2�cos

� 3��

2

�þ cos

� �þ

2

�� lr

�cos

� ��

2

�� 2 cos

� 3�þ

2

�þ cos

��þ 3

2

�� r2

�cos

� �þ

2

�þ cos

� 3ð�þ Þ

2

��sin��þ

2

��.f ðr; �; Þ;

�324 ¼ �3

42 ¼ ��2ðl2 þ 2lr cos �þ 3r2Þ cosþ lrð3þ cos 2Þ

�sin �

þ rð6r cos�þ lð4þ 2 cos 2�ÞÞ sin þ lðl þ r cos�Þ sin 2�=f ðr; �; Þ;

�333 ¼ ��3

23;

�344 ¼ �3

24; ð86Þ


�412 ¼ �4

21 ¼ �

�4�lðcos�� cosÞ þ rð3þ cosð�þ ÞÞ

�sin2

� �þ

2

��=f ðr; �; Þ;

�422 ¼ �

�2�l2ð1þ cosð�� ÞÞ þ 4lr cos þ r2ð3� cosð�þ ÞÞ sinð�þ Þ

��.f ðr; �; Þ;

�423 ¼ �4

32 ¼

�2�l2�cos

� 3��

2

�þ cos

� �þ

2

��þ lr

�5 cos

� ��

2

�þ cos

� 3�þ

2

�þ 2 cos

��þ 3

2

��þ 6r2 cos

��þ

2

��sin� �þ

2

��.f ðr; �; Þ;

�424 ¼ �4

42 ¼

�2�� l2

�cos

� �� 3

2

�þ cos

� �þ

2

��þ lr

�cos

� ��

2

�� 2 cos

��þ 3

2

�þ cos

� 3�þ

2

��þ r2

�cos

� �þ

2

�þ cos

� 3ð�þ Þ

2

��sin��þ

2

��.f ðr; �; Þ;

�433 ¼ �

�2r�3r cosþ lð2þ cos 2Þ

�sin �þ lðl þ r cosÞ sin 2�

þ�3lrþ lr cos 2�þ 2 cos �ðl2 þ 3r2 þ 2lr cosÞ

�sin

�=f ðr; �; Þ;

�444 ¼ �4

24;

ð87Þ

where

f ðr;�;Þ ¼ l2�10þ3cos2�þ2cosð��Þþ3cos2�2cosð�þÞ

�þ2lr

�7ðcos�þ cosÞ

þ cosð2�þÞþ cosð�þ2Þ�þ r2

�17� cosð2ð�þÞÞ

�:

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