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INTEGRABILITY AND STABILITY OF NONHOLONOMIC SYSTEMS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

By

Dmitry V. Zenkov, M.S.

*****

The Ohio State University 1998

Dissertation Committee: . , , Approved by Professor Anthony Bloch, Advisor Professor Alexander Dynin ~ , .... ' Advisor Professor Andrzej Derdzinski Department of Mathematis UMI Number: 9834105

UMI Microform 9834105 Copyright 1998, by UMI Company. All rights reserved.

This microform edition is protected against unauthorized copying under Title 17, United States Code.

UMI 300 North Zeeh Road Ann Arbor, MI 48103 ABSTRACT

In this thesis, methods of geometric mechanics are used to study the integrability and stability of nonholonomic systems (that is, mechanical systems with noninte- grable constraints such as rolling constraints). In the absence of external dissipation, such systems conserve energy, but nonetheless can exhibit neutrally stable and asymp­ totically stable, as well as linearly unstable relative equilibria. Unlike Hamiltonian dynamics, symmetries do not always lead to conservation laws as in the classical Noether theorem, but rather to a dynamic momentum equation. This momentum equation has the structure of a parallel transport equation for momentum corrected by additional terms and plays an important role in both integrability and stability analyses. For a number of systems, the momentum equation is pure transport. For such systems we show that the relative equilibria cannot be asymptotically stable and we find the energy-momentum conditions for stability of these equilibria. Often in this case of a pure transport momentum equation, the nonholonomic system is integrable. .A.mong these systems are the Routh problem and the rolling disk. We show that the invariant manifolds of the Routh problem are tori filled out with quasi-periodic motions. However, the way these tori are embedded in the phase space is quite different from that of the Liouville tori of integrable Hamiltonian systems. To carry out the stability analysis in the general non-pure transport case, we use a generalization of the energy-momentum method for holonomie systems combined with the Lyapunov-Malkin theorem and the center manifold theorem. We develop a

11 new energy-based approach, which allows one to find asymptotically stable equilibria. While this approach is consistent with the energy-momentum method for holonomie systems, it extends it in substantial ways. The theory is illustrated with several examples, including the rolling disk, the roller racer, and the rattleback top.

Ill To my parents

IV ACKNOWLEDGMENTS

I would like to thank my advisor, Tony Bloch, for taking the time to have numerous discussions with me and for his encouragement, support and patience. This thesis could never have been done without his enthusiasm. I am grateful to Jerry Marsden for his hospitality during my visit to Caltech, the interesting discussions, and his help. I learned a lot during this visit. It was my pleasure to work with Tony Bloch and Jerry Marsden on the iionholo- nomic stability project. I am thankful to Andrzej Derdzinski and Alexander Dynin for their discussions with me and to Greg Forest for his interest in my research and introducing me in 1993 to Tonv Bloch. VITA

October 12. 1962 ...... Bom—Moscow, Russia

1986 ...... M.S. Mechanics, Moscow State University

1986-1989 ...... Graduate School Department of Mechanics and Mathematics Moscow State University

1990-1993 ...... Assistant Professor, Moscow Technical State University 1993-present ...... Graduate Teaching Associate, Department of Mathematics. The Ohio State University

PUBLICATIONS

Research Publications

1 V.V. Kozlov and D.V. Zenkov, On Geometric Poinsot Interpretation for an n- dimensional Rigid Body, Tr. Semin. Vectom. Tenzom. Anal, 23 (1988). 202-204

2 D.V. Zenkov, On Asymptotic Stability of Periodic Motions in Nonholonomic Mechanics, Vestnik Moskov. Univ. Ser. I Math. Mekh., 3 (1989), 46-51

3 D.V. Zenkov, On the Routh Problem, Vestnik Moskov. Univ. Ser. I Math. Mekh., 3 (1991), 87-89

4 D.V. Zenkov, On the Problem of a Sphere Rolling over a Surface of Revolution, Vestnik Moskov. Univ. Ser. I Math. MeAh., 4 (1991), 94-96

VI 5 D.V. Zenkov, The Geometry of the Routh Problem, J. Nonlinear Sci. 5 (1995), 503-519

6 D.V. Zenkov, A.M. Bloch, and J.E. Marsden, The Energy-Momentum Method for Stability of Nonholonomic Systems, (to appear in Dynamics and Stability of Systems)

FIELDS OF STUDY

Major Field: Mathematics

Specialization: Applied Mathematics

Vll TABLE OF CONTENTS

Abstract ii

Acknowledgments v

V ita vi

List of Figures x

Introduction 1

1 Overview of Nonholonomic Mechanics and Stability Theory 5 1.1 The Hamilton Principle ...... 5 1.2 The Equations of Motion of Nonholonomic Systems with Symmetries 6 1.2.1 The Lagrange-d’Alembert Principle ...... 6 1.2.2 Sym m etries ...... 9 1.3 Examples of Nonholonomic System s ...... 9 1.3.1 The Rolling D isk ...... 10 1.3.2 The Routh Problem ...... 11 1.3.3 A Mathematical Exam ple ...... 11 1.3.4 The Roller R a c e r ...... 13 1.3.5 The Rattleback ...... 14 1.4 The Geometry of Nonholonomic Systems with Symmetry ...... 16 1.5 Lyapunov Stability T heory ...... 25 1.5.1 Main Definitions and T h eo rem s ...... 25 1.5.2 Center Manifold Theory in Stability Analysis ...... 27 1.6 The Energy-Momentum Method for Holonomie Systems...... 30 1.7 Invariant Manifolds ...... 32

2 The Routh Problem 35 2.1 Integrability of the Routh Problem ...... 35 2.2 Stability of Stationary Periodic Motions of the Routh Problem .... 40 2.3 Integral Manifolds of the Routh P ro b lem ...... 47

vm 3 An Energy-Momentum Method for Stability of Nonholonomic Systems 55 3.1 The Pure Transport Case ...... 57 3.1.1 The Nonholonomic Energy-Momentum M ethod ...... 58 3.1.2 The Rolling D i s c ...... 61 3.2 The Non-Pure Transport Case ...... 63 3.2.1 The Mathematical E x am p le ...... 63 3.2.2 The Nonholonomic Energy-Momentum M ethod ...... 67 3.2.3 The Roller R a c e r ...... 75 3.3 Nonlinear Stability by the Lyapunov-Malkin M ethod ...... 77 3.3.1 The Lyapunov-Malkin and the Energy-Momentum Methods 79 3.3.2 The R attleback ...... 82

Conclusion 86

Bibliography 87

IX LIST OF FIGURES

1.1 The geometry for the rolling disk ...... 10 1.2 The geometry for the roller racer ...... 14 1.3 The rattleback...... 15

2.1 Kinematic variables and axes of the Routh problem ...... 36

3.1 The manifolds Qto a-nd Qco ...... 69 INTRODUCTION

The theory of the motion of nonholonomic systems, which are mechanical systems subject to nonintegrable constraints, typified by rolling constraints, is remarkably rich. There is a large literature on nonholonomic systems and here we shall cite only a small part of it. For a more comprehensive listing, see Neimark and Fufaev [1972] and Bloch, Krishnaprasad, Marsden and Murray [1996], hereafter referred to as [BKMM]. As discussed in [BKMM] (and elsewhere), symmetries do not always lead to con­ servation laws as in the classical Noether theorem, but rather to an interesting mo­ mentum equation. This is one of the manifestations of the difference between the Euler-Lagrange equations for holonomie mechanical systems and the Lagrange-d’Al­ embert equations for nonholonomic systems, which are not variational in nature. Other work on symmetry and conservation laws may be found in Arnold [1988] and Bloch and Crouch [1992. 1995] for example. For the relationship between nonholo­ nomic systems and Hamiltonian structures, see Bates and Sniatycki [1993] and Koon and Marsden [1997a, b, c] and references therein. Another type of behavior which does not occur in unconstrained Hamiltonian and Lagrangian systems is that even in the absence of external forces and dissipation, nonholonomic systems may (but need not) possess asymptotically stable relative equilibria. This phenomenon was already known in the last century; cf. Walker [1896]. A key work on stability from our point of view may be found in Karapetyan [1980, 1983], which we shall specifically refer to in the course of this work. Integrable nonholonomic systems also demonstrate richer behavior than do Hamil­ tonian systems. Kozlov [1985] pointed out that invariant tori of integrable nonholo­ nomic systems are not necessarily filled out with quasi-periodic motions and stressed the importance of the presence of invariant measures in the theory" of integrable nonholonomic systems. Tatarinov [1988] discovered an example of an integrable non­ holonomic system with nontoric invariant manifolds. This work studies both integrability and stability of nonholonomic systems. In

Chapter 1 we review basics of nonholonomic mechanics and Lyapunov stability theorj'. In Chapter 2 we study the motion without sliding of a homogeneous sphere on a surface of revolution. Apparently Routh [1860] was the first to explore this problem. He described the family of stationary periodic motions and obtained a necessary condition for stability of these motions. Routh noticed as well that integration of the equations of motion may be reduced to integration of a system of two linear differential equations with variable coefficients and considered several cases when the equations of motion can be solved by quadratures. Solvability by quadratures was also considered in Bychkov [1965]. We discuss both integrability and stability of the Routh problem. In Section 2.1 we show that for an arbitrary surface of revolution the equations of motion have three integrals and are invariant under the action of the group 50(2). In Section 2.2 we prove (Theorem 2.2.2) that the necessary condition for stability of stationary periodic motions (relative equilibria) obtained by Routh is also a sufficient one. Routh in fact proves linear stability of these relative equilibria. We show that linear stability implies nonlinear orbital stability. In particular, we show that if the surface is an ellipsoid of revolution, then there is a critical value of the aspect ratio such that unstable solutions exist for ellipsoids with the aspect ratio greater then this critical value. It is important to stress that the Routh problem gives an example of a nonholonomic system which has stationary motions but does not have cyclic integrals. Stability analysis becomes more complicated in such situations (see articles Karapetyan and Rumyantsev [1990] and Mindlin and Pozharitskii [1965] and references therein). In Section 2.3 we prove that if the surface is compact and convex, and the curvature of the meridian is an analytic function of latitude, then almost all invariant manifolds are two-dimensional tori filled with quasi-periodic motions (Theorems 2.3.3, 2.3.6). As shown in Kozlov [1985] and Tatarinov [1988], this fact does not follow immediately from integrability, as it does in Hamiltonian mechanics. We also show (Theorem 2.3.10) that the rotation number of the flow on these tori is not a constant. Hermans [1995a, b] obtained a similar result using a different approach. He used the fact that the flow of the Routh problem is reversible to find invariant tori. We find these tori by studying the flow of the reduced system. This method is new. and it may be generalized for studying generic nonholonomic systems with symmetries. We intend to do this in future research. The main goal of Chapter 3 is to analyze the stability of relative equilibria for nonholonomic mechanical systems with symmetry using an energy-momentum analy­ sis for nonholonomic systems that is analogous to that for holonomie systems given in Simo, Lewis, and Marsden [1991]. The energy-momentum method has proven to be effective for stability analysis of Hamiltonian systems. The methods used by different authors for stability analysis of relative equilibria of nonholonomic systems may be split into two groups: energy-based methods and linearization methods. See Kara­ petyan and Rumyantsev [1990] for details. The energy methods were used for finding nonasymptotically stable (relative) equilibria of nonholonomic systems with cyclic variables. We generalize this approach for systems with non-abelian symmetries. The linearization methods were used for studying asymptotically stable relative equilibria. We develop here a new energy-based approach which allows us to find asymptotically stable equilibria and compare it with linearization methods. We combine the techniques of Chapter 2 and of [BKMM] and show that the stability types of the relative equilibria depend on the structure of the momentum equation. The momentum equation has the structure of a parallel transport equation for the momentum corrected by additional terms. This parallel transport occurs in a certain vector bundle over shape space. In some instances, such as the Routh problem (see Chapter 2), this equation is pure transport, and in fact is integrable (the curvature of the transporting connection is zero). This leads to nonexplicit conservation laws. In other important instances, the momentum equation is partially integrable in a sense that we shall make precise. Our goal is to make use of, as far as possible, the energy momentum approach to stability for Hamiltonian systems. This method goes back to fundamental work of Routh (and many others in this era), and in more modern works, that of Arnold [1966], Smale [1970], and Simo, Lewis and Marsden [1991] (see for example, Marsden [1992] for an exposition and additional references). Because of the nature of the momentum equation, the analysis we present is rather different from the holonomie case in several important respects. In particular, our energv- momentum analysis varies according to the structure of the momentum equation and. correspondingly, we divide our analysis into several parts. We list the underlying assumptions that we are making at the beginning of the relevant section. We illustrate our energy-momentum stability analysis with a low-dimensional model example, and then with several mechanical examples of interest, including the falling disk, the roller racer, and the rattleback top. We note also that not all nonholonomic systems fall into the categories discussed here. We indicate in the main body of the thesis precisely which systems are covered by our analysis. CHAPTER 1

OVERVIEW OF NONHOLONOMIC MECHANICS

AND STABILITY THEORY

In this chapter we discuss basics of the dynamics of nonholonomic systems and of the stability theory.

1.1 The Hamilton Principle

In this section we introduce unconstrained Lagrangian systems and obtain the equa­ tions of motion of these systems. An unconstrained Lagrangian system consists of a smooth configuration man­ ifold Q with coordinates q = ..., g") and a Lagrangian L : TQ M.. Co­ ordinates q induce coordinates (ç, q) on the phase space TQ. The dimension of the configuration manifold is referred to as the number of degrees of freedom of the Lagrangian system. The Lagrangian is a smooth function satisfying the condition

In mechanics, the Lagrangian equals the kinetic energy minus the potential energy. The Hamilton principle states that along the motion

b 5 J L{q{t),q{t))dt = 0. (1.1) More precisely, in (1.1) we consider all curves q : [a,h] Q connecting two fixed points Qa and % in the configuration space. The set of these curves is called the path space from Ça to % and is denoted fi(ça, %). We then define a map I : %) R by b I{l) = J L{'y{t)n{t))dt. a The condition (1.1) then states that the motions from Qa to % are critical points of the map I. Computing (1.1), we obtain

b /(I? “II?)

(see for example Marsden and Ratiu [1994] for details) and thus ( 1.1 ) is equivalent to the Euler~Lagrange equations

d dL d l , , ‘ = (1-2 )

1.2 The Equations of Motion of Nonholonomic Systems with Symmetries

In this section we briefly discuss the mechanics of nonholonomic systems with sym­ metries. We make use of the approach introduced in [BKMM]. The key equations are (1.12) and (1.13) in a body frame, given below. It is the form of these equations that determines the stability and dynamics of the systems analyzed here.

1.2.1 The Lagrange-d’Alembert Principle

We now describe briefly the equations of motion for a nonholonomic system, following the notation of [BKMM]. We confine our attention to nonholonomic constraints that are homogeneous in the velocity. Accordingly, we consider a configuration space Q and a distribution V that describes these constraints. Recall that a distribution D is a collection of linear subspaces of the tangent spaces of Q] we denote these spaces by Vg C TgQ, one for each q Q. A curve q{t) € Q will be said to satisfy the construints if q{t) € Vg(t) for all t. This distribution will, in general, be nonintegrable; i.e., the constraints are. in general, nonholonomic.

Consider a Lagrangian L : TQ —> R. In coordinates ç'.i = 1,... ,n, on Q with induced coordinates (g',ç‘) for the tangent bundle, we write L(ç',ç‘). The equations of motion are given by the following Lagrange-d’Alembert principle.

Definition 1.2.1 The Lagrange-d’Alembert equations of motion for the sys­ tem are those determined by b 6 J L(q\q’)dt = 0, where we choose variations ôq{t) of the curve q{t) that satisfy Sq(a) = 0q(b) = 0 and

6q{t) 6 T>g(t) for each t where a < t < b.

This principle is supplemented by the condition that the curve itself satisfies the constraints. Note that we take the variation before imposing the constraints; that is. we do not impose the constraints on the family of curves defining the variation. This is well known to be important to obtain the correct mechanical equations. If instead we impose the constraints first and then take the variation, i. e. apply Hamil­ ton principle to the constrained system, we obtain different equations of motion. The corresponding dynamics is known as vakonomic or variational nonholo­ nomic mechanics and is remarkably different from the dynamics determined by the Lagrange-d’.A.lembert principle—see Bloch and Crouch [1993]. However, if the con­ straint distribution is integrable, i.e. the constraints are nonholonomic, the Hamilton principle and the Lagrange-d’Alembert principle are equivalent. See Arnold [1988] and [BKMM] for a discussion and references. The usual arguments in the calculus of variations show that the Lagrange-d’Al­ embert principle is equivalent to the equations

= (IB " B) ° for all variations Sq such that Sq 6 Vq at each point of the underlying curve q{t). One can of course equivalently write these equations in terms of Lagrange multipliers.

Let (a;“,a = 1,... ,p} be a set of p independent one forms whose vanishing de­ scribes the constraints. Choose a local coordinate chart q = (r, s) € R"“P x W, which

we write as = (r“, s“), where 1 < a < n — p and 1 < a < p such that

cj“(ç) = ds“ + A“ (r, s)dr°

for all a = l,...,p . In these coordinates, the constraints are described by vectors

= (u“, r “) satisfying -I- = 0 (a sum on repeated indices over their range is understood). The equations of motion for the system are given by (1.3) where we choose variations Sq{t) that satisfy the constraints, i.e., a;“(ç) • Sq = Q, or equivalently, d.s“ -I- = 0, where Sq^ = (

dt df° dr°‘ ) “ \ dt 5s“ y

for all a = 1,..., n — p. Equation (1.4) combined with the constraint equations

s“ = -A “r “ (1.5)

for all a = 1,... ,p gives the complete equations of motion of the system. .\ useful way of reformulating equations (1.4) is to define a “constrained” Lagran­ gian by substituting the constraints (1.5) into the Lagrangian:

Lc(r“, s“, r“) := L(r“, s“, r“, -A “ (r, s)r“).

The equations of motion can be written in terms of the constrained Lagrangian in the following way, as a direct coordinate calculation shows: d dLc dLc _ ^ a dLc _ dL

where is defined by

t dr^ dr^ “ 9s“ ^ 5s“ Geometrically, the .4“ are the coordinate expressions for the Ehresmann con­ nection on the tangent bundle defined by the constraints, while the are the corresponding curvature terms (see [BKMM]).^

1.2.2 Symmetries

As we shall see shortly, symmetries play an important role in our analysis. We begin here with just a few preliminary notions. Suppose we are given a nonholonomic system with the Lagrangian L : TQ R, and a (nonintegrable) constraint distribution V. We can then look for a group G that acts on the configuration space Q. It induces an action on the tangent space TQ and so it makes sense to ask that the Lagrangian L be invariant. Also, one can ask that the distribution be invariant in the sense that the action by a group element g ^ G maps the distribution P , at the point q E Q to the distribution Vgq at the point gq. If these properties hold, we say that G is a symmetry group. In many examples, the symmetry group will be evident. For example, for systems rolling on the plane, the group of Euclidean motions of the plane, SE{2) will be appropriate.

1.3 Examples of Nonholonomic Systems

Here we introduce nonholonomic systems which illustrate the above definitions. We

will study stability of relative equilibria of these systems in Chapters 2 and 3.

^Strictly speaking, an Ehresmann connection requires one to have a bundle for which the distri­ bution is regarded as the horizontal space but in fact, the bundle structure is not required if one regards the connection one form to be not vertical valued, but rather TQ/D-valued. 1.3.1 The Rolling Disk

A classical example of a nonholonomic system is a disk rolling without sliding on the xy-plane, as in Figure 1.1.

Figure 1.1: The geometry for the rolling disk.

.As the figure indicates, we denote the coordinates of contact of the disk in the xy-plane by {x, y) and let 0 , 0 , and i) denote the angle between the plane of the disk and the vertical axis, the “heading angle” of the disk, and “self-rotation” angle of the disk respectively. In [BKMM], the vertical rolling disk was considered, but we consider a disk that can “fall”. .A classical reference for the rolling disk is Vierkandt [1892] who showed that on the reduced space V/SE{2)—the constrained velocity phase space modulo the action of the Euclidean group SE[2)—all orbits of the system are periodic. Modern references that treat this example are Hermans [1995a] and O’Reilly [1996]. For the moment, we just give the Lagrangian and constraints, and return to this example later on. As we will eventually show, this is a system which exhibits stability but not asymptotic stability. Denote the mass, the radius, and the moments of inertia of the disk by m. R, .4, B respectively. The Lagrangian is given by the kinetic minus

10 potential energies:

L = Y [(^ ~ iZ(0sin0 + i))Ÿ + sin^ ^ + (t^cos^ +

+ i \a {&^ + cos^ 0) + S (0 sin 0 — mgR cos 6. where Ç = x cos

X = —i/fR cos 0 ,

g = —'0i2sin0.

Note that the constraints may also be written as Ç = 0, 77 = 0.

1.3.2 The Routh Problem

Another classical example of a nonholonomic system is Routh’s problem of a sphere rolling inside of a surface of revolution. We study this system in Chapter 2. We obtain the equations of motion following the approach of Routh, which is different from the general procedure given below in Section 1.4. The Routh problem turns out to be integrable. We show that a generic motion is quasi-periodic on two-dimensional tori. We shall see that the Routh problem and the rolling disc both may have stable, but cannot have asymptotically stable, relative equilibria. We later employ the approach used for stability analysis of the Routh problem for deriving the general nonholonomic energy-momentum method (see Chapter 3).

1.3.3 A Mathematical Example

We now consider an instructive, but (so far as we know) nonphysical example. Un­ like the rolling disk, it has asymptotically stable relative equilibria, and is a simple example that exhibits the richness of stability in nonholonomic systems. Our general theorems presented later are well illustrated by this example and the reader may find it helpful to return to it again later.

11 Consider a Lagrangian on TK^ of the form

L(r\r^,s,f\r^,s) = i {(1 - - 2o(r^) 6(r^)f^r^

+ (1 - [6(r^)]2)(r2)2 + s2| _ (1.6 )

where a, b, and V are given real valued functions of a single variable. We consider the nonholonomic constraint

s = a(r^)r^ + 6(r^)r^. (1.7 )

Using the definitions, straightforward computations show that B 12 = drib = —B^i- The constrained Lagrangian is Lc = j {(f^)^ + (r^)^)} — V{r^) and the equations of motion, namely, (d/dt){draLc) — 5r<»Lc = —sBagr^ become

d dLc dLc _ .2 d d L c 1 d t 5ri 9ri “ ^ ’ d t d f^ ~ ' The Lagrangian is independent of and correspondingly, we introduce the non­ holonomic momentum defined by

dL c

We shall review the nonholonomic momentum later on in connection with general symmetries, but for now just regard this as a definition. Taking into account the constraint equation and the equations of motion above, we can rewrite the equations of motion in the form

^ (a(r^)r^ + 6(r^)p) p, (1.8)

p = ^ [a{r^)f^ + b{r^)p) . ( 1.9)

Observe that the momentum equation does not, in any obvious way, imply a conser­ vation law. A relative equilibrium is a point (ro,po) that is an equilibrium modulo the variable r"; thus, from the equations (1.8) and (1.9), we require = 0 and

+ ^^(^o)Po = 0 -

12 We shall see that relative equilibria are Lyapunov stable and in addition asymptoti­ cally stable in certain directions if the following two stability conditions are satisfied:

(i) the energy function E = -I- -f V, which has a critical point at (ro,po), has a positive definite second derivative at this point.

(ii) the derivative of E along the flow of the auxiliary system

1 dV 9b , I I Lf \ r = r “ ^ (a(r^)r^ + b{r^)p) p, p = ■^b{r^)pr^ dr^ dr^ ’ dr^

is strictly negative.

1.3.4 The Roller Racer

We now consider a tricycle-like mechanical system called the roller racer, or the Tennessee racer, that is capable of locomotion by oscillating the front handlebars. This toy was studied using the methods of [BKMM] in Tsakiris [1995]. The methods here may be useful for modeling and studying the stability of other systems, such as aircraft landing gears and train wheels. The roller racer is modeled as a system of two planar coupled rigid bodies (the main body and the second body) with a pair of wheels attached on each of the bodies at their centers of mass. We assume that the mass and the linear momentum of the second body are negligible, but that the moment of inertia about the vertical axis is not. See Figure 1.2. Let (x, y) be the location of the center of mass of the first body and denote the angle between the x-axis of the inertial reference frame and the line passing through the center of mass of the first body by 9, the angle between the bodies by 0 , and the distances from the centers of mass to the joint by d\ and dg. The mass of body 1 is denoted m and the inertias of the two bodies are written as I\ and Jg- The Lagrangian and the constraints are

L = -m(x^ 4- y^) + 4-

13 Figure 1.2: The geometry for the roller racer. and

\ sm 0 sm 0 / in A \ sin é sin é respectively. The configuration space is SE{2) x 50(2). The Lagrangian and the constraints are invariant under the left action of SE{2) on the first factor of the configuration space. We shall see later that the roller racer has a two-dimensional manifold of equilibria and that under a suitable stability condition some of these equilibria are stable modulo

5E(2) and in addition asymptotically stable with respect to 0 .

1.3.5 The Rattleback

.A. rattleback is a convex nonsymmetric rigid body rolling without sliding on a hori­ zontal plane. It is known for its ability to spin in one direction and to resist spinning in the opposite direction for some parameter values, and for other values, to exhibit multiple reversals. See Figure 1.3. Basic references on the rattleback are Walker [1896], Karapetyan [1980, 1981], Markeev [1983, 1992], Pascal [1983, 1986], and Bondi [1986]. We adopt the ideal model (with no energ}' dissipation and no sliding) of these references and within that context.

14 Figure 1.3: The rattleback.

no approximations are made. In particular, the shape need not be ellipsoidal. Walker did some initial stability and instability investigations by computing the spectrum while Bondi extended this analysis and also used what we now recognize as the momentum equation. (See Burdick, Goodwine and Ostrowski [1994]). Karapetyan carried out a stability analysis of the relative equilibria, while Markeev’s and Pascal’s main contributions were to the study of spin reversals using small parameter and averaging techniques.

Introduce the Euler angles 6, 0 , -ip using the principal axis body frame relative to an inertial reference frame. These angles together with two horizontal coordinates (x, Î/) of the center of mass are coordinates in the configuration space SO{3) x of the rattleback. The Lagrangian of the rattleback is computed to be

L = (i^ + ^ [a cos^ il) + B sin^ ip + m (7 i cos ^ ( sin 9Ÿ\ 9^

+ ^ [(.4 sin^ ip + B cos^ ij)) sin^ 9 -\-C cos^ 0] ^ [C + m'yl sin^ 0]

+ m(7 i cos 0 — C sin 0)72 sin 9 9ip + {A — B) sin 9 sin ip cos ip 9é

+ C cos 9^ip + mg{yi sin ^ 4- ( cos 9), where .4. B, and C are the principal moments of inertia of the body, m is the total mass of the body, (^, 77, C) are coordinates of the point of contact relative to the body frame, 71 = ^sin^ + rjcosip, and 72 = ^ cos ip — t] sin ip. The shape of the body is

15 encoded by the functions 77 and C- The constraints are

X = a i d + Q2'0 + 0:3 ÿ = + ,^20 + where

a i = — (71 sin 0 + C cos d) sin <^,

»2 = 72 cos 0 sin 0 + 7 i cos 0,

0:3 = 72 sin 0 + (71 cos 0 — C sin 0 ) cos 0 ,

/5jfc = A: = 1,2,3.

The Lagrangian and the constraints are 5f^(2)-invariant, where the action of an element (a, 6, a) 6 SE{2) is given by

(x, 7/, 0 ) (x cos a — y sin a + a, X sin Q + y cos q + 6 ,0 + a ) .

Corresponding to this invariance, 77, and C are functions of the variables d and xl) onb^

1.4 The Geometry of Nonholonomic Systems with Symme­ try

Consider a nonholonomic system with the Lagrangian L : TQ —» E, the (noninte- grable) constraint distribution P, and the symmetry group G in the sense explained previously.

Orbits and Shape Space. The group orbit through a point q, an (immersed) submanifold, is denoted

Orb(y) := {gq \ g e G}.

Let g denote the Lie algebra of the Lie group G. For an element ^ E g, we write Çg, a vector field on Q for the corresponding infinitesimal generator. The tangent space

16 to the group orbit through a point q is given by the set of infinitesimal generators at that point:

T,(Orb(ç)) = {^q (ç) K € g}.

Throughout this work, we make the assumption that the action of G on Q is free and proper. The quotient space M = Q/G, whose points are the group orbits, is called shape space. Under these circumstances, shape space is a smooth manifold and the projection map tt : Q -> Q/G is a smooth surjective map with a surjective derivative T,7t at each point. The kernel of the linear map Tqir is the set of infinitesimal generators of the group action at the point q, i.e.,

kevTgir = {^q (ç) | ( € g} , so these are also the tangent spaces to the group orbits.

Reduced Dynamics. .Assuming that the Lagrangian and the constraint distribu­ tion are G-invariant, we can form the reduced velocity phase space TQjG and the reduced constraint space V/G. The Lagrangian L induces well defined functions, the reduced Lagrangian

I : TQ/G -> R satisfying L = I o tttq where tttq : TQ -> TQjG is the projection, and the c o n ­ strained reduced Lagrangian

Ic '. D f G —y IR, which satisfies L\t> = Ic ° t^v where ttx) : V V/G is the projection. By general considerations, the Lagrange-d’Alembert equations induce well defined reduced La­ grange-d’Alem bert equations on V/G. That is, the vector field on the manifold V determined by the Lagrange-d’Alembert equations (including the constraints) is G-invariant, and so defines a reduced vector field on the quotient manifold V/G.

17 The nonholonomic m omentum map. Consider the vector bundle

Define, for each q E Q, the vector subspace g’ to be the set of Lie algebra elements in g whose infinitesimal generators evaluated at q lie in Sqi

8" = {( € g I € 5,}.

The corresponding bundle over Q whose fiber at the point q is given by g^, is denoted g^. Consider a section of the vector bundle S over Q; i.e., a mapping that takes q to an element of 5, = Vq n 7^(0rb(g)). Assuming that the action is free, a section of S can be uniquely represented as and defines a section of the bundle g^.

Definition 1.4.1 The n onholono m ic m o m en tu m m ap is the bundle map taking TQ to the bundle (g^)* whose fiber over the point q is the dual of the vector space g*^ that is defined by

(r>^(v,u) = where G g'^. Intrinsically, this reads

= {WL(v ,U q ) , where ¥L is the fiber derivative of L and where Ç € g“^. For notational convenience, especially when the variable Vq is suppressed, we will often write the left-hand side of this equation as

Notice that the nonholonomic momentum map may be viewed as giving just some of the components of the ordinary momentum map, namely along those symmetry directions that are consistent with the constraints. For a nonholonomic system, the momentum map need not be a constant of motion. The following theorem states instead that the momentum map satisfies a certain equation.

18 Theorem 1.4.2 Assume that the Lagrangian and the constraint distribution are G- invariant, and that is a section of the bundle Then any solution of the Lagran­ ge-d’Alembert equations for a nonholonomic system must satisfy, in addition to the given kinematic constraints, the m o m en tu m equation:

d L \d _ = _ Q When the momentum map is paired with a section in this way, we will just refer to it as the m o m en tu m .

The momentum in body representation. Let a local trivialization be chosen on the principle bundle tt : Q Q/G, with a local representation having components denoted (r,g). Let r, an element of shape space Q/G, have coordinates denoted r“, and let g be group variables for the fiber, G. In such a representation, the action of G is the left action of G on the second factor. Put = L{r,g,f,g).

Choose a ç-dependent basis ea{q) for the Lie algebra such that the first m elements span the subspace g^ in the following way. First, one chooses, for each r. such a basis at the identity element g = Id, say

ei (r), C2(r),..., e„i(r), e,n+i(r), et(r).

For example, this could be a basis whose generators are orthonormal in the kinetic energ}- metric. Now define the body fixed basis by

^a{r,g) = Adj,ea(r), then the first m elements will indeed span the subspace g'' since the distribution is invariant. Define the components of the momentum in body representation to be

Pb := .

19 Thus, we have g, f, g) = Ad;_ip(r, f, ().

This formula explains why p and are the body momentum and the spatial mo­ mentum respectively.

The Nonholonomic Connection. Assume that the Lagrangian has the form ki­ netic energy minus potential energy, and that the constraints and the orbit directions span the entire tangent space to the configuration space ([BKMM] call this the "di­ mension assumption”):

V, + TgiOxh{q))=T,Q. (1.10)

In this case, the momentum equation can be used to augment the constraints and provide a connection on Q —> Q/G.

Definition 1.4.3 Under the dimension assumption in equation (1.10), and the as­ sumption that the Lagrangian is of the form kinetic minus potential energies, the nonholonom ic connection A is the connection on the principal bundle Q Q/G whose horizontal space at the point q E Q is given by the orthogonal complement to the space Sq within the space Vq.

Let E(ç) : g® (s^)* be the locked inertia tensor relative to g^. defined by

(:(?)(, ^) = ^,7? G g'', where ((•. •)) is the kinetic energy metric. Define a map : TqQ Sq = Vq D r,(Orb(g)) given by

This map is equivariant and is a projection onto Sq. Choose Uq C T,(Orb(g)) such that 7)y(0rb(g)) = Sq®Uq. Let .4,'" : TqQ —^Uq he a. Uq valued form that projects Uq onto itself and maps Vq to zero; for example, it can be given by orthogonal projection relative to the kinetic energy metric (this will be our default choice).

20 Proposition 1.4.4 The nonholonomic connection regarded as an Ehresmann con­ nection is given by

A = A*'*" + ( 1.11)

When the connection is regarded as a principal connection (i.e., takes values in the Lie algebra rather than the vertical space) we will use the symbol A .

Given a velocity vector q that satisfies the constraints, we orthogonally decompose it into a piece in Sq and an orthogonal piece denoted f^. We regard as the horizontal lift of a velocity vector f on the shape space; recall that in a local trivialization. the horizontal lift to the point (r, g) is given by

f" = i t - A ^ f ) = (r“ ,

where A% are the components of the nonholonomic connection which is a principal connection in a local trivialization. We will denote the decomposition of q by

q = O.Q{q)+f‘^, so that for each point ç, Q is an element of the Lie algebra and represents the spatial angular velocity of the locked system. In a local trivialization, we can write, at a point {r,g).

n = .A.dg(f 2ioc), so that fiioc represents the body angular velocity. Thus,

f^Ioc — Aioc^ 4" ^ and. at each point q, the constraints are that f 2 belongs to i.e.,

fiioc e span{ei(r),e 2(r),...,e,„(r)}.

21 The vector r* need not be orthogonal to the whole orbit, just to the piece Sq. Even if q does not satisfy the constraints, we can decompose it into three parts and write

q = Qqiq) + = fîÿ(g) + 0 ^(9 ) +

where Qq and Qq are orthogonal and where E Sq. The relation Qioc = A\ocr+^ is valid even if the constraints do not hold; also note that this decomposition of corresponds to the decomposition of the nonholonomic connection given by .4 = ^km ^ ^sym given in equation ( 1.11). To avoid confusion, we will make the following index and summation conventions

1. The first batch of indices range from 1 to m corresponding to the symmetry directions along constraint space. These indices will be denoted a. b.c.d....

and a summation from 1 to m will be understood.

2. The second batch of indices range from m +1 to k corresponding to the symmetry' directions not aligned with the constraints. Indices for this range or for the

whole range 1 to A: will be denoted by a', 6', ( /,... and the summations will be given explicitly.

3. The indices a, 3,... on the shape variables r range from 1 to a. Thus, cr is the dimension of the shape space Q/G and so a = n — k. The summation convention for these indices will be understood.

.According to [BKMM]. the equations of motion are given by the next theorem.

Theorem 1.4.5 The following reduced nonholonom ic Lagrange-d’Alem bert

é q u a t i o n s hold for each 1 < a < cr and I < b < m:

22 Here -, f ° , Pa) is the constrained Lagrangian; r°, 1 < a < cr, are coordinates in

the shape space; Pa, 1 ^ ol < m, are components of the momentum map in the body representation, Pa = {die/dÇl\oc, edr)); are the components of the inverse locked inertia tensor; are the local coordinates of the curvature B of the nonholonomic connection A; and the coefficients Vapb, l^ad-r given by the formulae

k k -DL = '£ -c i,A i + yt,+ 2 2 a'=l a'=m+l k / k ^ A.,. o '= m + l \ bf = l k

a' = l where

\ I I A b ' A b ' Aa'a = la'a ~ 2 ^ U-b-A^ := ~ 2 ^ gc.'gc 6"^o y=i ^ 6'=! ^ ^ for a' = m + \,... ,k. Here are the structure constants of the Lie algebra defined by [ea'.Cc'] = C^l^ey, a', b',d — , k; and the coefficients 7^^ are defined by

d^b ^ o' (/ = !

A relative equilibrium is an equilibrium of the reduced equations; that is, it is a solution that is given by a one parameter group orbit, just as in the holonomie case (see. e.g., iVIarsden [1992] for a discussion).

The Constrained Routhian. This function is defined by analogy with the usual Routhian by R(r“, r“, pd = Zc(r", r “, / “% ) - / “W , and in terms of it, the reduced equations of motion become

'dt'di^ ~ d i^ ~ - '^ L ^ ‘"^PcPd - BlgPcT-^ - Vpabl’^^'Pcr^ - K.aSyf^f'', (1-12)

j^Pb = ClJ'"^PcPd + 'DI^PcT° + • (1.13)

23 The Reduced Constrained Energy. As in [BKMM], the kinetic energ>" in the variables (r“, r“, equals

1 1 ^ \ 1 ^ Oa'a-/aV^^)î^“V“ + ^ ^ (1.14) a'=m+l o',c'=Tn+l where Ça3 are coefBcients of the kinetic energy metric induced on the manifold Q/G. Substituting the relations f2“ = / “*p& and the constraint equations O"' = Q in (1.14) and adding the potential energy, we define the function E by

E = ^ W r" r^ + [/(r",P.), (1.15) which represents the reduced constrained energy in the coordinates (r, r,p), where U{r‘^,Pa) is the am ended potentiaZ defined by

C /(r",pJ = ^ / “VaP 6 + l^(r“), (1-16)

and V{ t°‘) is the potential energy of the system. Now, we show that the reduced constrained energy is conserved along the solutions of (1.12), (1.13).

Theorem 1.4.6 The reduced constrained energy is a constant of motion.

P ro o f One way to prove this is to note that the reduced energy is a constant of motion, because it equals the energy represented in coordinates (r, f, p, g) and because the energy is conserved, since the Lagrangian and the constraints are time-invariant. .Along the trajectories, the constrained energy and the energy are the same. Therefore, the reduced constrained energy is a constant of motion. One may also prove this fact by a direct computation of the time derivative of the reduced constrained energy (1.15) along the vector field defined by the equations of motion. ■

Skew Symmetry Assumption. IVe assume that the tensor is skew-symme­ tric in c. d.

24 We remark firstly that this assumption implies that the dimension of the fam ­ ily of the relative equilibria equals the number of components of the (nonholonomic) momentum map. This is important for our energetic approach to the analysis of stability. We note also that this assumption holds for most physical examples and certainly the systems discussed in this paper. (Exceptions include systems with no shape space such as the homogeneous sphere on the plane and certain cases of the Suslov problem of a nonhomogeneous rigid body subject to a linear constraint in the angular veloci­ ties.) It is an intrinsic (coordinate independent) condition, since represents an intrinsic bilinear map of ( 0^)* x ( 0®)* to

Under this assumption, the terms quadratic in p in the momentum equation van­ ish. and the equations of motion become

0 (1.17)

+ D a a if f '. (1.18)

In the case when = 0 the matrix vanishes, and the preceding equations of motion are the same as those obtained by Karapetyan [1983]. Note that = 0 when the indices a,6 , c range over an abelian part of the symmetry group. This occurs for all examples discussed here.

1.5 Lyapunov Stability Theory

In this section we state some key results of stability theory. We also discuss stability of nonhyperbolic equilibria.

1.5.1 Main Definitions and Theorems

Consider a system of differential equations

x = X(x), (1.19)

25 where z G R" and X{x) is a smooth vector field on R” . We say that Zg is an equilibrium of (1.19) if %(zg) = 0. Let |z| be the standard norm in R” .

Definition 1.5.1 An equilibrium Zg of system (1.19) is called Lyapunov stable if for any e > 0 there exists 5 > Q such that for any positive t and for any a satisfying |a — Zg| < S the solution of (1.19) with the initial condition z(0) = a exists and |z(t) — Zg| < e. If in addition x{t) —^ Zg as t —> oo, then the equilibrium Zg is said to be asymptotically stable.

Lyapunov developed two principal approaches to stability analysis of (isolated) equilibria of systems of differential equations. One approach is based on linearization of equations (1.19):

Theorem 1.5.2 If all eigenvalues of the linearization of system (1.19) at the equi­ librium Zg have negative real parts, then this equilibrium is asymptotically stable.

Another approach is baised on a construction of a special function called the L y a ­ punov function:

Theorem 1.5.3 If there exists a function V{x) such that

(i) V is positive definite in some neighborhood of the equilibrium Zg,

(ii) The flow derivative o fV along trajectories of (1.19) is non-positive, then the equilibrium Zg is Lyapunov stable. If the flow derivative is negative definite, then the equilibrium is asymptotically stable.

Lyapunov also developed a method for stability analysis of non-isolated equilibria. We need here a special case of his theory, namely, a case when some eigenvalues of the linearization of system (1.19) are equal to zero while the rest of the eigenvalues have non-zero real parts. We discuss this case in the next section.

26 1,5.2 Center Manifold Theory in Stability Analysis

Here we discuss center manifold theory and its applications to the stability analysis of nonhyperbolic equilibria. We begin by assembling some preliminary results on center manifold theory and show how they relate to the Lyapunov-Malkin theorem. The center manifold theorem provides new and useful insight into the existence of integral manifolds. These integral manifolds play a crucial role in our analysis. Lyapunov’s original proof of the Lyapunov-Malkin theorem used a different approach to proving the existence of local integrals, as we shall discuss below. Malkin extended the result to the nonautonomous case. Consider a system of differential equations

x = Ax + X{x,y), (1.20)

y = By + Y{x,y), (1.21) where x € US'", y 6 K", and ,4 and B are constant matrices. It is supposed that all eigenvalues of ,4 have nonzero real parts, and all eigenvalues of B have zero real parts.

The functions X, Y are smooth, and satisfy the conditions %(0,0) = 0 , (0, 0 ) = 0 , 1(0,0) = 0. d y (0,0) = 0. We now recall the following definition:

Definition 1.5.4 A smooth invariant manifold of the formx = h{y) where h satisfies h{0) = 0 and dh{0) = 0 is called a center m anifold.

We are going to use the following version of the center manifold theorem following the exposition of Carr [1981] (see also Chow and Hale [1982]).

Theorem 1.5.5 (The center manifold theorem) If the functions X (x, y), Y [x, y) are C*’, k >2, then there exists a (local) center manifold for (1.20), (1.21), x = h{y), |y| < 6. where h is . The flow on the center manifold is governed by the system

ÿ = By-\-Y{h{y),y). (1.22)

27 The next theorem explains that the reduced equation (1.22) contains information

about stability of the zero solution of ( 1.20), (1.2 1 ).

Theorem 1.5.6 Suppose that the zero solution of (1.22) is stable (asymptotically stable) (unstable) and that the eigenvalues of A are in the left half plane. Then the zero solution of (1.20), (1-21) is stable (asymptotically stable) (unstable).

Let us now look at the special case of (1.21) in which the matrix B vanishes. Equations (1.20), (1.21) become

x = Ax + X (i,y ), (1.23)

y = Y{x,y). (1.24)

Theorem 1.5.7 Consider the system of equations (1.23), (1.24). If X{0,y) = 0, l'(0, y) = 0. and all eigenvalues of the matrix A have negative real parts, then system (1.23), (1.24) hos n local integrals in the neighborhood o/x = 0, y = 0.

P ro o f The center manifold in this case is given by x = 0. Each point of the center manifold is an equilibrium of system (1.23), (1.24). For each equilibrium point (0. y) of our system, consider a stable manifold S^{y). The center manifold and these manifolds S^{y) can be used for a (local) substitution (x, y) -)■ (x,ÿ) such that in the new coordinates the system of diflferential equations become

i = Àx + X{x,ÿ), p = 0.

The last system of equations heis n integrals ÿ = const, so that the original equation has n smooth local integrals. Observe that the tangent spaces to the common level sets of these integrals at the equilibria are the planes y = yo. Therefore, the integrals are of the form y = /(x , k), where dxf{0,k) = 0 . ■

The following theorem gives stability conditions for equilibria of system (1.23), (1.24):

2 8 Theorem 1.5.8 (Lyapunov-Malkin) C o n s i d e r the system of differential equations

(1.23), (1 .2 4 ), where x € K"*, y € R", A is anm x- m-matrix, and X{x,y), Y[x,y) represent nonlinear terms. If all eigenvalues of the matrix A have negative real parts,

and X{x,y), Y{x,y) vanish when x = 0, then the solution x = 0, y = 0 of system

(1.23), (1 .2 4 ) is stable with respect to {x,y), and asymptotically stable with respect

to X. If a solution {x{t),y{t)) of (1.23), (1.24) close enough to the solution x = 0,

y = 0 , then

lim x{t) = 0 , lim y{t) = c. £—>00 £—>oo

P ro o f .A.ccording to Theorem 1.5.7, the phase space of system (1.23), (1.24) is

locally represented as a union of invariant leaves = {y = /(z , c)}. We use x as

local coordinates on these leaves. On each leaf we have a reduced system x = F c(x ), where F{x) = Ax + X{x, f{x,c)). Since detA # 0, each reduced system has an

isolated equilibrium x = 0 on a corresponding leaf. The equilibrium of the system

reduced to a leaf passing through x = 0 , y = 0 is asymptotically stable because the matrix of the linearization of this reduced system is A. To finish the proof, we notice that the equilibria of systems on nearby leaves are asymptotically stable as well because corresponding matrices Ac have all eigenvalues in the left half plane for

|c|

Historical Note. The proof of the Lyapunov-Malkin theorem uses the fact that the system of differential equations has local integrals, as discussed in Theorem 1.5.7. To prove existence of these integrals, Lyapunov uses a theorem of his own about the existence of solutions of PDE’s. He does this assuming that the nonlinear terms on the right-hand sides are series in x and y with time-dependent bounded coefficients. Malkin generalizes Lyapunov's result for systems in which the matrix A is time- dependent. We consider the nonanalytic case, and to prove existence of these local integrals, we use center manifold theory.

The following lemma specifies a class of systems of differential equations that satisfies the conditions of the Lyapunov-Malkin theorem:

29 Lemma 1.5.9 Consider a system of differential equations of the form

ii = Au +By-^U{u,y), ÿ = y{u,y), (125)

where u 6 y € R” , detv4 ^ 0, and where U and y represent higher order nonlinear terms. There is a change of variables of the form u = x + d>{y) such that

(i) in the new variables {x,y) system (1.25) becomes

x = Ax + X{x,y), y = Y{x,y),

(ii) if V'(0 , y) = 0 , then X (0 , y) = 0 a5 well.

P ro o f Put u = X + é{y), where ^(y) is defined by

.40(y) + By + U{(f){y),y) = 0.

System (1.25) in the variables (x, y) becomes

X = .4x +A"(x,y), y = r(x,y), where

X{x,y) = .Adiy) + By + U{x + d>{y),y) - ^ F ( x , y),

y{^,y) =y{x+ (p{y),y).

Note that }'(0, y) = 0 implies A'(0, y) = 0. ■

1.6 The Energy-Momentum Method for Holonomie Systems

As mentioned above, we use here an approach to stability which generalizes the energy-momentum method for Hamiltonian systems. Of course the energy-momentum method has a long and distinguished history going back to Routh, Riemann, Poincaré, Lyapunov, Arnold, Smale and many others. The main new feature provided in the

30 more recent work of Simo, Lewis and Marsden [1991] (see Marsden [1992] for an exposition) is to obtain the powerful block diagonalization structure of the second variation of the augmented Hamiltonian as well as the normal form for the sym- plectic structure. This formulation also allowed for the proof of a converse of the energy-momentum method in the context of dissipation induced instabilities due to Bloch. Krishnaprasad, Marsden and Ratiu [1994, 1996]. Recall that the key idea for analyzing the stability of relative equilibria in the holonomie setting is to use the energy plus a function of other conserved quantities such as the momentum as a Lyapunov function. In effect, one is analyzing stability subject to the systems lying on a level surface of the momentum. In a body frame and in the special case of Lie-Poisson systems, the momentum often can be written in terms of a Casimir, a function that commutes with every function under the Poisson bracket, and the method is sometimes called the energy- Casimir method. While the energy is conserved, it does not provide sufficient information on sta­ bility since its second variation will be only semidefinite at a stable equilibrium in general. The algorithm for analyzing stability is thus as follows:

1. write the equations of motion in Hamiltonian form and identify the critical point of interest,

2. identify other conserved quantities such as momentum,

3. choose a function He such that the energy plus the function of other conserved quantities has a critical point at the chosen equilibrium and

4. show that He is definite at the given equilibrium. This proves nonlinear stability in the sense of Lyapunov.

Of course in special circumstances one has to interpret stability modulo the sym­ metry or a similar space in order to obtain stability. A good example is the study of two-dimensional .ABC flows, as in Chern and Marsden [1990].

31 In the nonholonomic case, while energy is conserved, momentum generally is not. As indicated in the discussion of the three principle cases above, in some cases how­ ever the momentum equation is integrable, leading to invariant surfaces which make possible an energy-momentum analysis similar to that of the Hamiltonian case. When the momentum equation is not integrable, one can get asymptotic stability in certain directions and the stability analysis is rather different from the Hamiltonian case. Nonetheless, to show stability we will make use of the conserved energy and the dynamic momentum equation.

1.7 Invariant Manifolds

For a Lagrangian system L : TQ R consider a Legendre transform FL :TQ T*Q defined by

Put p, = dLjdcf. Since the matrix dqtdqjL is nondegenerate, FL is a diffeomorphism. Using Q and p as independent coordinates, we obtain a Hamiltonian system associated with our Lagrangian system. The phase space of this Hamiltonian system is cotangent bundle T'Q. It has a natural structure of a symplectic manifold where Q = dç' A dpi is the canonical symplectic form. The main result of the theory of integrable Hamiltonian systems is given by the following theorem (Nekhoroshev [1972] and Mischenko and Fomenko [1978]):

Theorem 1.7.1 (Liouville-Arnold-Nekhoroshev) Suppose that a Hamiltonian system on a symplectic manifold has (n + k) integrals Fi, F 2 ...., Fn+h such that on the common level set = {Fj = Cj} the functions F, are independent and in the neighborhood of Me the rank of the matrix {{Fi, Fj}) is a constant not ex­ ceeding 2k. Then the compact connected components of Me are diffeomorphic to an (n — k)-dimensional torus. The flow on this torus is quasi-periodic.

Since each holonomie system may be viewed as a Hamiltonian system, the non- critical compact invariant manifolds of integrable holonomie systems are tori. The

32 motions of integrable nonholonomic systems are represented by quasi-periodic trajec­ tories on these invariant tori. The above property of integrable holonomie systems is not true in general for integrable nonholonomic systems. The theory of integrable nonholonomic systems is not completely developed, and we mention here a few properties demonstrating how different the dynamics of integrable nonholonomic systems can be. One of the results is due to Kozlov [1985]. Consider a nonholonomic system with an n-dimensional

phase space, and suppose that this system has (n — 2 ) integrals and in addition has an integral invariant with positive smooth density. Kozlov proves that compact connected components of common level sets of the integrals are diffeomorphic to 2- dimensional tori provided that the flow does not have equilibria on these components. He shows then using the Kolmogorov theorem (see Kolmogorov [1953]) that there are angular coordinates x and y on these tori such that the flow in these coordinates becomes A // ^ y = where 0 (r, y) represents the density of the integral invariant of the system on a torus. For almost all pairs (A, //) it is possible to find angular coordinates u and v such that the flow becomes

2n2n A . y u = V = —: u V ~ 4^ / / (1 26) 0 0

.\n example of the system demonstrating the dynamics mentioned above is the Chaplygin problem of a non-homogeneous sphere rolling on a horizontal plane. If the center of mass of the sphere coincides with its geometric center, we obtain an integrable system. According to Kozlov [1985], the flow on some of the invariant tori cannot be represented as (1.26). The dynamics of a rigid body with a fixed point subject to the following nonholo­ nomic constraint: the body angular velocity vanishes along one of the inertia axis—is even further away from Hamiltonian behavior. Tatarinov [1988] showed that some of

33 the invariant manifolds of this system are spheres with several handles. Almost all trajectories on these invariant manifolds are closed. The integrability approach developed in this work is different from that of Kozlov. We use elements of nonholonomic reduction (see [BKMM]) to find the invariant tori of the Routh problem and to show that the flow on these tori is quasi-periodic. This approach may be generalized, and we intend to do this in a future publication.

34 CHAPTER 2

THE ROUTH PROBLEM

In this chapter we study the motion of a homogeneous sphere rolling without sliding inside of the surface of revolution. We first show that the corresponding dynamical system is integrable. Then we obtain stability conditions for the stationary horizontal motions of the sphere. In the last section of this chapter we study the integral manifolds of this problem.

2.1 Integrability of the Routh Problem

In this section we follow Routh [1860] and Neimark and Fufaev [1972] in our derivation of the equations of motion of a homogeneous sphere on a surface of revolution. We then discuss integrability of this mechanical system. Consider a surface of revolution S obtained by rotation of a smooth curve about a vertical axis Z, and a homogeneous sphere rolling on the inner side of this surface without sliding. Without loss of generality we can assume the mass of the sphere is unity. Let a denote the radius of the sphere. We are going to explore the case when the sphere and the surface S have only one point in common during all motion, so the curvature of the meridian of S (i.e. an intersection of S and a vertical plane containing the line Z) is smaller then 1/a. In that case, the set S of all possible positions of the center of the sphere is also a smooth surface of revolution with the same axis Z (see Bruce and Giblin [1984]). The position of the sphere’s center 0 can

35 CD »

Figure 2.1: Kinematic variables and axes of the Routh problem. be described by geographical coordinates Q and 0. The latitude d is an angle between the inner normal vector to S and the plane orthogonal to the line Z (see Fig. 2 .1).

The longitude 0 is the angle between the fixed plane containing the line Z and the mobile plane which goes through the line Z and the point O.

Introduce a moving orthogonal basis 616263. The vector 6 % is a tangent vector to a meridian of S at O (i.e. an intersection of S and a vertical plane containing the line Z and the point O). 63 is the inner normal vector to S at O, and 62 is equal to e X 61. Let us denote the velocity of the sphere’s center by u, and the angular velocities of the sphere and the basis 616263 by w and O respectively. The components of v . w. and fi in the basis 616263 are (u, u,0 ), (p, 9 , r), and (fii,fi2,fi3):

U= U6i+ U 62, O; = p 6 i + 962 + T63, fi = fii 6 i + fi2 62 + fis 63.

Note that u = c{6 )è, v = 6(^)0, where l/c(0) is the curvature of the meridian and b{9) is the distance between the point O and the line Z (Routh [I860]). Suppose that the interaction of the sphere and the surface is modeled by the reaction force i2 = /?i 6 i + R262 + ^363 without dissipation. The principles of linear

36 and angular momentum give us the equations of motion

dv du} — = v + Clxv = F-\-R, = J{ûi + n X u>) = —ac3 x R. dt dt

Here d/dt stands for the time derivative with respect to the inertial frame, the dot

stands for the time derivative with respect to the moving frame 616263, J = 2a^/5 is

the central axial inertial momentum of the sphere, and F is the gravitational force. In coordinates these equations become

il — Q^v = Fi + Ri, p + = C1R2/J,

V + Çl'^u = i^2 d- i?2i 9 4- ~ f i i r = —aR i/J, (2-1)

filU — fi2W = F3 + R 3 , T + fiiÇ — ÇI2 P = 0.

In order to close these equations we should specify the reaction force R. Since the

sphere is rolling without sliding, we have two nonholonomic constraints u — aç = 0 ,

V + ap = Q. Using the constraints we can eliminate i?i, R2, p, q in (2.1) and write down the equations of motion as follows:

Ù — Q 3V = oFi/7 + 2aSlir/7, v + ÇI3U = ÔF2/I + 2af22r/7,

. -uÇl,+vÇl 2 ^ ^ ^ r — ------, \liv — ÇI2 U = Fz-\- R 3 . a

The angular velocity Cl is expressed in terms of 0 and 9 and their derivatives by the formulae

Cli = 0COS0, CI2 = —9, fis = —(psm9.

Substituting these expressions into (2.2) and taking into account the fact that F is the gravitational force we obtain the equations of motion in the form

Ù = —v0sin9 + (2ar0cos0 — og cos9)/7,

V = u^sin9 — 2ar9/7, v9 — cos 9 r = ------, a i ?3 = v

37 In order to complete these equations we should use the formulae u = c{9)0, v = b{d)é. Let us choose 6, 6,v, w = or, and 0 as local coordinates for the phase space which is diffeomorphic to TS^{6,(f>) x R(iy). The equations of motion in these coordinates become

w — vé—^^^cos9, (2.5)

R 3 = cos 9 + u9 — g sin d. (2.7)

The last equation of system (2.3)-(2.7) enables one to compute the normal component of the reaction force. The sphere rolls on the surface while R 3 > 0. When i?3 = 0 the sphere leaves the surface. Since equations (2.3)-(2.5) do not contain the variable , system (2.3)-(2.6) is invariant under action of the group S0{2). Therefore, if (2.3)-(2.5) is an integrable system, i.e. it has three independent integrals, then (2.3)-(2.6) is also integrable; after 9{t). v{t), w{t) are determined, 0(t) can be calculated by integration of the function u{t)/b{9(t)). Equations (2.3)-(2.5) have the energ}' integral (Routh [I860])

{c{9)9)'^ +U{v,w,9) = h, (2.8) where ^ e U{v.w,9) = v~ + J c{x) cos X dx. a In order to obtain this integral consider the kinetic and the potential energy of the sphere:

V{9) = g J c(x) cos X dx.

38 Using the constraint equations we obtain = {v} + + w^)/a^. Therefore,

^ ^ v} + v'^ + w^ 7(v? + v^) ^ vp- 2 5 ÏÔ T '

Since u = c{9)6, the energy integral is

l{{c{6)èŸ + v^) w r + — + ^ / c{x) cos xdx = H. 10

If we multiply this formula by 10/7 we will obtain this energy integral in the form

(2.8).

Consider a system of two linear equations with coefficients depending on 6 \

dv c(6 ) sin9 2w dw f c{9) cos 9\

We can obtain these equations if we formally divide equations (2.3)-(2.5) by 9 and replace v/9, w/9, with dv/d9, dw!d9. Since the coefficients of equations (2.9) are continuous on / = | b{9) ^ 0}, the solutions of (2.9) are defined everywhere in I.

The general solution v = A:iUi(0) + k2 V2 {9 ), w = kiWi{9) + k2 W2 {9 ) of (2.9) gives us two integrals of equations (2.3)-(2.5):

^i't'i(^) + k2V2{9) — u = 0, k\Wi{9) + k2W2{9) — w = 0, (2.10)

where {vi,wi), {v2 ,W2 ) are independent solutions of system (2.9). In order to prove independence of integrals (2.8) and (2.10), consider the Jacobian matrix f2c(9)c'(9)»’ + 12£ £ ( ^ 2c=(«)9 2» ( ( kiv[{9) + k2V2{9) 0 —1 0 kiw[{9) + k2W2{9) 0 0 —ly consisting of the first order partial derivatives of the above integrals. The integrals are independent when the rank of this matrix equals the number of integrals, i.e. 3 (see .A.rnold [1989]). In our case this happens if 0 ^ 0. Therefore integrals (2.8) and (2.10) are independent on the dense subset of the four-dimensional phase space

39 of equations (2.3)-(2.5). As mentioned above, this implies integrability of system (2.3)-(2.6). Hence the Routh problem is an integrable nonholonomic system. Note that here we distinguish integrability and solvability by quadratures, because integrals (2.10) in the general case cannot be written down explicitly. Integrability of the reduced system (2.3)-(2.5) means that the solutions of (2.3)-(2.5) are given by common level lines of integrals (2.8) and (2.10). We prove in Section 2.3 that under certain restrictions integrability of the Routh problem implies quasi-periodic behavior of the phase flow. There exist only a few cases when equations (2.9) can be solved by quadratures. Among them are: 5 is a sphere, 5 is a paraboloid of revolution (Neimark and Fufaev [1972] and Routh [I860]. More complicated cases of solvability by quadratures of the equations (2.9) are discussed in Bychkov [1965].

2.2 Stability of Stationary Periodic Motions of the Routh Problem

In this section we prove that the necessary condition for stability of stationary periodic motions of the sphere obtained by Routh [1860] is also a sufficient one. By stability we mean nonlinear orbital stability (see Marsden [1992]). Roughly speaking, a periodic solution is orbitally stable if after a small enough perturbation of the initial conditions we obtain a solution which remains in a neighborhood of our periodic solution. It is possible for the sphere to perform a stationary periodic motion of the form

0 = a, 9 = 0 . 0 = u)t, r = n, (2.11) in coordinates 6 , 0 , r. If we substitute a solution of this form into equations (2.3-2.7), we obtain

76(Q)a,'^sina — 2aa;ncosQ + Sycosa = 0 , ~ T7 â\' R3 = vu>cosa — gsina. b{6 ) The condition R 3 > 0 may be transformed into an inequality

6(a)w^ cos a — (/sina > 0 .

40 This inequality must hold throughout the motion, otherwise the sphere leaves the surface. It is an additional condition for the existence of a motion above the equator. The condition for linear stability of stationary periodic motions described above

, , 5b(a)uj^cosa 2ui^cos^a lOo sin a cos a 25o^cosa „ + — ------W)— + 496(a)c(aV^ > ° was obtained by Routh [I860]. We use here a slightly different representation of this condition then Routh does. Notice that for the Routh problem the three eigenvalues of the linearized equations are always equal to zero because the system is autonomous and has a two parameter family of stationary solutions. Condition (2.12) means that the other two eigenvalues are pure imaginary. In this case nonlinear terms may cause instability as well as stability. We prove below (Theorem 2.2.2) that for the Routh problem linear stability implies nonlinear stability. Equilibrium points

6 = a. 6 = 0, V = b{a)ui, w = m (2.13) of equations (2.3)-(2.5) correspond to stationary motions (2.11). Here m = an. System (2.3)-(2.5) has equilibrium points (2.13) if a, w and m satisfy the condition

76(0 )0;“ sin Q = 2o;m cos o — S^coso.

Stationary periodic motion (2.11) is orbitally stable if the corresponding equilibrium (2.13) is stable (Marsden [1992]). Let us restrict system (2.3)-(2.5) to the common level set Qk of integrals (2.10) which contains equilibrium (2.13). To do so one should compute the constants A:i and k-2 corresponding to equilibrium (2.13). It gives expressions for v and w in terms of 0. Substituting these expressions into U{v,w,6) we get the reduced potential energ}' Uk{6) which, when added to (c(6)6)^, gives us the energy integral of the restricted system. As we shall see from the following lemma, equilibria (2.13) of equations (2.3)-(2.5) corresponds to the critical point of the reduced potential energy Uk{9).

41 Lem m a 2 .2 .1 If the sphere performs a stationary motion 6 = a, 4) = uit, v = b{a)ui, w = m, then Uj^ioc) = 0 , and Uj^{a) equals the left-hand side of inequality (2.12) multiplied by 2c^{a).

Proof Let us calculate the derivatives U'{9) and U"{6)\

C/iW = 2«g + ^ ^ + 12££(^. (2,14)

Using (2.9) we obtain

Since the sphere is performing a stationary motion, v = b{a)u. Therefore, (2.14) implies £/;(a)/2 = c(a) sin a - 5gcosa\ ^ ^

The derivative of (2.14) divided by 2 gives us Uf({d)/2\

U'^{6)/2 = vv" 4- + {2ww" + 2w'^ + 5gc'{6) cos 6 — ogc{9) sin0)/7. (2.15)

The second derivatives of v and w can be found from equations (2.9):

c{9) sine ( m sinfl 2 ^ 'i / 9c(g) cos^ I f :in g b{9) V m 7j \ TH9) 7^

= (* - (^) - t ) • (Si)'™’ .A.fter substituting the formulae for v" and w" into (2.15), replacing 9 with a, and eliminating m we obtain

7c(a) 2w^ cos^ o lOp sin a cos o ^ 2ôg^cosa)sa \

76(a) 496(a)c(t ' ^ J •

42 T heorem 2 .2 .2 A stationary periodic motion 6 — oi, w = m, (p = u)t is orbitally stable if a Ç: I and if a and w satisfy condition (2.12) for linear stability.

P ro o f As mentioned above, it is enough to show stability of equilibrium (2.13). .A.ccording to Lemma 2.2.1 and the conditions of the theorem, U'^id) has a nondegen­

erate minimum at q. Here A:i and k2 are determined by the periodic solution 6 = a.

w = m, 0 = Lût. Therefore equations (2.3)-(2.5), restricted to the manifold Qk, have a stable equilibrium point 9 = a on Qk- If k' is sufficiently close to k, then by the properties of families of Morse functions (see Milnor [1963]) the function Uk>: Qk' R has a nondegenerate minimum at a

point q! which is close to a. This means that for all k' sufficiently close to k system (2.3), (2.4), (2.5) restricted to Qk' has a stable equilibrium 9 = a'. Therefore, the equilibrium 9 = a, 9 = 0, v = b{a)uj, w = m of equations (2.3)-(2.5) is stable, and the corresponding stationary periodic motion is orbitally stable. ■

Corollary 2.2.3 Motions which are close to a stable stationary motion are oscilla­ tory with respect to 9.

Let us consider now a couple of examples. Imagine that the sphere is rolling on the inner side of a sphere of radius c -f- a. In this case 5 is a sphere of the radius c. Condition for stability (2.12) becomes the inequality

2 lOosina 25 o“

" “ 7?“ 49Â7 ^ °

wiiich holds for all values of or except a = ±7t/2. Therefore all stationary periodic motions (2.11) are stable. This case is of great interest, because the reduced system is exactly the same as the one in the motion of the heavy symmetric top (Lagrange’s top) (Neimark and Fufaev [1972]). When the sphere is rolling inside of a sphere equations (2.9) become

dv ^ 2 w dw ^ , - _ v t a n 6l - — , — = 0 . (2.16)

43 The solutions of (2.16) can be expressed in terms of elementary functions: sin 9 + 1 v{9) = _T^sin = m. COSP Substituting these functions into (2.8) we obtain 2/Î2 (—fmsin0 + Z)2 2m^ + 10^c(sin0 — sin a) _ ^ 7 Therefore the dynamics of the reduced system is described by the equations

= (A' - Gu)(l - u^) - (L - Muf, ^ = {L- Mu)/{1 - u^).

Here u = sin0, G = 10^/7c, K = {h — 2m^ + 10^csinor)/7c^, M = 2m/7c, L = Z/c. The reduced dynamics of Lagrange’s top is described by the equations (see Arnold [1989])

= {JC — Gx){l - x^) — (£ - M xŸ , 0 = {C — M x)/{1 - x^),

where x = cos 9: 9 and é are Euler’s angles. This phenomenon admits the following mechanical explanation. When the sphere rolls inside of a sphere, then the energy, the vertical component M. of the momentum, and the normal component A/„ of the momentum are preserved (Neimark and Fufaev [1972]):

Esphere = ^ cos^ + ^ÿcsin 9 = const,

Mz = 0 COS' 9 + A/„ sin9 = const,

Mji = const.

Since A/„ is conserved, system (2.3)-(2.6) reduces to a system on TS^ describing the motion of the center of the sphere. Note that the energy, the vertical, and the normal component of the angular momentum relative to the fixed point for Lagrange’s top are also constants of the motion:

Etop ~ ^ sin^ 9^ + + mgl cos 9 = const,

Hz = Ji0 sin^ 9 + Hji cos 9 = const,

Hn = const.

44 and the motion of the axis of the top is also governed by the equations on Both sets of integrals have the same structure. The following proposition explains why those two reduced systems are the same.

Proposition 2.2.4 If a dynamical system onTS^ has two integrals of the form

-4i (0^ + cos^ 9) + A2 sin 6 = Ci, Bi0 cos^ 9 + B2 sin 9 = Co

or

Ai{9‘^ + (é^sin^ 9) 4- A2 cos 9 = C\, Bi0sin^ 9 B2 cos9 = C2 depending on the choice of spherical angles 9, (f>, then the reduced system is governed by the equations

ir = (ai — a2u)(l - v}) — (03 - a^u^), é = {03 - 04U^)/(1 — u~). where u = sin 9 (u = cos 9).

Proof The integrals may be rewritten as

.4.1 + 0^(1 — + .42^(1 — u^) = Ci(l — u^), B i0 (l — u~) + B2U = Co.

Solving these equations for and 0 and denoting Ci/.4i, . 42/.4i, C2/B 1, B 1/B 2 by oi, 02- 03, Ü4 respectively, we obtain

Û- = (ai - a2u)(l — u^) - (ü 3 - 04^^), 0 = (03 - a4U^)/{l — u“). ■

Now consider the case when S is an ellipsoid of revolution with horizontal axis .4 and vertical axis B. We can choose the parameterization

X = A cos r, z = B sin T of the meridian of that ellipsoid. Then . (.4^ sin^ r + B^ cos^ c{r) = ------— ------, b{r) = A COST, ^ sin r B cos r sin Q = 7——-7-0----- —---=—tttZ) COSO = (A^ sin^ T 4- B^ cos^ r)^/^ ’ sin^ r 4- B^ cos^

45 Introduce a parameter e = AjB. The left-hand side of condition for stability (2.12) becomes

cos^ r 4- 25^^ ^ le^Bu^ sin^ r + 2Buj^ cos^ r — 10^ sin r sin^ r -f- cos^ r)^ 7(f^ sin^ r 4- cos^ r) Denote the numerator of the sum above by G(w^):

G{ui^) = (7B^{s^ sin^ r 4- cos^ T)(7s^ sin^ r 4- 2 cos^ r) 4- Soe^B^ cos'* r)w'*

- (70Bÿsin r(£^ sin^ r 4- cos^ r))w^ 4- 25^^.

The surface G{uj^) = 0 separates stable and unstable stationary motions. Since the first and the last coefficients of this biquadratic equation are positive, either all stationary" periodic motions are stable or, for certain values of w, stationary motions are unstable. The condition of the existence of such motions is the positivity of the discriminant of G(w^). It can be written as F{r,s^) < 0, where

F (r, e^) = 7 tan"* r s'* 4- (19 tan^ r 4- 5)e^ 4- (2 — 5 tan^ r).

The biquadratic inequality F(r, s^) < 0 has solutions only if the corresponding

quadratic equation has at least one positive root. The coefficient (19 tan* r 4- 5) is positive for all |r| < ir/2. Hence if 2 — 5tan^r is positive then both roots of the quadratic equation are negative, and the biquadratic inequality has no solutions. If 2 -5 tan^ r < 0. then the quadratic equation has both positive and negative roots, and the solutions of the biquadratic inequality are [ 0 ,ci), where ej is a positive root of the equation F(r, = 0. So. there exists a critical value of the aspect ratio £c such that for s > Sc unstable motions do not exist. In the case t < Si we should check the condition F 3 > 0 , which for the case under consideration becomes n o sin r > B cos^ r Calculations show that the motion is unstable if w, r, e satisfy the following condi­ tions:

2/5 < tan^ r < 25/31, 0 <

46 or

u t < ü / < u \, 2/5 < tan^ r < 10/7, e\ < < e\. or

< u^ < 25/31 < tan^ r, 0 < f^ < si. B cos-^ T

Here is a root of the equation 49 tan® r £■'*+7 tan^ r(4 tan^ r+5)e^+(25—31 tan^ r) = 0, and ut, are roots of G(w^) = 0.

2.3 Integral Manifolds of the Routh Problem

Here we are going to show that if 5 is a closed compact surface and c{6) is an analytic function satisfying the condition c(±7t/2) 7^ 0 then almost all invariant manifolds of the Routh problem are tori filled with almost periodic motions. The integrability of the Routh problem does not imply this fact directly, because there exist exam­ ples of integrable nonholonomic systems with invariant manifolds which are not tori (Tatarinov [1988]). To accomplish our goal, we show (see Lemma 2.3.1 below) that for almost all initial conditions the trajectory of the center of the sphere does not cross the poles of S (i.e. points of intersection of S and its axis of revolution). Then we prove (Theorem 2.3.3) that the motions just mentioned fill out two-dimensional tori in the phase space unless these motions belong to the critical level of the en­ ergy^ integral, and that almost all levels of energy are not critical (Lemma 2.3.5 and Theorem 2.3.6). The phase flow restricted to the common level set of integrals (2.8) and (2.10) is described by the equations

c^e)è^ = h - Uk{9). 0 = 0(g), where 0(g) = v/b{9).

Independence of the right-hand sides of these equations on 0 reflects the fact that 50(2) is a group of symmetries for our problem.

47 Lemma 2.3.1 If c { ± tt/ 2 ) ^ 0, t h e n for almost all ki, the function Uk{Q) —>■ oo as

6 -4- ±7t/2.

Proof Consider the Wronskian W(d) of general solution (2.10) of equations (2.9). By Liouville’s formula (Arnold [1989]),

c{d) sin 9 dO W{9) = iy(0) exp J 6(g) 0

If 5 is a compact closed surface of revolution, then 6 (±7t / 2 ) = 0, but 6 '(±7t / 2 ) # 0, which imply B c{6) sin 9 d9 GO / 6(g) 0 as 6 ±7t/2. Thus. l'V"(g) —> oo as g -4 ±7t/2, and at least one of functions i>i(g), U2{9), wi{9), W2{9) tends to infinity as g -4 ±r/2. Therefore, for almost all ki, Ajg, v^{9) + 2vP-{9)l~ -4- oc as g -4 ±7t2, and so does Uk{9). ■

Corollary 2.3.2 For almost all initial conditions there exist Ai, .4g E (—7t/2. 7r/2), such that Ai < 9{t) < .42-

Theorem 2.3.3 If Uk{9) —¥ oo as 9 ±7t/2 and h is not a critical value ofUk{9), then connected components of the invariant manifold are diffeomorphic to two- dimensional tori. There exist angular coordinates {x,y) which linearize the equations of the phase flow on these tori.

Proof Suppose h is not a critical value of Uk{9). If ai, Q2 are two consecutive roots of the equation Uk{9) = h such that Uk{9) < h îor < 9 < 0 2 . then cki < 9{t) < 02 for any t. and the time of motion r from to 02 equals (Arnold [1989])

_c{9)_d9_ V h-U ki9)' ctlJ

48 This integral converges, because Qi, Og are simple roots of the equation Uk(d) = h. Therefore the corresponding trajectory of the equation c?{6)&^ = h — Uk{9) is diffeo­ morphic to the circle 5^ (see Arnold [1989]). Since 50(2) is a group of symmetries, the component of the invariant manifold containing the above trajectory is diffeomorphic to T2. Define a new variable x by

7T f c{9) dd X 7 y/h-Uk{e) Ql Since and og are simple roots of Uk{6) = h this integral converges. This new variable x is an angular variable because x increases by 2;r when 9 changes from Qi

to 02 and then again to oi. The equations of the phase flow in angular coordinates (x, 0) are of the form z = wi, 0 = F(x), where ij\ = tt/t, F{ x ) = $(^(x)). Introduce a new variable y by

X 0 = y H / T{x) dx, J 0 where F[x) = F(x) — (F) and

2% (F) = ^ J F {x)dx. 0 Since (F) = 0. the angle 0 depends periodically on x and y:

x+2ir 0 (x + 27t, y) = y H / F{x)dx UJi J 0 X i+27r X = yH / F(x) dx 4- — I F{x)dx = y+ — fF{x)dx = d>{x,y), J J UJl J 0 I 0

0{x. y+ 2w) = y + 27T + — J F{x)dx = I y -I J F(x) dx j mod 27t = 0(x, y).

49 Therefore (x, y) are angular coordinates on T^. The phase flow in these coordinates is described by the equations

r = Wi, y = uJ2,

since

0 = ÿ + = ÿ + T{x) = {F) + F{x). ■

Corollary 2.3.4 In the neighborhood of a stable stationary periodic motion, integral manifolds of the Routh problem are diffeomorphic to two-dimensional tori.

Proof .According to Lemma 2.2.1, we get a stable stationary motion 9 = a. 0 =

V{a)t if a is the critical point of Uk{9). If |/i—C/;fc(a)| «C 1 then the equation Uk{9) = h

has exactly two simple roots oi, 0 3 . ■

Lemma 2.3.5 Let c{6) be an analytic function, and k\, Azg be such that Uk{9) —> oc as 9 ±7r/2. Then the set {h € R|Lfc(0) = h has multiple root(s)] is finite and non-empty.

Proof The equation Uk{9) = h has multiple root(s) iff"/i is a critical value of Uk{9). If c{9) is an analytic function, then Uk{9) also is an analytic function. Therefore. Uk{9) has a finite number of critical points on [/li, .Ag]. To complete the proof, note that

Uk{9) has at least one critical point on [.4i, . 42], because Uk{9) 00 as 9 —¥ ±7t/2. ■

Theorem 2.3.6 If c{9) is an analytic function and c(± 7t/2 ) ^ 0 then almost all invariant manifolds are diffeomorphic to unions of two-dimensional tori.

P ro o f Since c(± 7t/2 ) ^ 0, then by Lemma 2.3.12 and Theorem 2.3.3, for almost all ki, /cg. connected components of the invariant manifold are diffeomorphic to T~ unless h is a critical value of Uk{9). By Lemma 2.3.5 the number of critical points of the function Uk{9) on [>li,.42] is finite, which implies the conclusion of the theorem.

50 If the ratio wg/wi of frequencies is an irrational number then the invariant torus is the closure of any solution with initial conditions in this torus. If wg/wi € Q then the invariant torus is filled with periodic solutions of the same period (Arnold [1989]). We prove below that generically is not a constant, and consequently the phase space is filled with both resonant and nonresonant tori. To do this, we show that the derivative of wg/wi with respect to h differs from zero. Since 2ir Q2 Q2

U ai ai and UJl = 7r/r, = l f $(g)c(g) de UJl IT J V h -u ,{ e ) a\ Let 9 = a, 4> = V{a)t be a stable stationary periodic motion, and let T be its period. Without loss of generality we may redefine Uk{9) in such a way that Uk{oc) = 0. According to Lemma 2.2.1 the function Uk{9) has a nondegenerate minimum at a. Therefore by Morse theor}' (Milnor [1963]) there exists a variable A such that

Uk{9{\)) = A^. The equation Uk{9) = h becomes A^ = h. The roots qi and Q 2 of the first equation correspond to roots ± \/h of the second one: 6[—\/h) = a i, 6{\/h) = ctg. Hence 0(g(A))c(g(A)) dO dX. UJl IT J \/h — dX — Vh The above integral is improper. It becomes proper after the substitution A = \/hsm^:

n/2

— = — f V{9{\/hsin^))9'{\/hsin^) d^. U J l T T J — t t /2

Here V{9) = $(g)c(g) = v{d)c{e)/b{e), 9' = d9/du, V = dV/d9.

Lemma 2.3.7 If h 0, then

^ = V(0(O))0'(O) I (É/(ü))6/'(ü) + + ^ ^ (W'(0(O))0'^O) +3W(0(O))0"(O)5'(O) + V{9{0))9"'{0)') +o{h). ^1

51 Proof The Taylor expansion for K(0(A))0'(\/hsinÇ) is

V'(0(O))5'(O) + y/hsin^ (y'(0(O))g'^(O) + V^(0(O))0"(O))

+ (/i/2) sin^^ (l/"(^(O))0'"(O) +3r(0(O))e"(O)0'(O) + F(0(O))r'(O)) +o(/i).

After integration we get the statement of lemma. ■

C orollary 2.3.8 / / 1/"(0(O))0'^(O) + 3V'(0(O))r(O)0'(O) + F(0(O))0'"(O) # 0, then

A. (ws/wi) ^ 0. dh A=0

Denote the coefficients of Taylor expansions for V{9) and 0(A) by V], 0,:

V'(0) = V'o + Vi(0 - a) + V2(0 - a f 4- 0(0 - o ) \ (2.17)

0(A) = Q + 01A + 02A^ + 03 A^ + 0(A"). (2.18)

Since

V"'(0(O))0'\O) + 3r'(0(O))0"(O)0'(O) + V'(0(O))0"'(O) = 3 VÔ03 + 3110102 + V'20i, then (W2/W1) 7^0 if

3V'o03 + 3V10102 + V'20? 7^ 0. (2.19)

If we substitute (2.17) and (2.18) into Taylor expansion

Uk{0) = 112(0 - a f + 11/3(0 - a f + 114(0 - o)" + 0 ( 0 - 0 )= we obtain

A^H'20^ + A^(211'20i02 + II30?) + A \H 2(0| + 20103) + 31Hi0?02 + l%(0f) + O(A^) = A \ since &\(0(A)) = A^. This gives us a system of equations for 0^:

11201 = 1, 21120102 + ll'30f = 0, H^(0| + 20103) + 311^30102 + H401 = 0 .

52 Hence.

If we divide (2.19) by 9i and use (2.20) we obtain

loVoWl - 12FoPr2H'-4 - 12VW2M^3V. +4- SVRV^Wl2WI # 0. 8 Wl

Therefore condition (2.19) is satisfied if P # 0 where

P = ISV'olT^ - 12VÎ,PT2H'4 - I 2 F1H 2H 3 + 8 V2WI (2.21)

The coefficients Vq, Vi, V 2, VV'2, W3, VV4 can be expressed through the coefficients Vi, Wi, li of Taylor expansions for functions v{a + x), w{q + x), l{a + x). Here l{x) = c{a + x)/b{a + x) = Iq + l\x + l2x"^ + l^x^ + ..., x = 9 — a.

Let us show that P ^ 0. Since u, and Wi are polynomials in the variables vq, wq, Ij, sin a, cosQ, g then V, and Wi also are polynomials in the same variables. Therefore it is enough to check that at least one of the terms of the polynomial P is not equal to zero.

Lemma 2.3.9 In the case g = 0 the polynomial Wi does not depend on and li.

P ro o f In the case ^ = 0

Uk{9) = v\9) + 2w\9)l7.

Since c, and Wi do not depend on /, (this follows immediately from (2.9)), then VV', does not depend on /, either. Calculations show that the terms of the polynomial Wj which depend on /,_i are

2 ^fglx-isinQ - “^vowoli-i cos= 2uo/t-i ^wosina - ^ wqCOSo^ .

The difference in parentheses equals zero because the sphere performs a stationary motion. Therefore H', does not depend on Z,_i. ■

53 Theorem 2.3.10 In the general case # 0. dh A=0

Proof The definition of V{9) implies that Vq, V\ do not depend on I2 but does. Taking into account Lemma 2.3.9 we can conclude that the only terms of (2.21)

which depend on I2 are -I 2 V0W2W4 + Since W2 ^ 0 v;e need to prove that

—SlokTl, + 2V2W2 ^ 0. Let us use the sign = for marking terms depending on I,:

- 3 V0W4 + 214^2 = vo{-31oW4 + 2 I2W2), (2 .22 ) W 4 = 2{V i V3 + V0V4 + 2{ w i W3 + W qW 4 )/7 ).

The coefficients V{, Wi of the Taylor expansions for v{a + x), w{a + x) can be found from (2.9). Substituting these coefficients into (2.22), we obtain

If we multiply the last expression by —3Zo and add 2/2 we obtain

5I0I2VI cos a.

Therefore in the general case —I2 VQW2W4 + 8 V2W 2 depends nontrivially on I2. M

Corollary 2.3.11 If c{9) is an analytic function then in the general case U 2/U1 is not a constant.

Corollary 2.3.12 In the general case the neighborhood of the stable stationary peri­ odic motion 9 = a, 0 = ujt, w = m contains infinitely many nonstationary periodic motions.

54 CHAPTER 3

AN ENERGY-MOMENTUM METHOD

FOR STABILITY OF NONHOLONOMIC SYSTEMS

In this chapter we develop an energy-momentum approach to the stability of relative equilibria of nonholonomic systems that satisfy the dimension assumption and the skew symmetry assumption (see Section 1.4). Recall that the equations of motion are d dR dR

^P i = where r“ are the shape coordinates and pa are the components of the nonholonomic momentum. Below, three principal cases will be considered:

1. Pure Transport Case In this case, terms quadratic in r are not present in the momentum equation, so it is in the form of a transport equation— i.e. the momentum equation is an equation of parallel transport and the equation itself defines the relevant connection.

Under certain integrability conditions (see below) the transport equation defines invariant surfaces, which allow us to use a type of energy-momentum method for stability analysis in a similar fashion to the manner in which the holonomie case uses the level surfaces defined by the momentum map (see Section 1.6). The key difference is that in our case, the additional invariant surfaces do not arise from conservation of momentum. In this case, one gets stable, but not

55 asymptotically stable, relative equilibria as we shall see in §3.1.1. Examples include the rolling disk, a body of revolution rolling on a horizontal plane, and the Routh problem.

2. Integrable Transport Case In this case, terms quadratic in r are present in the momentum equation and thus it is not a pure transport equation. However, here we assume that the transport part is integrable. As we shall also see. in this case relative equilibria may be asymptotically stable. We are able to find a generalization of the energy-momentum method which gives conditions for asymptotic stability (see §3.2.2). An example is the roller racer.

3. Nonintegrable Transport Case Again, the terms quadratic in r are present in the momentum equation and thus it is not a pure transport equation. How­ ever, the transport part is not integrable. Again, we are able to demonstrate asymptotic stability using the Lyapunov-Malkin theorem and to relate it to an energy-momentum type analysis under certain eigenvalue hypotheses, as we will see in §3.3. .-Vn example is the rattleback top, which we discuss in §3.3.2. .A.nother example is a nonhomogeneous sphere with a center of mass lying off the planes spanned by the principal axis body frame. See Markeev [1992].

In some examples, such as the nonhomogeneous (unbalanced) Kovalevskaya sphere rolling on the plane, these eigenvalue hypotheses do not hold. We intend to investigate this case in future work.

.A.S indicated above, the key difference between cases 1 and 2 is the existence of the nontransport terms in the momentum equation. These terms cause the momentum to drift between the invariant manifolds that arise from the integrable pure transport term. As a result, in case 2 relative equilibria may be asymptotically stable. Similar qualitative behavior occurs in case 3 where even the transport part of the momentum equation does not define invariant manifolds.

56 In the sections below where these différent cases are discussed we will make clear at the beginning of each section what the underlying hypotheses on the systems are by listing the key hypotheses and labeling them by Hi, H2, and H3.

3.1 The Pure Transport Case

In this section we assume that

HI Pq (36 are skew-symmetric in q, 0. Under this assumption, the momentum equa­ tion can be written as the vanishing of the connection one form defined by dp6 - T>l^pcdr°.

H2 The curvature of the preceding connection form is zero.

A nontrivial example of this case is that of Routh’s problem of a sphere rolling in a surface of revolution. See Chapter 2. The proof of the main Theorem in this section generalizes the approach to stability of the Routh problem. Under the above two assumptions, the distribution defined by the momentum equation is integrable, and so we get invariant surfaces, which makes further reduction possible. This enables us to use the energy-momentum method in a way that is similar to the holonomie case, as we explained above. Note that if the number of shape variables is one. the above connection is inte­ grable, because it may be treated as a system of linear ordinary differential equations with coefficients depending on the shape variable r;

As a result, we obtain an integrable nonholonomic system, because after solving the momentum equation for pb and substituting the result in the equation for the shape variable, the latter equation may be viewed as a Lagrangian system with one degree of freedom, which is integrable.

57 3.1.1 The Nonholonomic Energy-Momentum Method

We now develop the energy-momentum method for the case in which the momentum equation is pure transport. Under the assumptions HI and H2 made so far, the equations of motion become

= (31)

J^Pi = (3.2)

A relative equilibrium is a point {r,f,p) = (ro,0,po) which is a fixed point for the dynamics determined by equations (3.1) and (3.2). Under assumption Hi the point (ro,Po) is seen to be a critical point of the amended potential. Because of our zero curvature assumption H2, the solutions of the momentum equation lie on surfaces of the form Pa = Pa{T°,kb), a,b = l,...,m , where kb are constants labeling these surfaces. Using the functions = Pai^^^kb) we introduce the reduced am ended potential

Ut(r") = f/(r,a(r",A;6)).

We think of the function Uk{r°) as being the restriction of the function U to the invariant manifold

Qk = {(r“,Pa) I Pa = Pa{r°.kb)}.

Theorem 3.1.1 Let assumptions Hi and H2 hold and let (ro,po), where po =

P{'’o

Proof First, we show that the relative equilibrium

— ^0 : p2 ~-^o(^o 5 (3.3) of system (3.1), (3.2) is stable modulo perturbations consistent with Qko- Consider the phase flow restricted to the invariant manifold Qko, where ko corresponds to the

58 relative equilibrium. Since has a nondegenerate minimum at Tq , the function

E\ q^^ is positive definite. By Theorem 1.4.6 its derivative along the flow vanishes.

Using E\ q^^ as a Lyapunov function, we conclude that equations (3.1), (3.2), restricted to the manifold Qko, have a stable equilibrium point r “ on Qko- To finish the proof, we need to show that equations (3.1), (3.2), restricted to nearby invariant manifolds Qk, have stable equilibria on these manifolds. If k is sufficiently close to ko, then by the properties of families of Morse functions (see Milnor [1963]), the function Uk'- Qk has a nondegenerate minimum at the point r “ which is close to Tq. This means that for all k sufiSciently close to ko system (3.1), (3.2) restricted to Qk has a stable equilibrium r“. Therefore, equilibrium (3.3) of equations (3.1), (3.2) is stable. The stability here cannot be asymptotic, since the dynamical systems on Qk have a positive definite conserved quantity—the reduced energy function. ■

Remark. Even though in general Pair‘d, kb) can not be found explicitly, the types of critical points of Uk may be explicitly determined as follows. First of all, note that

dpb dr^ — ^taPc as long as (r“,pa) e Qk- Therefore

dUk

3 t ° where

(3.4)

(cf. Karapetyan [1983]). The operators Vq may be viewed as covariant derivatives in the vector bundle T>/G —» T{Q/G) with fibers (s^)*- They arise from the connec­ tion defined by the transport term in the momentum equation. We note that these derivatives commute in cases 1 and 2. The relative equilibria satisfy the condition

V qU = 0,

59 while the condition for stability

dr^ {i.e., is positive definite) becomes the condition

» 0.

In the commutative case this was shown by Karapetyan [1983]. Now we give the stability condition in a form similar to that of the energy- momentum method for holonomie systems given in Simo, Lewis, and Marsden [1991].

Theorem 3.1.2 (The nonholonomic energy-momentum method)Under as­ sumptions Hi and H2, the point = (rQ, 0,p°) is a relative equilibrium if and only if there is a ^ such that q^ is a critical point of the augmented energy : V/G R (i.e., Eç is a function of {r,f,p)), defined by

E^ = E - (p- P{r,k),^).

This equilibrium is stable if S^E^ restricted to Tq^Qk is positive definite (here 6 denotes differentiation with respect to all variables except f,).

Proof .A. point Çe € Qjt is a relative equilibrium if dr" 14 = 0. This condition is equivalent to d{E\Q,f) = 0. The last equation may be represented as

d{E - ip - P{r,k),f)) = 0 for some ( E g^'. Similarly, the condition for stability cPUk 3> 0 is equivalent to d~ [E\ q^) » 0, which may be represented as {S^E^) |r,„Qfc ^ 0 . ■

Note that if the momentum map is preserved, then the formula for becomes

E^ = E - {p- k,^), which is the same as the formula for the augmented energy Eç for holonomie systems.

60 3.1.2 The Rolling Disc

There are several examples which illustrate the ideas above. For instance the falling disk, Routh’s problem, and a body of revolution rolling on a horizontal plane are systems where the momentum equation defines an integrable distribution and we are left with only one shape variable. Since the stability properties of all these systems are similar, we consider here only the rolling disk. For the body of revolution on the plane see Chaplygin [1897a] and Karapetyan [1983]. For the Routh problem see Chapter 2.

The Rolling Disk. Consider again the disk rolling without sliding on the xy-plane. Recall that we have the following: Denote the coordinates of contact of the disk in the xy-plane by (x, y). Let 6, 0, and ip denote the angle between the plane of the disk and the vertical axis, the “heading angle” of the disk, and “self-rotation” angle of the disk respectively, as was introduced earlier. The Lagrangian and the constraints in these coordinates are given by

L = y - R(0sin 0 -f- 0))^ + sin^ ^ + ivcos 0 -f-

+ - -f- 0^ cos^ 6) + B {^sin 6 + — m gR cos 6,

X = —0i?COS0,

ÿ = —lùRsinp, where ^ = xcos0 + ysin0 + Rip, y = —xsin 0 + ÿ cos 0. Note that the constraints may be written as ,^ = 0, y = 0. This system is invariant under the action of the group G = SE{2) x 50(2); the action by the group element (a, 6, a, 0) is given by

{9, 0, Ip, X , y) (0,0 - t - Q, 0 -f /?, xcosor — y sin Q - t - a, X sin a - I - y cos a -f- h).

Obviously, r,0rb(5) - span |-) ,

61 and D , = span -B eos^ - iîsin ÿ + A ) ,

which imply d d d d ^ ( -i2 cos 0 — - i2sin 0 — + — ^

Choose vectors (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1) as a basis of the Lie algebra g of the group G. The corresponding generators are dx, dy, -ydx+xdy-^-d^. dy.. Taking into account that the generators d^, —i2cos05x — i2 sin 0 + 5^. correspond to the elements {y, —x, 1,0), (—i?cos 0, —/2sin0,0,1) of the Lie algebra g, we obtain the following momentum equations:

Pi = mR^ cosOàijj. (3.5) P2 = —mR COS 9 64), where

Pi = .40 cos^ 6 + [mR? + B) (0 sin 0 + 0) sin 6, , . . (3G) P2 = {mR + 5)(0sin0 -r 0).

One may notice that dly die ~ V i ' Solving (3.6) for 0 and 0 and substituting the solutions back into equations (3.5) we obtain another representation of the momentum equations:

& = m R ^cose ( - e.

The right-hand sides of (3.7) do not have terms quadratic in the shape velocity 9. The distribution, defined by (3.7), is integrable and defines two integrals of the form

Pi = Pi{9,ki,k2), P2 = P2{9.ki,k2). It is known that these integrals may be written down explicitly in terms of the hypergeometric function. See Appel [1900], Chaplygin [1897a], and Korteweg [1899] for details.

62 To carry out stability analysis, we use the remark following Theorem 3.1.1. Using formulae (3.6), we obtain the amended potential (Pi -P2sing)^ pj + mg R COS 9. A cos^ 9 B + mE? Straightforward computation shows that the condition for stability 2> 0 of a

relative equilibrium 9 = 9q, pi = p°, p 2 = p° becomes

- ^ “■ Note that this condition guarantees stability here relative to (0.^,Pi,Pz); in other words we have stability modulo the action of SE(2) x 50(2). The falling disk may be considered as a limiting case of the body of revolution which also has an integrable pure transport momentum equation (this example is treated in Chaplygin [1897a] and Karapetyan [1983]). The rolling disc has also been analyzed recently by O’Reilly [1996] and Cushman, Hermans and Kemppainen [1996]. O’Reilly considered bifurcation of relative equilibria, the stability of vertical station­ ary motions, as well as the possibility of sliding.

3.2 The Non-Pure Transport Case

In this section we consider the case in which the coefficients are not skew sym­ metric in Q. Ô and the two subcases where the transport part of the momentum equation is integrable or is not integrable, respectively. In either case one may obtain asymptotic stability.

3.2.1 The Mathematical Example

T he Lyapunov-M alkin conditions. Recall from §1.3.3 that the equations of mo­ tion are

r = -^{a{r)T+ b{T)p)p, p = (a(r)r-h 6(r)p) r; (3.8)

63 here and below we write r instead of Recall also that a point r = tq, p = po is a relative equilibrium if tq and po satisfy the condition

a y , , db^. . 2 ^ -^(^o) + ^6(ro)Po = 0.

Introduce coordinates (ui,U 2,v) in the neighborhood of this equilibrium by

r = To + , f =U2, p = pq + v .

The linearized equations of motion are

111 = U2,

112 = A u2 + 3ui + Cu,

V = T)U2, where

96 A - - ^ a p o ,

9^6 f db Po,

e - - 2 — bpa,

® and where I', a. 6, and their derivatives are evaluated at tq. The characteristic polynomial of these linearized equations is calculated to be

A[A2 - yiA - (3 + 63)].

It obviously has one zero root. The two others have negative real parts if

3 + 63 < 0, < 0. (3.9)

These conditions imply linear stability. We discuss the meaning of these conditions later.

64 Next, we make the substitution v = y + Dui, which defines the new variable y. The (nonlineeir) equations of motion become

lii = U2,

Û2 = Au2 + (3 + C3)ui + Qy + U{u,y),

ÿ = y{u,y),

where y), y{u, y) stand for nonlinear terms, and y{u, y) vanishes when u = 0. By Lemma 1.5.9 there exists a further substitution u = x + {y) such that the equations of motion in coordinates (x, y) become

x = Px + X(x, y), y = F(x,y),

where X{x,y) and Y{x,y) satisfy the conditions %(0, y) = 0, F(0, y) = 0. Here,

This form enables us to use the Lyapunov-Malkin theorem and conclude that linear stability implies nonlinear stability and in addition that we have asymptotic stability

with respect to (xi,X 2).

The Energy-Momentum Method. To find a Lyapunov function based approach for analyzing the stability of the mathematical example, we introduce a modified dynamical system and use its energ\' function and momentum to construct a Lyapunov function for the original system. This modified system is introduced for the purpose of finding the Lyapunov function and is not used in the stability proof. We will generalize this approach below and this example may be viewed as motivation for the general approach. Consider then the new system obtained from the Lagrangian (1.6) and the con­ straint (1.7) by setting a{r) = 0. Notice that Lc stays the same and therefore, the equation of motion may be obtained from (3.8): dV db 2 • ^6 .

65 The condition for existence of the relative equilibria also stays the same. However, a crucial observation is that for the new system, the momentum equation is now integrable, in fact explicitly, so that in this example p = A:exp(6^(r)/2). Thus, we may proceed and use this invariant surface to perform reduction. The amended potential, defined by U{r,p) = T/(r) + jp^, becomes

Uk{r) = V{r) + i {kexp{b^{r)/2)Ÿ .

Consider the function

Wk = + Uk{r) + e(r - ro)f.

If e is small enough and C4 has a nondegenerate minimum, then so does IT*. Suppose that the matrix P has no eigenvalues with non-negative real parts. Then by Theorem 1.5.7 equations (3.8) have a local integral p = V{r,f,c). Differentiate Wk along the vector field determined by (3.8). We obtain

— —o(ro)por^ + + {higher order terms}.

Therefore, Wk is a Lyapunov function for the flow restricted to the local invariant manifold p = V{r, r. c) if

+ + ( ^ W ) ) jpo>0 (3.10) and

— a(ro)por^ > 0. (3.11)

Notice that the Lyapunov conditions (3.10) and (3.11) are the same as conditions (3.9). Introduce the operator d d b d

66 Then condition (3.10) may be represented as

> 0 ,

which is the same as the condition for stability of stationary motions of a nonholo- nomic system with an integrable momentum equation (recall that this means that there are no terms quadratic in r, only transport terms defining an integrable distri­ bution). The left-hand side of formula (3.11) may be viewed as a derivative of the energ}' function

along the flow

dV 9b . 9b r = - — (o(r)r -t- b{r)p) p, p = — b{r)pr.

or as a derivative of the amended potential U along the vector field defined by the nontransport terms of the momentum equations

p• = —a(r)r / \2 .

3.2.2 The Nonholonomic Energy-Momentum Method

We now generalize the energy-momentum method discussed above for the mathe­ matical example to the general case in which the transport part of the momentum equation is integrable. Here we assume hypothesis H2 in the present context, namely:

H2 The curvature of the connection form associated with the transport part of the momentum equation, namely dpb — VljPcdr°‘, is zero.

The momentum equation in this situation is

^^Pb = T^laVci-'^ + 'Da0bT‘^f^-

67 Hypothesis H2 implies that the form due to the transport part of the momentum equation defines an integrable distribution. Associated to this distribution, there is a family of integral manifolds

Pa = P a {r°,kb )

with Pa satisfying the equation dPb = V^Pcdr^^. Note that these mtmifolds are not invariant manifolds of the full system under consideration because the momentum equation has non-transport terms. Substituting the functions kb = const, into E{r,f,p), we obtain a function

V h{r^,n = E{r^,r‘^,Pair‘d,kb)),

that depends only on (r“, r“) and parametrically on k. This function will not be our final Lyapunov function but will be used to construct one in the proof to follow. Pick a relative equilibrium r “ = rj, Pa = p°. In this context we introduce the following definiteness assumptions:

H3 the equilibrium r“ = rj, Pa = Pa the two symmetric matrices and i'Paâb + 'Pgab)P'^Pc are positive definite.

Theorem 3.2.1 Under assumptions H2 and H3, the equilibrium r'^ = r^, pa = p° is Lyapunov stable. Moreover, the system has local invariant manifolds that are tangent to the family of manifolds defined by the integrable transport part of the momentum equation at the relative equilibria. The relative equilibria, that are close enough to (^o,Po), are asymptotically stable in the directions defined by these invariant man­ ifolds. In addition, for initial conditions close enough to the equilibrium r “ = Tq, Pa. = Pa, the perturbed solution approaches a nearby equilibrium.

P roof The substitution Po = P° + 2/a + T>„£,(ro)p°u“, where u°‘ = r° — r j , eliminates the linear terms in the momentum equation. In fact, with this substitution, the

68 equations of motion (1.17), (1.18) become

+ ( % - % (r.))p:f- + ©„«r“r«.

We will show in §3.3.1 that H3 implies the hypotheses of Theorem 1.5.7. Thus, above equations have local integrals j/a = A (r,f, c), where the functions fa are such that 9rfa = dr fa = 0 at the equilibria. Therefore, the original equations (1.17), (1.18) have n local integrals

Pa = 'Pair°,f°, Cb), Cb = const. (3.12)

where Va are such that dV dP &P = 0 5 r“ ’ ôf “ at the relative equilibria. We now use 14(r“,r“) to construct a Lyapunov function to determine the condi­ tions for asymptotic stability of the relative equilibrium r‘^ = Tq , pa = Pa- We will do this in a fashion similar to that used by Chetaev [1959] and Bloch, Krishnaprasad, .Marsden. and Ratiu [1994].

Figure 3.1: The manifolds Q*o and Qco-

Consider the following two manifolds at the equilibrium (see Figure 3.1) ( r j, 0,p°): the integral manifold of the transport equation

Qko — {Pa — P i(^°, ^o)} ,

69 and the local invariant manifold

Qco = {Pa = r“ , Co)} .

Without loss of generality, suppose that ^Q^(ro) = 5q0. Introduce the function

~ (ro ) = V^U{To,p„)=0

and rW) = V„V«C/(r„, po) > 0, dr^dr^ the function is positive definite in some neighborhood of the relative equilibrium (rg, 0) € Qco- The same is valid for the function Wk^ if e is small enough. Now we show that Wto (as a function on Qq,) is negative definite. Calculate the derivative of Wka along the flow:

#to = + \gaar^r^ + I’^Pah (T 1 • + -r'^PaPb + V + ^J 2 • (3.13) a=l Using the explicit representation of equation (1.17), we obtain

- (3.14)

Therefore.

gcar°r^ + \g .a r‘^r‘^ + /""P .A + \ p ‘^PaPb + V

l a M 2 dr°

70 Using skew-symmetry of and K.qBi with respect to a, ^ and canceling the terms

+ l>. we obtain

+ \gair“f‘‘ + r'& A + \i°^P.Pi + V

= (P .P . - p .p .) r“ . (3.15)

Substituting (3.15) in (3.13) and determining r “ from (3.14), we obtain

O’ W-'b. = + ( ^ ( f » ) "

- + P£,/“P.P.)

7=1 '

+ é (-9.«/ + 7 = 1

- V 0abl‘”'V cr^ -

1 9 /“* + 2 (f.fk - ^ » n ) r “ + / “ P L (PaA - ?.?&) r “. (3.16)

Since ' 1 a ra6 at the equilibrium and the linear terms in the Taylor expansions of V and P are the same. dV 1 3 /“* + g + '^ta^^‘^'PcPd = FaOU^ + {nonlinear terms}, (3.17) where

= _ËÜ1 + i J!£Ü-p P + ^ P ©= p 3r“3r^ 2 3r“ ^ + %/'") + PL-f“ + D^P.P.)

= VqV^C/.

71 In the last formula all the terms are evaluated at the equilibrium.

Taking into account that gag = Sa0 + 0{u), that the Taylor expansion of PaPb - VaVb starts from the terms of the second order, and using (3.17), we obtain from (3.16)

a = l

- 6 {V0 abl'’‘'{ro)p° + B l0 {ro)pl) + {cubic terms}.

Therefore, the condition {Va 0 b + T^ 0 ab)I^^V°c ^ 0 implies that Wk^ is negative definite if e is small enough and positive. Thus, Wk^ is a Lyapunov function for the flow on Qco, and therefore the equilibrium (rg, 0) for the flow on Qco is asymptotically stable. Using the same arguments we used in the proof of Theorem 3.1.1, we conclude that the equilibria on the nearby invariant manifolds Qk are asymptotically stable as well. ■

There is an alternative way to state the above theorem, which uses the basic intu­ ition we used to find the Lyapunov function.

Theorem 3.2.2 (The nonholonomic energy-momentum method) Under the assumption that H2 holds, the point = (r^, 0,p°) is a relative equilibrium if and only if there is a ^ £ g'^' such that Çg is a critical point of the augmented energy = E — {p — P{r,k),^). Assume that

(i) 5‘^E^ restricted to Tq^Qk is positive definite (here 5 denotes differentiation by all variables except E,):

(ii) the quadratic form defined by the flow derivative of the augmented energy is negative definite at q^.

Then H3 holds and this equilibrium is Lyapunov stable and asymptotically stable in the directions due to the invariant manifolds ( 3 .1 2 ).

72 P ro o f We have already shown in Theorem 3.1.2 that positive definiteness of is equivalent to the condition VqV^C/ » 0. To complete the proof, we need to show that requirement (ii) of the theorem is equivalent to the condition {VaSb + 'Ddab)^‘“'{T^o)Pc > 0- Compute the flow derivative of E f

È^ = È - { p - p , o = È -

Since at the equilibrium p = P. ^“ = I'^Pbi and P = 0 (Theorem 1.4.6), we obtain

4 = -Pa^a/“'’(ro)pgr“r^.

The condition 0 is thus equivalent to {Va^b + D^a6)/'"^(ro)Pc » 0. ■

For some examples, such as the roller racer, we need to consider a degenerate case of the above analysis. Namely, we consider a nongeneric case, when U = j / “*(r)poP6 (the original system has no potential energy), and the components of the locked inertia tensor / “* satisfy the condition 1 prai

Recalling our definition of the covariant derivatives in formula (3.4), we observe that the covariant derivatives VqP of the amended potential are equal to zero, and further that the equations of motion (1.17), (1.18) become

^ (3.19)

+ © ,» r V . (3.20)

Thus, we obtain an (m + a)-dimensional manifold of equilibria r = tq, p = po of these equations. Further, we cannot apply Theorem 3.2.1 because the condition 3> 0 fails. However, we can do a similar type of stability analysis as follows. -A.S before, set

Vk = P(r, r, P(r, k)) = + i/“'’(r)P„(r, k)Pb{r, k).

73 Note that P satisfies the equation

dr°— = which implies that

AI I =

Therefore \ l ‘''‘PaPb = const and Vk = ^9a0r‘'r^

(up to an additive constant). Thus, Vk is a positive definite function with respect to r. Compute Vk'.

I:; = + 0(r")-

Suppose that [VaBb + î^/3û6)(^o)-1'*‘^(^o)p2 3> 0. Now the linearization of equations (3.19) and (3.20) about the relative equilibria given by setting f = 0 has (m + a) zero eigenvalues corresponding to the r and p directions. Since the matrix corre­ sponding to the f-directions of the linearized system is of the form D + G, where

D is positive definite and symmetric (in fact, D = |(I?a/36 + T>^a6)(^o)-^*‘^(^o)Pc) ^.nd G is skew-symmetric, the determinant of D + G is not equal to zero. This follows from the observation that r'(D 4- G)x = x‘Dx > 0 for D positive-definite and G skew-symmetric. Thus using Theorem 1.5.7, we find that the equations of motion have local integrals r = 7Z{r,k), p = V{r,k).

Therefore Vk restricted to a common level set of these integrals is a Lyapunov function for the restricted system. Thus, an equilibrium r = tq, p = po is stable with respect

74 to (r, f, p) and asymptotically stable with respect to r if

{V^0b + Vg^t)I^{To)pl » 0. (3.21)

Summarizing, we have:

Theorem 3.2.3 Under assumptions H2 if V = 0 and conditions (3.18) and (3.21) hold, the nonholonomic equations of motion have an {m + a)-dimensional manifold of equilibria parametrized by r and p. An equilibrium r = tq, p = po ^ stable with respect to {r,f,p) and asymptotically stable with respect to r.

3.2.3 The Roller Racer

The roller racer provides an illustration of Theorem 3.2.3. Recall that the Lagrangian and the constraints are

L = im (i= ' + f ) + i/,«= + + i f

and

\ sin 0 sin 0 y

\ Sin 0 sin 0 J The configuration space is SE{2) x 50(2) and, as observed earlier, the Lagrangian and the constraints are invariant under the left action of SE{2) on the first factor of the configuration space. The nonholonomic momentum is

p = m{d\ cos0 4- d2){xcos0 + ijsmd) + [(/i + fg)^ + l 2 ^\ sin0.

See Tsakiris [1995] for details of this calculation. The momentum equation is

((/i + /2) COS0 - mdi(di COS0 + ^2)) sin0 • ^ m(di COS0 + ^2)^ + (fi + ^2) sin^0 ^ m(di + d2 cos 0) (72^1 cos 0 — I\ 8 2 ) ^2 m[di cos 0 + d2 Y + (7i + I 2 ) sin^ 0

75 Rewriting the Lagrangian using p instead of 6, we obtain the energy function for the roller racer: E =

where ^ [é] — j [m(di cos + dgjdg 4- 72 sin^

■ .r . r\_: (3-22) m(di cos (j) + + (/i + 1 2 ) sin^ (f>

The amended potential is given by

r? U = 2[m(di cos 0 + d2 Y + (A + h ) sin^ 0] ’ which follows directly from (1.16) and (3.22). Straightforward computations show that the locked inertia tensor /(0) satisfies condition (3.18), and thus the roller racer has a two-dimensional manifold of relative equilibria parametrized by 0 and p. These relative equilibria are motions of the roller racer in circles about the point of intersection of lines through the axles. For such motions, p is the system momentum about this point scaled by a factor of sin 0, where o is the relative angle between the two bodies. Therefore, we may apply the energy-momentum stability conditions (3.21) ob­ tained in §3.2.2 for the degenerate case. Multiplying the coefficient of the nontrans­ port term of the momentum equation, evaluated at 0o, by /(0)po and omitting a positive factor, we obtain the condition for stability of a relative equilibrium 0 = 0o, p = Po of the roller racer:

(01 -f 02 cos 0 o) (^201 cos 00 - /l 02)po > 0 .

Note that this equilibrium is stable modulo SE{2) and in addition is asymptotically stable with respect to 0.

76 3.3 Nonlinear Stability by the Lyapunov-Malkin Method

Here we study stability using the Lyapunov-Malkin approach; correspondingly, we do not a priori assume hypotheses HI (skewness of Vaffb in a,/9), H2 (a curvature is zero) or H3 (definiteness of second variations). Rather, at the end of this section we will make eigenvalue hypotheses. We consider the most general case, when the connection due to the transport part of the momentum equation is not necessary flat and when the nontransport terms of the momentum equation are not equal to zero. In the case when is commutative, this analysis was done by Karapetyan [1980]. Our main goal here is to show that this method extends to the noncommutative case as well. We start by computing the linearization of equations (1.17) and (1.18). Introduce coordinates (u“, Wa) in the neighborhood of the equilibrium r = ro, p = po by the formulae r° = r^ + u®, r “ = Pa = P° + '^a-

The linearized momentum equation is

To find the linearization of (1.17), we start by rewriting its right-hand side explicitly. Since R = - ^I^^PaPb ~ ! ’• equation (1.17) becomes

+ ga„r°f^ - ^

Keeping only the linear terms, we obtain

aZT - 1 ffijab a Tab

= -C;„/=''(ro)p>, - - -^g-(n)plvU ‘

- C s ../“ (ro)pS T,".

77 Next, introduce matrices A, !B, C, and V by

.

, (3.23)

e; = - (^(ro)pS +PL^“ (’-o)p2 + K .I’^(r„)pl) . (3.24)

'Daa=Vl,{ro)pl (3.25)

Using these notations and making a choice of r“ such that gapiro) = we can represent the equation of motion in the form

û“ = u“, (3.26)

û“ = A^v^ + + V°{u, V, w), (3.27)

tha = 2)aaU° + Wa(u,z;,u;), (3.28)

where V and W stand for nonlinear terms, and where

>1^ =

gaa ^ jan-q

(Or yig = = 9°'^'^-y0, G““ = if ^o^(ro) ^ Sap.) Note that

"Wa = (%^L(Pc + ^c) - I>aû) + VaPaV^V^. (3.29)

The next step is to eliminate the linear terms from (3.28). Putting

U^a — 25oQ^ "P 2(1,

(3.28) becomes Za = Za{u,V,z), where Za{u.v,z) represents nonlinear terms. Formula (3.29) leads to

Za{u, V, z) = Zaa{u, V, z)v°‘.

78 In particular, Zg(u, 0, z) = 0. Equations (3.26), (3.27), (3.28) in the variables {u,v.z) become

11° = v°,

i’° = A $ V ^ + ( 3 g + e ° ° T > a 0) u ^ + e ° ° Z a + V°{u, V, 2 . +

Za = Za(u,V,w).

Using Lemma 1.5.9, we find a substitution x° = u° +4>°[z), y° = v° such that in the variables (x, y, z) we obtain

x° = y° + X °{x,y ,z),

r = A%y^ + (3g + Q°°Va0 )x^ + Y°{x, y, z), (3.30)

Za = Za{x,y,z), where the nonlinear terms X (x, y,z), Y (x, y ,z ,), Z(x, y, z) vanish if x = 0 and y = 0. Therefore, we can apply the Lyapunov-Malkin theorem and conclude:

Theorem 3.3.1 The equilibrium x = 0 , y = 0 , z = 0 of system (3.30) is stable with respect to [x.y.z) and asymptotically stable with respect to (x,y) if all eigenvalues of the matrix

0 l\ (3.31) /B -kG T A. have negative real parts.

3.3.1 The Lyapunov-Malkin and the Energy-Momentum Methods

Here we introduce a forced linear Lagrangian system associated with our nonholo­ nomic system. The linear system will have matrix (3.31). Then we compare the Lyapunov-Malkin approach and the energ}-momentum approach for systems satisfy­ ing hypothesis H2.

79 Thus, we consider the system with matrix (3.31)

(3.32) y = A y + {T, + QT>)x.

According to Theorem 3.3.1, the equilibrium a: = 0, y = 0, 2 = 0 of (3.30) is stable with respect to (x, y, z) and asymptotically stable with respect to (x, y) if and only if the equilibrium x = 0, y = 0 of (3.32) is asymptotically stable. System (3.32) may be viewed as a linear unconstrained Lagrangian system with additional forces imposed on it. Put

c = ((3 + e v ) + (® + e v Y ) ,

F = 1 ((3 + en) - (3 + e v Y ) ,

D = (A + A ^),

G = i(A -A ‘).

The equations become

x“ = -C^x^ + F^x^ - Dgx^ + Ggx^. (3.33)

These equations are the Euler-Lagrange equations with dissipation and forcing for the Lagrangian i = l - G'SxOf - with the Rayleigh dissipation function

and the nonconservative forces F aV .

Note that = {VaSb + '^ 0 ab)F‘’{ro)pl. The next theorem explains how to compute the matrices C and F using the amended potential of our nonholonomic system.

80 Theorem 3.3.2 The entries of the matrices C and F in the dissipative forced system (3.33), which is equivalent to linear system (3.32), are

Cad = ^ 0^aW {ro,P o), Faff = ~(VaV^ - V;3 Va)C/(ro, Po)-

Proof Recall that the operators of covariant dilferentiation due to the transport equation are (see (3.4)) „ d d V. = g;:j+I>„Pc^-

Consequently,

VflVa = Vg + KaPc- dpa

dr^dr°‘ dr^ Therefore, for the amended potential U = V + ^I'^^PaPb we obtain

1 Q2jab g V flV af/ = Q ^ J g ^ + ) PcPd grab + g;xP«Pi + + KT>Lr“PcPé-

Formulae (3.23), (3.24), and (3.25) imply that

g2\~ 1 g2 Tab g (» e + ®)=o = + * 7 (--oIpM grab + ï)w^(ro)p:p3 + + %Z)L/-(ro)p2p3

= VgVaf/(ro,po).

Therefore

Cog = ^(VaVg +VgVa)[/(ro,Po), = ^(VaV^ - VflVa)t/(ro,Po). I

Observe that the equilibrium x = 0, y = 0, 2 = 0 of (3.30) is stable with respect to (x, y, z) and asymptotically stable with respect to (x, y) if and only if the equilibrium

81 I = 0, y = 0 of the above linear lagrangian system is asymptotically stable. The

condition for stability of the equilibrium r = Tq, p = po of our nonholonomic system becomes: all eigenvalues of the matrix

0 I VgV^[/(ro,Po) -{V0abI^{To) + B%0{ro))pl^

have negative real parts. If the transport equation is integrable (hypothesis H2), then the operators Vq and Vg commute, and the corresponding linear Lagrangian system (3.33) has no nonconservative forces imposed on it. In this case the sufficient conditions for stability are given by the Thompson theorem (Thompson and Tait [1987], Chetaev[1959|): the equilibrium x = 0 of (3.33) is asymptotically stable if the matrices C and D are positive definite. These conditions are identical to the energy-momentum conditions for stability obtained in Theorem 3.2.1. Notice that if C and D are positive definite, then matrix (3.31) is positive definite. This implies that the matrix A in Theorem 1.5.7 has spectrum in the left half plane. Further, our coordinate transformations here give the required form for the nonlinear terms of Theorem 1.5.7. Therefore, the above analysis shows that hypothesis H3 implies the hypotheses of Theorem 1.5.7.

Remark. On the other hand (cf. Chetaev [1959]), if the matrix C is not positive definite (and thus the equilibrium of the system x = —Cx is unstable), and the matrix D is degenerate, then in certain cases the equilibrium of the equations x =

—Cx — Dx 4- Gx may be stable. Therefore, the conditions of Theorem 3.2.1 are sufficient, but not necessary.

3.3.2 The Rattleback

Here we outline the stability theory of the rattleback to illustrate the results discussed above. The details may be found in Karapetyan [1980, 1981] and Markeev [1992].

82 Recall that the Lagrangian and the constraints are

L = i [i4 cos^ ii> + B sin^ 4- m(7i cos 0 — C sin d)^]

4- ^ [(.4 sin^ ip + B cos^ sin^ 6 + C cos^ 0\

4- ^ (C 4- m'yl sin^ d) ijx^ 4- {x^ 4- y^)

4- 7n(7i cos 0 — C sin 0)72 sin ÔÔ-iI; + {A — B) sin 9 sin ip cos ib è(p

4- C cos 6 ^ip 4- mg{'yi sin 9 + Ç cos 9)

and

X = ai 9 4 - Q21P 4- 0130, ÿ = 0 i9 4- /hip 4-

where the terms were defined in §1 .3 .5 . Using the Lie algebra element corresponding to the generator = ocsdx+Pjdy+d^ we find the nonholonomic momentum to be

p = I (9, ip)o 4- [(-4 — B) sin 9 sm Ip cos Ip — m(7i sin0 4- C cos 0)72] 9

4- [C cos 9 4- m(72 cos 0 4 - 71 (71 cos 0 — C sin 0 ))] ii\

where

I {9, Ip) = (.4 sin^ V 4- Scos^ ip) sin^ 9 + C cos^ 9 4- 01(7! 4- (71 cos 9 — Ç sin0 )^).

The amended potential becomes

p^ f-' = ^-y - T^ailx sin 0 4- c cos 9).

The relative equilibria of the rattleback are

9 = 9q. Ip = ipo, p = Po where 9q, ipo, and po satisfy the conditions

mp(7i cos 00 - Csin 0 o)/^( 0 o,i/;o) 4 - [(Asin^i/)o 4- Bcos^ ipo — C) sin 00 cos 00

- m(7i cos00 - (sin 00)(71 sin00 4- ( cos 0 q)] Po = 0

83 and

mg'y2p{9o, ipo) + [(^ - B) sin 6q sint^o cos rpo - 77272(71 sin 0o + Ccos0o)] Po = 0-

which are derived from VgC/ = 0, V^C/ = 0. In particular, consider the relative equilibria

7T P = Po,

that represent the rotations of the rattleback about the vertical axis of inertia. For such relative equilibria ^ = 0, and therefore the conditions for existence of relative equilibria are trivially satisfied with an arbitrary value of po. Omitting the computa­ tions of the linearized equations for the rattleback, which have the form discussed in §3.3.1 (see Karapetyan [1980] for details), and the corresponding characteristic poly­ nomial, we just state here the Routh-Hurwitz conditions for all eigenvalues to have negative real parts:

(3.34) S2 J 5 2

(.4 - C)(r2 - ri)posinacosQ > 0. (3.35)

If these conditions are satisfied, then the relative equilibrium is stable, and it is asymptotically stable with respect to {6,9, tp, tp). In the above formulae and Tq stand for the radii of curvature of the body at the contact point, a is the angle between horizontal inertia axis and the r;-curvature direction, and the quantities P, R, and S are

P = (.4-1- ma^){C + ma^),

R=[{A + C-B + 2ma^Ÿ

— [A-^C — B + 2mc?)ma{ri + T2 ) -I- m^a^rir2] ^

Po Po (.4 - B )^ + m{a - sin^ a - T2 cos^ a) -f- (v4 -h 7720^)

(C - B) ^ -f- m{a - T2 sin^ a - ti cos^ a) -t- ( C A 7720^),

84 1•X / 2 s = (*4 — B){C — B ) ^ + m^(a — ri)(a — T2) B*

+ [A(a - ri cos^ a - V2 sin^ a)

+ C (a — ri sin^ a — cos^ a) — B(2a — — r 2)] -

Condition (3.34) imposes restrictions on the mass distribution, the magnitude of the angular velocity, and the shape of the rattleback only. Condition (3.35) dis­ tinguishes the direction of rotation corresponding to the stable relative equilibrium. The rotation will be stable if the largest (smallest) principal inertia axis precedes the largest (smallest) direction of curvature at the point of contact. The rattleback is also capable of performing stationary rotations with its center of mass moving at a constant rate along a circle. A similar argument gives the stability conditions in this case. The details may be found in Karapetyan [1981] and Markeev [1992].

85 CONCLUSION

We have given a general energy-momentum method for analyzing the stability of relative equilibria of a large class of nonholonomic systems. We have also shown that for systems to which the classical Lyapunov-Malkin theorem applies, one can interpret and even verify the hypotheses in terms of definiteness conditions on the second variation of energy-momentum functions. We have also studied stability of some systems (in the pure transport case) for which energetic arguments give stability, but the hypotheses of the Lyapunov-Malkin theorem fail (because eigenvalues are on the imaginar}' axis). We also developed an approach for studying integrable nonholonomic systems, using the Routh problem as an example. This method allows one to find quasi- periodic motions and study the frequency properties of these motions. A similar method could be applied to other systems such as the rolling disk and the solid of revolution on the horizontal plane. The results obtained in this thesis may be used for further development of non­ holonomic mechanics. We intend to develop the general theory of integrable non­ holonomic systems based on the approach of Chapter 2. We also plan to apply the energy-momentum method to problems of control and feedback stabilization of non­ holonomic systems with symmetry. As indicated in the text, not all nonholonomic systems satisfy the assumptions made in this paper and we intend to consider these in future research.

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