Reduction, Linearization and stability of relative equilibria for mechanical systems on riemannian manifolds

REDUCTION, LINEARIZATION, AND STABILITY OF RELATIVE EQUILIBRIA FOR MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS FRANCESCO BULLO AND ANDREW D. LEWIS Abstract. Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory “commute.” As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory. 1. Introduction Mechanical systems with symmetry have been a focus of an enormous research effort during the past few decades. This reflects of the importance of the notion of symmetry in physics. Of particular importance are those trajectories of a dynamical system that are also orbits for the symmetry group of the problem; these are relative equilibria. The stability of relative equilibria has both theoretical and practical importance. From a theoretical point of view, the relative equilibria, and their associated stability analysis, often give important insight into global behavior of solutions. In practical applications, relative equilibria arise in such diverse areas as fluid mechanics and underwater vehicle dynamics. In such problems, one often desires stability of a given relative equilibrium. Should such a relative equilibrium be naturally unstable, one must then develop ways of stabilizing it using control theory. Date: 28/02/2005. 1 2 FRANCESCO BULLO AND ANDREW D. LEWIS This paper is concerned with the analysis of relative equilibria for simple mechanical systems, i.e., those that are Lagrangian with kinetic energy minus potential energy Lagrangians. More specifically, in the paper we explore some of the Riemannian geometry associated with a system with symmetry in general, and a relative equilibrium in particular. Of course, much work has been done in this area, so let us locate our work in this body of literature. When talking about reduction using symmetry (as opposed to the more specific discussion of relative equilibria), one can work in a Hamiltonian or Lagrangian setting. The Hamiltonian setting is perhaps the more developed, going back to work of Arnol’d [1966], and the symplectic reduction work of Marsden and Weinstein [1974] and Meyer [1973]. This work has been presented in a fully developed manner in the books [Abraham and Marsden 1978, Guillemin and Sternberg 1984, Marsden 1992, Marsden and Ratiu 1999], for example. More recent research on Hamiltonian reduction theory is concerned with so-called singular reduction theory, where the regularity assumptions of the earlier work are relaxed. We refer the reader to [Ortega and Ratiu 2004] for the literature in this area. The Lagrangian theory of reduction is more recent, and we refer to the presentation in the book [Marsden and Ratiu 1999], and to the papers [Cendra, Marsden, Pekarsky, and Ratiu 2003, Cendra, Marsden, and Ratiu 2001] as representative of the work in this area. In the Hamiltonian theory of reduction using symmetry, the symplectic, or more generally Poisson, structure plays the prominent role. Indeed, much of the work in this area is concerned with general symplectic manifolds rather than cotangent bundles. On the Lagrangian side, the emphasis has been on the reduction of variational principles, the motivation for this being that variational principles are fundamental for Lagrangian mechanics. In this paper we focus on Lagrangians that are of the kinetic energy minus potential energy form. For such Lagrangians, an important role is played by the kinetic energy Riemannian metric and its attendant Levi-Civita connection. With this as motivation, we study reduction and relative equilibria strictly in terms of Riemannian geometry. We do not wish to assert that the Riemannian geometry approach we give here is superior to the variational approach; we too believe in the primacy of the variational principle in Lagrangian mechanics. However, the extra structure offered by the Riemannian metric does lead one to naturally ask whether a theory of reduction based purely on Riemannian geometry is possible and, if so, revealing. We show that it is possible, and we believe that it is revealing. This is presented in Section 3. Another emphasis of this paper is linearization. We start this in a rather general way. Since the stabilization of relative equilibria is of interest [Bullo MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 3 2000, Jalnapurkur and Marsden 2000, 2001], we develop our theory of linearization in the setting of control systems. One of the main contributions of the paper is to explain the geometry that connects the unreduced and reduced linearizations. Since the unreduced linearization is about a trajectory and not an equilibrium point, we first present a rather general geometric theory of linearization of a control-affine system about an arbitrary controlled trajectory. We then specialize this to so-called affine connection control systems; these are control-affine systems whose state manifold is the tangent bundle of a configuration manifold, whose drift vector field is the geodesic spray for an affine connection, and whose control vector fields are vertical lifts of vector fields on the configuration manifold. As is explained in [Bullo and Lewis 2004, Chapter 4], affine connection control systems arise in the modeling of a large class of mechanical systems, including those with nonholonomic constraints. The linearization of an affine connection control system is related to the Jacobi equation of geodesic variation, and we explicitly develop this relationship in Proposition 5.6; this association is not surprising, but it does not appear to have been presented before. We then specialize the affine connection control system setup to the special case where the affine connection is the Levi-Civita connection. In Theorem 5.10 we explicitly show how the general linearization for an affine connection control system is related to the linearization of the reduced system. Again, it is not surprising that this should be possible. However, the explicit development does not seem to have been presented, and is not entirely trivial. The final topic of the paper is stability of relative equilibria. We present here two stability theorems, one linear and one nonlinear, the latter relying on the Lyapunov Stability Criterion and the LaSalle Invariance Principle. As with our study of linearization, one of the essential contributions here is an understanding, in the context of Riemannian geometry, of the relationship between the unreduced linearized energy and the reduced linearized energy. Here again, that there should be a relationship is expected, but the explicit development, culminating in Proposition 6.6, is new and not completely trivial. In particular, we utilize the Sasaki metric in our presentation, something that has not been done before to the best of our knowledge. 2. Simple mechanical (control) systems with symmetry In this section we review some well-known concepts in geometric mechanics, and mechanical systems with symmetry. We refer the reader to [Bloch 2003, Bullo and Lewis 2004, Marsden and Ratiu 1999] for additional discussion. Our presentation is in the context of systems with controls, since the development of Sections 3 and 5 is also done in this context. Our notation mostly follows [Bullo and Lewis 2004]. 4 FRANCESCO BULLO AND ANDREW D. LEWIS Notation. In this section we quickly present some of the concepts and notation we use. If Uand Vare R-vector spaces, L(U;V) denotes the set of linear maps from Uto V. Throughout the paper we work with manifolds and geometric object that are of class C∞, unless stated to the contrary. If Mis a manifold, then C∞(M) denotes the set of C∞-functions on M. The tangent bundle of Mis denoted by πTM :TM →M. If (x1, . . . , xn) are coordinates for M, then natural tangent bundle coordinates for TM are denoted by ((x1, . . . , xn),(v1, . . . , vn)), or sometimes more compactly by (x,v). The derivative of a map φ:M→Nis denoted by Tφ:TM →TN, and the restriction of the derivative to the tangent space TxMis denoted by Txφ. If π:E→Bis a vector bundle, then Γ∞(E) denotes the set of C∞-sections of E. A vector field X:E→TE on the total space of a vector bundle π:E→B is a linear vector field over a vector field X0on Bif Xis π-related to X0 and if the following diagram commutes: EX// π ²² TE Tπ ²² BX0 //TB The zero section of a vector bundle Eis denoted by Z(E). If Bis a (0,2)- tensor field on a manifold M, then B[:TM →T∗Mis the vector bundle map defined by hB(ux); vxi=B(vx, ux) for ux, vx∈TM. If B[in an isomorphism, then its inverse is denoted by B]. If Gis a Riemannian metric on Mand if f∈C∞(M), then gradfdenotes the vector field gradf= G]◦df. Here dfis the differential of f. Also, k·kGdenotes the norm on the fibers of TM defined by G. Typically, I⊂Rwill denote an interval. If γ:I→Mis a differentiable curve on M, then its tangent vector field is denoted by γ0:I→TM. A curve γ:I→Mis locally absolutely continuous (abbreviated LAC ) if f◦γis locally absolutely continuous for every f∈C∞(M). If γ0is LAC, then γis locally absolutely differentiable (abbreviated LAD). For a vector field X, the flow of Xis denoted by (t, x)7→ ΦX 0,t(x), so that the integral curve through xis t7→ ΦX 0,t(x). We will be considering time-varying vector fields at various points in the paper, and since we wish to allow fairly general time-dependence, we should be precise about how we do this. To this end, by a time-dependent vector field on Mwe shall mean a map X:I×M→TM with the following properties: (1) for each t∈I, the map Xt:x7→ X(t, x) defines a C∞-vector field; MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 5 (2) for each x∈M, the map t7→ X(t, x) is measurable (meaning its components are measurable in some, and so any, set of coordinates); (3) for each k∈\, each collection X1, . . . , Xkof C∞-vector fields on M, each C∞-one-form αon M, and each compact subset K⊂M, there exists a positive locally integrable function ψ:I→Rsuch that |LX1···LXkhα;Xti(x)| ≤ ψ(t), x ∈K. The usual Carath´eodory theory shows that time-dependent vector fields defined in this fashion have LAC integral curves, and that their flows are of class C∞with respect to initial conditions. We shall use the notation (t, x)7→ ΦX 0,t(x) to denote the flow of a time-dependent vector field X, i.e., we use the same notation for the flow of both time-independent and time-dependent vector fields. 2.1. Simple mechanical (control) systems. Aforced simple mechanical control system is a 5-tuple Σ = (Q,G, V, F, F={F1, . . . , Fm}) where (1) Qis the configuration manifold, here assumed to be of pure dimension n, (2) Gis the kinetic energy metric, which is a Riemannian metric on Q, (3) Vis the potential energy, a function on Q, (4) F:TQ →T∗Qis a bundle map over idQ, which is the external force, and (5) {F1, . . . , Fm}are the control forces, which are one-forms on Q. The presentation in [Bullo and Lewis 2004] also includes an extra piece of data in the form of a set U⊂Rmwhere the control takes values. However, for the purposes of this paper, this is not important, so we do not include it (that is to say, we take U=Rm). The equations governing a simple mechanical control system are G ∇γ0(t)γ0(t) = −gradV(γ(t)) + G]◦F(γ0(t)) + m X a=1 ua(t)G]◦Fa(γ(t)),(1) where G ∇is the Levi-Civita connection associated with G. We also denote by G Γi jk the Christoffel symbols for G ∇. A controlled trajectory for Σ is thus a pair (γ, u) where u:I→Rmis locally integrable and defined on an interval I⊂R, and where γ:I→Qis LAD and satisfies (1). Simple mechanical control systems are examples of a more general class of systems which we now introduce. A forced affine connection control system is a 4-tuple (Q,∇, Y, Y={Y1, . . . , Ym}), where (1) Qis a manifold as above, 6 FRANCESCO BULLO AND ANDREW D. LEWIS (2) ∇is an affine connection on Q, (3) Y:TQ →TQ is a bundle map over idQ, and (4) {Y1, . . . , Ym}are vector fields on Q. The equations governing such a system are ∇γ0(t)γ0(t) = Y(γ0(t)) + m X a=1 ua(t)Ya(γ(t)). We can define the notion of a controlled trajectory for an affine connection system in the same way as we did for a forced simple mechanical control system. Moreover, note that a forced simple mechanical control system is also a forced affine connection control system by taking ∇= G ∇,Y= gradV◦πTQ +G]◦F. In [Bullo and Lewis 2004, Chapter 4] it is shown that affine connection control systems model a large class of mechanical systems, including those with nonholonomic constraints. Parts of the paper will be concerned with systems without forces, so let us introduce notation for these. A forced simple mechanical system is a quadruple (Q,G, V, F), where all the data are as for a forced simple mechanical control system. Of course, the governing equations are G ∇γ0(t)γ0(t) = −gradV(γ(t)) + G]◦F(γ0(t)). If the external force Fis omitted from list of data, the resulting triple (Q,G, V ) is a simple mechanical system. 2.2. Systems with symmetry. Now let us add symmetry to the above formulations. In this paper we shall only consider a single infinitesimal symmetry, rather than the more usual situation where one considers symmetry of the system under a Lie group G. The extension of the results in the paper to this more general setup of a Lie group symmetry is currently ongoing. Thus we consider a forced simple mechanical system Σ = (Q,G, V, F, F={F1, . . . , Fm}) and a vector field Xon Qhaving the following properties: (1) Xis an infinitesimal isometry for G, i.e., LXG= 0; (2) LXV= 0; (3) LX(F(Y)) = F(LXY) for all vector fields Yon Q; (4) LXFa= 0 for a∈ {1, . . . , m}. A vector field Xhaving these properties is an infinitesimal symmetry for Σ. The infinitesimal symmetry Xis complete if Xis a complete vector field. A vector field having only property 1 is an infinitesimal isometry for G. Let us give some useful properties of infinitesimal isometries of G. MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 7 Proposition 2.1 (Characterization of infinitesimal isometry).Let (Q,G) be a Riemannian manifold. The following statements hold: (i) the vector field Xis an infinitesimal isometry if and only if the covariant differential G ∇Xis skew-symmetric with respect to G; that is, for all Y, Z ∈Γ∞(TQ), G(Y, G ∇ZX) + G(Z, G ∇YX) = 0; (2) (ii) if the vector field Xis an infinitesimal isometry, then the function q7→ kXk2 G(q)is X-invariant and satisfies 1 2grad kXk2 G=− G ∇XX. (3) Proof. Regarding part (i), we compute, for Y, Z ∈Γ∞(TQ), G(Y, G ∇ZX) + G(Z, G ∇YX) = G(Y, G ∇XZ+ [Z, X]) + G(Z, G ∇XY+ [Y, X]) =LX(G(Y, Z)) −G(Y, [X, Z]) −G(Z, [X, Y ]) = (LXG)(Y, Z). The result follows since this equality holds for all Y, Z ∈Γ∞(TQ). To prove equation (3), let Z∈Γ∞(TQ) and compute G(Z, grad kXk2 G) = LZkXk2 G= 2G(X, G ∇ZX) = −2G(Z, G ∇XX). To show that q7→ kXk2 G(q) is X-invariant, note that LXkXk2 G=−2G(X, G ∇XX) = 0. ¤ Associated to an infinitesimal isometry Xfor Gis the X-momentum map JX:TQ →Rdefined by JX(vq) = G(X(q), vq). One can show [Bullo and Lewis 2004, Theorem 5.69] that the X-momentum map satisfies the evolution equation d dtJX(γ0(t)) = ¯ Fu(t, γ0(t)); X(γ(t))® along a controlled trajectory (γ, u), where ¯ Fu(t, vq) = F(vq) + X a=1 ua(t)Fa(vq). In particular, if F= 0 and ua(t) = 0, a∈ {1, . . . , m}, for all t, then the X-momentum is conserved; this is the Noether Conservation Law in this particular setup. 8 FRANCESCO BULLO AND ANDREW D. LEWIS 2.3. Relative equilibria. For mechanical systems with symmetry, relative equilibria are important since they often give initial insights into the behavior of a system. In control theory, relative equilibria are often elementary mechanical motions that can be used as a basis for, for example, motion planning. For additional discussion we refer to [Bloch 2003, Marsden 1992, Marsden and Ratiu 1999]. Definition 2.2 (Relative equilibrium).Let Xbe a complete infinitesimal symmetry for Σ=(Q,G, V, F, F)and let χ:R→Qbe a maximal integral curve of X. (i) The curve χis a relative equilibrium for Σif it is a solution to the equation of motion (27). (ii) The relative equilibrium χis regular if χis an embedding. • To characterize relative equilibria, it is convenient to have the following notion. We define the energy for a forced simple mechanical control system Σ = (Q,G, V, F, F) by E(vq) = 1 2G(vq, vq) + V(q). Definition 2.3 (Effective energy).Let Xbe a complete infinitesimal symmetry for Σ=(Q,G, V, F, F), and let JX:TQ →Rbe the associated momentum map. (i) The effective potential VX:Q→Ris the function VX(q) = V(q)−1 2kXk2 G(q). (ii) The effective energy EX:TQ →Ris the function EX(vq) = E(vq)−JX(vq).• The following result characterizes the effective energy. Lemma 2.4. EX(vq) = VX(q) + 1 2kvq−X(q)k2 G. Proof. The lemma is a consequence of the following “completing the square” computation: E(vq)−JX(vq) = V(q) + 1 2kvqk2 G−G(vq, X(q)) =V(q)−1 2kXk2 G(q)+ 1 2kXk2 G(q)+ 1 2kvqk2 G−G(vq, X(q)) =VX(q)+ 1 2kvq−X(q)k2 G, as desired. ¤ Using this result, we can characterize relative equilibria as follows. Proposition 2.5 (Existence of relative equilibria).Consider a forced simple mechanical system Σ=(Q,G, V, F)and let Xbe a complete infinitesimal symmetry for Σ. Then a maximal integral curve χ:R→Qof Xis a relative equilibrium for Σif and only if dVX(χ(t)) = F(χ0(t)) for some (and hence MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 9 for all) t∈R. In particular, if F(χ0(t)) = 0 for some (and hence for all) t∈R, then χis a relative equilibrium for Σif and only if VXhas a critical point at χ(t)for some (hence for all) t∈R. Proof. We proceed in a direct way. Assume that χis a maximal integral curve of X. The curve χsatisfies the equations of motion (27) if and only if 0 = G ∇χ0(t)χ0(t) + gradV(χ(t)) −G]◦F(χ0(t)) = G ∇X(χ(t))X(χ(t)) + gradV(χ(t)) −G]◦F(χ0(t)) = grad¡−1 2kXk2 G+V¢(χ(t)) −G]◦F(χ0(t)) =G]◦dVX(χ(t)) −G]◦F(χ0(t)). Therefore, χis a relative equilibrium if and only if dVX(χ(t)) = F(χ0(t) for all t∈R. Because VXis X-invariant, dVX(χ(t)) is independent of t. Since Xis an infinitesimal symmetry for Σ, LXF(X) = F[X, X] = 0. Therefore, F(χ0(t)) is also independent of t, and so the result follows. ¤ 3. Reduction by an infinitesimal symmetry In this section we shall investigate the dynamics of a simple mechanical system with a complete infinitesimal symmetry. As mentioned in the introduction, our treatment is limited in scope; for a more comprehensive treatment we refer the reader to the related work in [Cendra, Marsden, and Ratiu 2001]. 3.1. Preliminary constructions. We begin with an assumption on the nature of an infinitesimal isometry Xthat will hold throughout the paper. Assumption 3.1. Let Xbe a complete infinitesimal symmetry for a simple mechanical system (Q,G, V ). We assume that the set of X-orbits is a manifold, say B, and that the projection πB:Q→Bis a surjective submersion.• In Figure 1 we provide an illustration of how the reader should think about Assumption 3.1. Now we introduce some useful definitions. The vertical distribution VQ is the distribution on Qgenerated by the vector field X. The horizontal distribution HQ is the G-orthogonal complement of VQ so that TQ = HQ ⊕VQ. At each q∈Q, the linear map TqπB:TqQ→TπB(q)Bis a surjection. Therefore, the map TqπB|HqQ:HqQ→TπB(q)Bis a linear isomorphism. The horizontal lift of the tangent vector vb∈TbBat point q∈Qis the tangent vector in TqQgiven by hlftq(vb) = (TqπB|HqQ)−1(vb).(4) 16 FRANCESCO BULLO AND ANDREW D. LEWIS where Γi jk,i, j, k ∈ {1, . . . , n}are the Christoffel symbols for ∇. This defines “hlft” as the horizontal lift map for the connection we describe here. (Note that we make an abuse of notation here by using “hlft” both for the horizontal lift on πB:Q→Band on πTQ :TQ →Q.) We can also define the vertical lift map by vlft³∂ ∂qi´=∂ ∂vi. This definition can be written intrinsically as vlftvq(wq) = d dt¯ ¯t=0(vq+twq), defining an isomorphism from TqQto VvqTQ. Note that we have S(vq) = hlftvq(vq). Also note that this splitting TvqTQ ≃TqQ⊕TqQextends the natural splitting T0qTQ ≃TqQ⊕TqQthat one has on the zero section away from the zero section. We depict the situation in Figure 2 to give the reader vq TvqTQ TqQ Q VvqTQ (canonical) HvqTQ (defined by ∇) 0q0 Tq0Q V0q0TQ (canonical) H0q0TQ (canonical) Figure 2. A depiction of the Ehresmann connection on πTQ :TQ →Qassociated with an affine connection on Q some intuition for what is going on. In this section we “lift” the preceding construction to construct an Ehresmann connection on πTTQ :TTQ →TQ. This requires an affine connection on TQ. It turns out that there are various ways of lifting an affine connection on Qto its tangent bundle, and the one suited to our purposes is defined as follows [Yano and Ishihara 1973]. Lemma 4.2. If ∇is an affine connection on Q, then there exists a unique affine connection ∇Ton TQ satisfying ∇T XTYT= (∇XY)T for vector fields Xand Yon Q. This affine connection is the tangent lift of ∇. MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 17 Now we may construct the Ehresmann connection on πTTQ :TTQ →TQ using the affine connection ∇T. Let us denote this connection by H(TTQ). Note that this connection provides a splitting TXvqTTQ ≃TvqTQ ⊕TvqTQ (7) for Xvq∈TvqTQ. Also, the Ehresmann connection HTQ on πTQ :TQ →Q described above gives a splitting TvqTQ ≃TqQ⊕TqQ. Therefore, we have the resulting splitting TXvqTTQ ≃TqQ⊕TqQ⊕TqQ⊕TqQ.(8) In this splitting, the first two components are the horizontal subspace and the second two components are the vertical subspace. Within each pair, the first part is horizontal and the second is vertical. Let us now write a basis of vector fields on TTQ that is adapted to the splitting (8). To obtain a coordinate expression for the Ehresmann connection on πTTQ :TTQ →TQ, we use the coordinate expression for ∇T. We can then write a basis that is adapted to the splitting of TvqTQ. Let us skip the messy intermediate computations, and simply present the local bases since these are all we shall need. The resulting basis vector field for H(TTQ) are hlftT³∂ ∂qi−1 2(Γj ik + Γj ki)vk∂ ∂vj´=∂ ∂qi−1 2(Γj ik + Γj ki)vk∂ ∂vj −1 2(Γj ik + Γj ki)uk∂ ∂uj−1 2³∂Γj i` ∂qku`vk+∂Γj `i ∂qku`vk+ (Γj ik + Γj ki)wk −1 2(Γk i` + Γk `i)(Γj km + Γj mk)umv`´∂ ∂wj, i ∈ {1, . . . , n}, hlftT³∂ ∂vi´=∂ ∂vi−1 2(Γj ik + Γj ki)uk∂ ∂wj, i ∈ {1, . . . , n}, with the first nbasis vectors forming a basis for the horizontal part of HXvq(TTQ), and the second nvectors forming a basis for the vertical part of HXvq(TTQ), with respect to the splitting HXvq(TTQ)≃TqQ⊕TqQ. Note that we use the notation hlftTto refer to the horizontal lift for the connection on πTTQ :TTQ →TQ. We also denote by vlftTthe vertical lift on this vector bundle. We may easily derive a basis for the vertical subbundle of πTTQ :TTQ → TQ that adapts to the splitting of TvqTQ ≃TqQ⊕TqQ. We may verify 18 FRANCESCO BULLO AND ANDREW D. LEWIS that the vector fields vlftT³∂ ∂qi−1 2(Γj ik + Γj ki)vk∂ ∂vj´=∂ ∂ui−1 2(Γj ik + Γj ki)vk∂ ∂wj, i∈ {1, . . . , n}, vlftT³∂ ∂vi´=∂ ∂wi, i ∈ {1, . . . , n}, have the property that the first nvectors span the horizontal part of VXvqTTQ, and the second nspan the vertical part of VXvqTTQ. 4.3. The Jacobi equation and the tangent lift of the geodesic spray. As we have stated several times already, we will be looking at controlaffine systems whose drift vector field is the geodesic spray Sfor an affine connection. One way to frame the objective of this section is to think about how one might represent STin terms of objects defined on Q, even though STis itself a vector field on TTQ. That this ought to be possible seems reasonable as all the information used to describe STis contained in the affine connection ∇on Q, along with some canonical tangent bundle geometry. It turns out that it is possible to essentially represent STon Q, but to do so requires some effort. What is more, it is perhaps not immediately obvious how one should proceed. To understand the meaning of ST, consider the following construction. Let γ:I→Qbe a geodesic for the affine connection ∇. Let σ:I×J→Q be a variation of γ. Thus (1) Jis an interval for which 0 ∈int(J), (2) s7→ σ(t, s) is differentiable for t∈I, (3) for s∈J,t7→ σ(t, s) is a geodesic of ∇, and (4) σ(t, 0) = γ(t) for t∈I. If one defines ξ(t) = d ds¯ ¯s=0σ(t, s), then it can be shown (see Theorem 1.2 in Chapter VIII of volume 2 of [Kobayashi and Nomizu 1963]) that ξsatisfies the Jacobi equation: ∇2 γ0(t)ξ(t) + R(ξ(t), γ0(t))γ0(t) + ∇γ0(t)(T(ξ(t), γ0(t))) = 0, where Tis the torsion tensor and Ris the curvature tensor for ∇. Thus the Jacobi equation tells us how geodesics vary along γas we vary their initial conditions. With this and Remark 4.1–4 as backdrop, we expect there to indeed be a concrete relationship between STand the Jacobi equation. This relationship involves the Ehresmann connections on πTQ :TQ →Qand πTTQ :TTQ → TQ presented in the preceding section. MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 19 First recall that the connection H(TTQ) on πTTQ :TTQ →TQ and the connection HTQ on πTQ :TQ →Qcombine to give a splitting TXvqTTQ ≃TqQ⊕TqQ⊕TqQ⊕TqQ, where Xvq∈TvqTQ. Here we maintain our convention that the first two components refer to the horizontal component for a connection H(TTQ) on πTTQ :TTQ →TQ, and the second two components refer to the vertical component. Using the splitting (7), let us write Xvq∈TvqTQ as uvq⊕wvq for some uvq, wvq∈TqQ. Note that we depart from the usual notation of writing tangent vectors in TqQwith a subscript of q, instead using the subscript vq. This abuse of notation is necessary (and convenient) to reflect the fact that these vectors depend on where we are in TQ, and not just in Q. We may use this representation of STto obtain a refined relationship between solutions of the Jacobi equation and integral curves of ST. To do so, we first prove a simple lemma. We state a more general form of this lemma than we shall immediately use, but the extra generality will be useful in Section 5. Lemma 4.3. Let Ybe a time-dependent vector field on Q, suppose that γ:I→Qis the LAD curve satisfying ∇γ0(t)γ0(t) = Y(t, γ(t)), and denote by Υ: I→TQ the tangent vector field of γ(i.e., Υ = γ0). Let X:I→ TTQ be an LAC vector field along Υ, and denote X(t) = X1(t)⊕X2(t)∈ Tγ(t)Q⊕Tγ(t)Q≃TΥ(t)TQ. Then the tangent vector field to the curve t7→ X(t)is given by γ0(t)⊕Y(t, γ(t)) ⊕˜ X1(t)⊕˜ X2(t), where ˜ X1(t) = ∇γ0(t)X1(t) + 1 2T(X1(t), γ0(t)), ˜ X2(t) = ∇γ0(t)X2(t) + 1 2T(X2(t), γ0(t)). Proof. In coordinates, the curve t7→ X(t) has the form (qi(t),˙qj(t), Xk 1(t), X` 2(t)−1 2(Γ` mr + Γ` rm) ˙qm(t)Xr 1(t)). The tangent vector to this curve is then given a.e. by ˙qi∂ ∂qi+(Yi−Γi jk ˙qj˙qk)∂ ∂vi+˙ Xi 1 ∂ ∂ui+³˙ Xi 2−1 2 ∂Γi jk ∂q`˙qk˙q`Xj 1−1 2 ∂Γi kj ∂q`˙qk˙q`Xj 1 −1 2(Γi jk + Γi kj)(Yk−Γk `m ˙q`˙qm)Xj 1−1 2(Γi jk + Γi kj) ˙qk˙ Xj 1´∂ ∂wi. A straightforward computation shows that this tangent vector field has the representation γ0(t)⊕Y(t, γ(t)) ⊕¡∇γ0(t)X1(t) + 1 2T(X1(t), γ0(t))¢⊕ ⊕¡∇γ0(t)X2(t) + 1 2T(X2(t), γ0(t))¢, 20 FRANCESCO BULLO AND ANDREW D. LEWIS which proves the lemma. ¤ ¤ We may now prove our main result that relates the integral curves of ST with solutions to the Jacobi equation. Theorem 4.4 (Tangent lift of Sand Jacobi equation).Let ∇be an affine connection on Qwith Sthe corresponding geodesic spray. Let γ:I→Qbe a geodesic with t7→ Υ(t),γ0(t)the corresponding integral curve of S. Let a∈I,u, w ∈Tγ(a)Q, and define vector fields U, W :I→TQ along γby asking that t7→ U(t)⊕W(t)∈Tγ(t)Q⊕Tγ(t)Q≃TΥ(t)TQ be the integral curve of STwith initial conditions u⊕w∈Tγ(a)Q⊕Tγ(a)Q≃TΥ(a)TQ. Then Uand Whave the following properties: (i) Usatisfies the Jacobi equation ∇2 γ0(t)U(t) + R(U(t), γ0(t))γ0(t) + ∇γ0(t)(T(U(t), γ0(t))) = 0; (ii) W(t) = ∇γ0(t)U(t) + 1 2T(U(t), γ0(t)). Proof. This follows most directly, although very messily, from a coordinate computation using Lemma 4.3. We refer the reader to [Bullo and Lewis 2005]. ¤ 4.4. The Sasaki metric. When the constructions of this section are applied in the case when ∇is the Levi-Civita affine connection associated with a Riemannian metric Gon Q, there is an important additional construction that can be made. Definition 4.5 (Sasaki metric).Let (Q,G)be a Riemannian manifold, and for vq∈TQ, let TvqTQ ≃TqQ⊕TqQdenote the splitting defined by the affine connection G ∇. The Sasaki metric is the Riemannian metric GTon TQ given by GT(u1 vq⊕w1 vq, u2 vq⊕w2 vq) = G(u1 vq, u2 vq) + G(w1 vq, w2 vq).• The Sasaki metric was introduced by Sasaki [1958, 1962]. Much research has been made into the properties of the Sasaki metric, beginning with the work of Sasaki who studied the curvature, geodesics, and Killing vector fields of the metric. Some of these results are also given in [Yano and Ishihara 1973]. 5. Linearization along relative equilibria This long section contains many of the essential results in the paper. Our end objective is to relate the linearization of equilibria of the reduced equations of Theorem 3.3 to the linearization along the associated relative equilibria of the unreduced system. We aim to perform this linearization in MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 21 a control theoretic setting so that our constructions will be useful not just for investigations of stability, but also for stabilization. To properly understand this process, we begin in Section 5.1 with linearization of general control-affine systems about general controlled trajectories. Then we specialize this discussion in Section 5.2 to the linearization of a general affine connection control system about a general controlled trajectory. We then, in Section 5.3, finally specialize to the case of interest, namely the linearization of the unreduced equations along a relative equilibrium. Here the main result is Theorem 5.10 which gives the geometry associated with linearization along a relative equilibrium. In particular this result, or more precisely its proof, makes explicit the relationship between the affine differential geometric concepts arising in the linearization of Section 5.2 and the usual concepts arising in reduction of mechanical systems, such as the curvature of the mechanical connection and the effective potential. Then, in Section 5.4 we turn to the linearization, in the standard sense, of the reduced equations about an equilibrium. The main result here is Theorem 5.13 which links the reduced and unreduced linearizations. We let Σ = (Q,G, V, F, F) be a forced simple mechanical control system with Ftime-independent, let Xbe a complete infinitesimal symmetry of Σ, and let χ:R→Qbe a regular relative equilibrium for (Q,G, V, F ). Thus χis an integral curve for Xthat is also an uncontrolled trajectory for the system. We let Bdenote the set of X-orbits, and following Assumption 3.1, we assume that Bis a smooth manifold for which πB:Q→Bis a surjective submersion. We let Ya=G]◦Fa,a∈ {1, . . . , m}, and if u:I→Rmis a locally integrable control, we denote Yu(t, q) = m X a=1 ua(t)Ya(q), t ∈I, q ∈Q,(9) for brevity. Define vector fields YB,a,a∈ {1, . . . , m}, by YB,a(b) = TqπB(Ya(q)) for q∈π−1 B(b). Since the vector fields Yare X-invariant, this definition is independent of q∈π−1 B(b). Similarly to (9), we denote YB,u(t, b) = m X a=1 ua(t)YB,a(b). 22 FRANCESCO BULLO AND ANDREW D. LEWIS In Theorem 3.3 we showed that the reduced system has TB ×Ras its state space, and satisfies the equations GB ∇η0(t)η0(t) = −gradB¡Veff X,v(t)¢B(η(t)) + v(t)CX(η0(t)) +TπB◦G]◦F(γ0(t)) + YB,u(t, η(t))), ˙v(t) = −v(t)hd(kXk2 G)B(η(t)); η0(t)i (kXk2 G)B(η(t)) +hF(γ0(t)); X(γ(t))i (kXk2 G)B(η(t)) +G(Yu(t, γ(t)), X(γ(t))) (kXk2 G)B(η(t)) , (10) where (γ, u) is the controlled trajectory, η=πB◦γ, and vis defined by ver(γ0(t)) = v(t)X(γ(t)). The relative equilibrium χcorresponds to the equilibrium point (TπB(χ0(0)),1) of the reduced equations (10). Therefore, linearization of the relative equilibrium χcould be defined to be the linearization of the equations (10) about the equilibrium point (TπB(χ0(0)),1). This is one view of linearization of relative equilibria. Another view is that, since χis a trajectory for the unreduced system, we could linearize along it in the manner described when describing the Jacobi equation. In this section we shall see how these views of linearization of relative equilibria tie together. We build up to this by first considering linearization in more general settings. 5.1. Linearization of a control-affine system along a controlled trajectory. In order to talk about linearization along a relative equilibrium, we first discuss linearization along a general controlled trajectory. In order to do this, it is convenient to first consider the general control-affine case, then specialize to the mechanical setting. Let us first recall that a control-affine system is a pair (M,C= {f0, f1, . . . , fm}) where Mis a manifold and f0, f1, . . . , fmare vector fields on M. The drift vector field is f0and the control vector fields are f1, . . . , fm. The governing equations for a time-dependent control-affine system (M,C) are then γ0(t) = f0(γ(t)) + m X a=1 ua(t)fa(γ(t)), for a controlled trajectory (γ, u). We shall also consider the notion of atime-dependent control-affine system, by which we mean a controlaffine system where the vector fields f0, f1, . . . , fmare time-dependent. We shall also require the standard notion of a linear control system, by which we mean a triple (V, A, B), where Vis a finite-dimensional R- vector space, A∈L(V;V), and B∈L(Rm;V). The equations governing a MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 23 linear control system are ˙x(t) = A(x(t)) + B(u(t)). Suppose that we have a (time-independent) control-affine system (M,C= {f0, f1, . . . , fm}), and a controlled trajectory (γ0, u0) defined on an interval I. We wish to linearize the system about this controlled trajectory. Linearization is to be done with respect to both state and control. Thus, speaking somewhat loosely for a moment, to compute the linearization, one should first fix the control at u0and linearize with respect to state, then fix the state and linearize with respect to control, and then add the results to obtain the linearization. Let us now be more formal about this. If we fix the control at u0, we obtain the time-dependent vector field fu0 on Mdefined by fu0(t, x) = f0(x) + m X a=1 ua 0(t)fa(x). We call fu0the reference vector field for the controlled trajectory (γ0, u0). The linearization of the reference vector field is exactly described by its tangent lift, as discussed in Remark 4.1–4. Thus one component of the linearization is fT u0. The other component is computed by fixing the state, say at x, and linearizing with respect to the control. Thus we consider the map R×Rm3(t, u)7→ f0(x) + m X a=1 (ua 0(t) + ua)fa(x)∈TxM, and differentiate this with respect to uat u=0. The resulting map from T0Rm≃Rmto Tfu0(t,x)(TxM)≃TxMis simply given by u7→ m X a=1 uafa(x). In order to add the results of the two computations, we regard TxMas being identified with Vfu0(t,x)TM. Thus the linearization with respect to the control yields the linearized control vector fields vlft(fa), a∈ {1, . . . , m}. In this way, we arrive at the time-dependent control-affine system ΣT(γ0, u0) = (TM,{fT u0,vlft(f1), . . . , vlft(fa)}), whose controlled trajectories (ξ, u) satisfy ξ0(t) = fT u0(t, ξ(t)) + m X a=1 ua(t)vlft(fa)(ξ(t)).(11) The following result gives an important property of these controlled trajectories. 24 FRANCESCO BULLO AND ANDREW D. LEWIS Lemma 5.1. For every locally integrable control t7→ u(t), the timedependent vector field (t, vx)7→ fT u0(t, vx) + m X a=1 ua(t)vlft(fa)(vx) is a linear vector field over fu0. Proof. This is easily proved in coordinates. ¤ From Remark 4.1–1, we then know that, if (ξ, u) is a controlled trajectory for ΣT(γ0, u0), then πTM ◦ξis an integral curve for fu0. In particular, if (ξ, u) is a controlled trajectory for ΣT(γ0, u0) that satisfies πTM ◦ξ(t) = γ0(t) for some t∈I, then ξis a vector field along γ0. To formally define the linearization along (γ0, u0), we need an additional concept, following Sussmann [1997]. Definition 5.2 (Differential operator along a curve).Let Mbe a manifold, let γ:I→Mbe an LAC curve, and let π:E→Mbe a vector bundle. A differential operator in Ealong γassigns, to each LAC section ξof E along γ, a locally integrable section L(ξ)along γ, and the assignment has the property that, if f∈C∞(M)and if Ξ∈Γ∞(E), then L(f◦γ(Ξ ◦γ))(t) = f◦γ(t)L(Ξ ◦γ)(t)+(Lγ0(t)f)(γ(t))Ξ ◦γ(t).• Thus a differential operator simply “differentiates” sections of Ealong γ, with the differentiation rule satisfying the usual derivation property with respect to multiplication with respect to functions. Sussmann [1997] shows that, in coordinates (x1, . . . , xn) for M, if t7→ (ξ1(t), . . . , ξk(t)) are the fiber components of the local representative of an LAC section ξof E, then the fiber components of the local representative of L(ξ) satisfy (L(ξ))a(t) = ˙ ξa(t) + k X b=1 La b(t)ξb(t), a ∈ {1, . . . , k}, for some locally integrable functions t7→ Lb a(t), a, b ∈ {1, . . . , k}. If γ:I→ Mis an integral curve of a time-dependent vector field X, then there is a naturally induced differential operator in TM along γ, denoted by LX,γ , and defined by LX,γ (ξ) = [Xt,Ξ](γ(t)),a.e. t∈I, where Ξ is a vector field satisfying ξ= Ξ ◦γ, and where Xtis the vector field defined by Xt(x) = X(t, x). In coordinates this differential operator satisfies LX,γ (ξ)i(t) = ˙ ξi(t)−∂Xi ∂xj(γ(t))ξj(t), i ∈ {1, . . . , n}. MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 25 This differential operator is sometimes referred to as the “Lie drag” (see [Crampin and Pirani 1986, Section 3.5]). A coordinate computation readily verifies the following result, and we refer to [Lewis and Tyner 2003, Sussmann 1997] for details. Proposition 5.3 (Tangent lift and differential operator).Let X:I×M→ TMbe a time-dependent vector field, let vx0∈TxM, let t0∈I, and let γ:I→Mbe the integral curve of Xsatisfying γ(t0) = x0. For a vector field ξalong γsatisfying ξ(t0) = vx0, the following statements are equivalent: (i) ξis an integral curve for XT; (ii) there exists a variation σof Xalong γsuch that d ds¯ ¯s=0σ(t, s) = ξ(t)for each t∈I; (iii) LX,γ (ξ) = 0. With the preceding as motivation, we can define the linearization of a control-affine system. Definition 5.4 (Linearization of a control-affine system).Let Σ = (M,C={f0, f1, . . . , fm})be a control-affine system with (γ0, u0)a controlled trajectory. The linearization of Σabout (γ0, u0)is given by {LΣ(γ0, u0), bΣ,1(γ0, u0), . . . , bΣ,m(γ0, u0)}, where (i) LΣ(γ0, u0)is the differential operator in TM along γ0defined by LΣ(γ0, u0) = Lfu0,γ0, and (ii) bΣ,a,a∈ {1, . . . , m}, are the vector fields along γ0defined by bΣ,a(γ0, u0)(t) = vlft(fa(γ0(t))), a ∈ {1, . . . , m}.• The equations governing the linearization are LΣ(γ0, u0)(ξ)(t) = m X a=1 ua(t)bΣ,a(γ0, u0), which are thus equations for a vector field ξalong γ0. By Proposition 5.3, these equations are exactly the restriction to image(γ0) of the equations for the time-dependent control-affine system in (11). In the special case where f0(x0) = 0x0,u0=0,γ0=x0for some x0∈M, one can readily check that we recover the linearization of the system at x0in the usual sense. 5.2. Linearization of a forced affine connection control system along a controlled trajectory. After beginning our discussion of linearization in the context of control-affine systems, we next specialize to affine connection control systems. We let Σ = (Q,∇, Y, Y) be a forced 32 FRANCESCO BULLO AND ANDREW D. LEWIS where no summation is intended over “n.” To simplify the terms involving the external force, we note that G ∇ξ(t)(G]◦A[(X(χ(t)))) = = ( G ∇ξ(t)(G]◦A[))(X(χ(t))) + G]◦A[( G ∇ξ(t)X(χ(t))). Thus, using the fact that G ∇is torsion-free, we have − G ∇ξ(t)(G]◦A[◦X)(χ(t)) + ( G ∇ξ(t)(G]◦A[))(χ0(t)) +G]◦A[( G ∇χ0(t)ξ(t)) = G]◦A[([X, Ξ](χ(t))). Using (17) and (18) we arrive at − G ∇ξ(t)(G]◦A[◦X)(χ(t)) + ( G ∇ξ(t)(G]◦A[))(χ0(t)) +G]◦A[( G ∇χ0(t)ξ(t)) = G]◦A[(hlftχ(t)( ˙x(t))) + ˙ ξnG]◦A[(X(χ(t))),(23) where x(t) = Tχ(t)πB(ξ(t)). Finally, for a vector field ξalong χ, let x(t) = TπB(ξ(t)) and let ν(t)X(t) = ver(LX,χ(ξ)). In terms of our coordinates above, ν(t) = ˙ ξn(t). One now combines equations (15), (16), (17), (19), (20), (21), (22) and (23) to get the result. ¤ Remark 5.11. The preceding theorem is not obvious; in particular, the equivalence of equations (13) and (14) is not transparent. Indeed, the relationship between the curvature tensor and the components of the system that appear in the theorem statement, C,Hess(VX), and gradV, is rather subtle. In this respect, the proof of the theorem bears study, if these relationships are to be understood. • 5.4. Linearization of the reduced equations along a relative equilibrium. We again consider a forced simple mechanical control system Σ=(Q,G, V, F, F) satisfying Assumption 5.9, take Xto be a complete infinitesimal symmetry for Σ satisfying Assumption 3.1, and let χbe a relative equilibrium. In this section we provide the form of the linearization along a relative equilibrium by linearizing, in the usual manner, the reduced MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 33 equations, which we reproduce here for convenience: GB ∇η0(t)η0(t) = −gradB¡Veff X,v(t)¢B(η(t)) + v(t)CX(η0(t)) +TπB◦G]◦A[¡γ0(t)−X(γ(t))¢+YB,u(t, η(t)), ˙v(t) = −v(t)hd(kXk2 G)B(η(t)); η0(t)i (kXk2 G)B(η(t)) +hA[(γ0(t)−X(γ(t))); X(γ(t))i (kXk2 G)B(η(t)) +G(Yu(t, γ(t)), X(γ(t))) (kXk2 G)B(η(t)) . (24) Here (γ, u) is a controlled trajectory for Σ, η=πB◦γ, and vis defined by ver(γ0(t)) = v(t)X(γ(t)). The reduced equations are straightforward to linearize, since we are merely linearizing about an equilibrium point. To compactly state the form of the linearization requires some notation. Define a (1,1)-tensor field AB on Bby AB(vb) = TqπB◦G](q)◦A[(q)◦hlftq(vb), for q∈π−1 B(b). This definition can be shown to be independent of the choice of q∈π−1 B(b) by virtue of the X-invariance of A. Define a vector field aB on Bby aB(b) = TqπB◦G](q)◦A[(q)(X(q)), where q∈π−1 B(b), and again this definition can be shown to be well-defined. Finally, define a one-form αBon Bby hαB(b); vbi=hA[(hlftq(vb)); X(q)i (kXk2 G)B(b), where q∈π−1 B(b) and vb∈TbB. This definition, too, is independent of the choice of q. We may now state the form of the linearization of the reduced equations. The proof of the following result is by fairly simple direct computation. Proposition 5.12 (Linearization after reduction).Let Σ = (Q,G, V, F, F) be a forced simple mechanical control system, let Xbe a complete infinitesimal symmetry of Σsatisfying Assumption 3.1, and let χbe a regular relative equilibrium with b0=πB◦χ(0). 34 FRANCESCO BULLO AND ANDREW D. LEWIS The linearization of equations (24) about (0b0,1) is the linear control system (Tb0B⊕Tb0B⊕R, AΣ(b0), BΣ(b0)), where AΣ(b0) =    0 idTb0B −GB(b0)]◦Hess(VX)B(b0)[CX(b0) + AB(b0) 0−2dVB(b0) (kXk2 G)B(b0)+αB(b0) 0 2gradBVB(b0) + aB(b0) hA[(X(q0));X(q0)i (kXk2 G)B(b0)   , BΣ(b0) =   0 BΣ,2(b0) BΣ,3(b0)  , where BΣ,2(b0)∈L(Rm;Tb0B)is defined by BΣ,2(b0)(u) = m X a=1 uaYB,a(b0), and where BΣ,3(b0)∈L(Rm;R)is defined by BΣ,3(b0)(u) = m X a=1 uaG(Ya(χ(0)), X(χ(0))) (kXk2 G)B(b0). The equations governing controlled trajectories for the linearization of the reduced system are ¨x(t) = −GB(b0)]◦Hess(VX)B(b0)[(x(t)) + CX(b0)( ˙x(t)) + 2ν(t)gradBVB(b0) + AB(b0)( ˙x(t)) + ν(t)aB(b0) + BΣ,2(b0)·u(t), ˙ν(t) = −2hdVB(b0); ˙x(t)i (kXk2 G)B(b0)+αB(b0)( ˙x(t)) +ν(t)hA[(X(q0)); X(q0)i (kXk2 G)B(b0)+BΣ,3(b0)·u(t). The following result gives the relationship between the reduced and the unreduced linearization. Theorem 5.13 (Relationship between linearizations).Let Σ = (Q,G, V, F, F)be a forced simple mechanical control system with Fsatisfying Assumption 5.9, let Xbe a complete infinitesimal symmetry of Σ satisfying Assumption 3.1, and let χbe a regular relative equilibrium with b0=πB◦χ(0). MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 35 For a curve t7→ x(t)∈Tb0B, a vector field ξalong χ, a function ν:R→ R, and a locally integrable control t7→ u(t), the following statements are equivalent: (i) t7→ (x(t)⊕˙x(t)⊕ν(t), u(t)) is a controlled trajectory for the linearization of the equations (24) about (0b0,1), and in turn hor(ξ(t)) = hlftχ(t)(x(t)), and ν(t) = ˙ ζ(t), where ver(ξ(t)) = ζ(t)X(χ(t)); (ii) (ξ, u)is a controlled trajectory for the linearization of Σabout (χ, 0), and in turn x(t) = TπB(ξ(t)), and ν(t) = ˙ ζ(t), where ver(ξ(t)) = ζ(t)X(χ(t)). Proof. This follows easily from Theorem 5.10 and Proposition 5.12. ¤ The theorem is an important one, since it will allow us to switch freely between the reduced and unreduced linearizations. In some cases, it will be convenient to think of certain concepts in the unreduced setting, while computations are more easily performed in the reduced setting. 6. Linearized effective energies In many existing results concerning stability of relative equilibria, a central role is played by Hessian of the energy. This is a consequence of the fact that definiteness of the Hessian, restricted to certain subspaces, can easily deliver stability results in various forms. In this section we study the Hessian of the effective energy for a relative equilibria. In particular, we consider the interplay of the various natural energies with the reduction process and with linearization. Specifically, we spell out the geometry relating the processes of linearization and reduction. 6.1. Some geometry associated to an infinitesimal isometry. The utility of the constructions in this section may not be immediately apparent, but will become clear in Proposition 6.6 below. In this section we let (Q,G) be a Riemannian manifold with Xan infinitesimal isometry satisfying Assumption 3.1. We denote by TTQXthe restriction of the vector bundle πTTQ :TTQ →TQ to image(X). Thus TTQXis a vector bundle over image(X) whose fiber at X(q) is TX(q)TQ. We denote this fiber by TTQX,X(q). In like manner, HTQXand VTQX denote the restrictions of HTQ and TQ, respectively, to image(X). The Ehresmann connection on πTQ :TQ →Q, defined by G ∇as in Section 4.2, gives a splitting of each fiber of TTQXas TTQX,X(q)=HTQX,X(q)⊕VTQX,X(q). 36 FRANCESCO BULLO AND ANDREW D. LEWIS This gives a vector bundle isomorphism σX:TTQX→HTQX⊕VTQX. Denote by ΠB:TQ →TQ/Rthe projection onto the set of XT-orbits. We define φB:TQ →TB ×Rby φB(wq) = (TπB(wq), νX(wq)), where νX(wq) is defined by ver(wq) = νX(wq)X(q). Note that φB◦X(q) = (0πB(q),1). Indeed, one can easily see that φB,X , φB|image(X): image(X)→Z(TB)× {1}is a surjective submersion. We next define a vector bundle map ψB:TTQ →T(TB ×R) over φBby ψB=TφB. We denote ψB,X =ψB|TTQX, noting that this is a surjective vector bundle map from TTQXto the restricted vector bundle T(TB ×R)|(Z(B)× {1}). Next we wish to give a useful description of the vector bundle T(TB ×R)|(Z(B)× {1}). We think of TB ×Ras a vector bundle over B×R, and we let RB×Rbe the trivial vector bundle (B×R)×R over B×R. We then note that T0bTB ≃Tb⊕TbB, where the first component in the direct sum is tangent to Z(TB) (i.e., is horizontal) and the second component is tangent to TbB(i.e., is vertical). Thus we have a natural identification T(TB ×R)|(Z(B)× {1})≃(TB ×R)⊕(TB ×R)⊕RB×R(25) of vector bundles over Z(TB)× {1} ≃ B× {1}. The fiber over (b, 1) is isomorphic to TbB⊕TbB⊕R. We shall implicitly use the identification (25) in the sequel. Next, we define a vector bundle map ιB:HTQ ⊕VTQ → (TB ×R)⊕(TB ×R)⊕RB×Rby ιB(uvq⊕wvq) = ¡TqπB(uvq), TqπB(wvq− G ∇X(uvq)), νX(wvq− G ∇X(uvq))¢. We then let ιB,X be the restriction of ιBto HTQX⊕VTQX. The following result summarizes and ties together the above constructions. Lemma 6.1. The following statements hold: (i) σXis a vector bundle isomorphism over idimage(X)from TTQX to HTQX⊕VTQX; (ii) φB,X is a surjective submersion from image(X)to B× {1}; (iii) ψB,X is a surjective vector bundle map over φB,X from TTQXto (TB ×R)⊕(TB ×R)⊕RB×R; (iv) ιB,X is a surjective vector bundle map over φB,X from HTQX⊕ VTQXto (TB ×R)⊕(TB ×R)⊕RB×R; MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 37 (v) the following diagram commutes: TTQX σX// ψB,X )) T T T T T T T T T T T T T T T HTQX⊕VTQX ιB,X ttiiiiiiiiiiiiiiiii (TB ×R)⊕(TB ×R)⊕RB×R Proof. This is most easily proved in an appropriate set of coordinates. Take coordinates (q1, . . . , qn) for Qwith the following properties: (1) X=∂ ∂qn; (2) for times tfor which χ(t) is in the chart domain, {∂ ∂q1(χ(t)), . . . , ∂ ∂qn(χ(t))}is an orthogonal basis for Tχ(t)Q. This means that (q1, . . . , qn−1) are coordinates for B. These also form, therefore, coordinates for Z(TB) and thus also for Z(TB)× {1}. Since a typical point in image(X) has the form ((q1, . . . , qn),(0, . . . , 0,1)) in natural coordinates for TQ, we can use (q1, . . . , qn) as coordinates for image(X). We denote natural coordinates for TTQ by ((q,v),(u,w)). Then (q,u,w) form a set of coordinates for TTQX. The map φBfrom TQ to TB ×Rhas the form ((q1, . . . , qn),(v1, . . . , vn)) 7→ ((q1, . . . , qn−1),(v1, . . . , vn−1), vn). In the coordinates for image(X) and for Z(TB)× {1}, the map φB,X has the form (q1, . . . , qn)7→ (q1, . . . , qn−1). The coordinate form of ψBis then (((q1 , . . . , qn),(v1 , . . . , vn)),((u1 , . . . , un),(w1 , . . . , wn))) 7→ (((q1 , . . . , qn−1),(v1 , . . . , vn−1), vn),((u1 , . . . , un−1),(w1 , . . . , wn−1), wn)) and the coordinate form for ψB,X is given by ((q1, . . . , qn),(u1, . . . , un),(w1, . . . , wn)) 7→ ((q1, . . . , qn−1),(u1, . . . , un−1),(w1, . . . , wn−1), wn).(26) In coordinates, the map σXis given by ((q1, . . . , qn),(u1, . . . , un),(w1, . . . , wn)) 7→ ((q1, . . . , qn),(u1, . . . , un),(w1+ G Γi njuj, . . . , wn+ G Γn njuj)). 38 FRANCESCO BULLO AND ANDREW D. LEWIS Finally, in our above coordinates, the form of the map ιB,X is ((q1 , . . . , qn),(u1 , . . . , un),(w1 , . . . , wn)) 7→ ((q1 , . . . , qn−1),(u1 , . . . , un),(w1 − G Γ1 njuj , . . . , wn−1− G Γn−1 nj wj), wn − G Γn njuj). All statements in the statement of the lemma follow directly from the preceding coordinate computations. ¤ We shall see in Proposition 6.6 that ιB,X relates two natural energies associated to a relative equilibrium. 6.2. The effective energies and their linearizations. We let Σ = (Q,G, V, F) be a simple mechanical system with Xa complete infinitesimal symmetry for Σ satisfying Assumption 3.1. First recall from Lemma 2.4 that the effective energy for a forced simple mechanical system Σ = (Q,G, V, F) with complete infinitesimal symmetry Xis EX(vq) = 1 2kvq−X(q)k2 G+VX(q), where VX=V−1 2kXk2 Gis the effective potential. The relative equilibria for Σ are then characterized by the critical points of EX, as in Proposition 2.5. As we shall see in Section 7, the Hessian of the effective energy at such critical points is useful for determining the stability of the corresponding relative equilibrium. The following result characterizes this Hessian in terms of the splitting of the fibers of TTQ using the Ehresmann connection on πTTQ :TTQ →TQ associated with G ∇. Lemma 6.2. Let Σ=(Q,G, V, F )be a forced simple mechanical system and let Xbe a complete infinitesimal symmetry for Σ. Let vqbe a critical point for the effective energy and let TqQ⊕TqQbe the splitting of TvqTQ associated with G ∇, as described in Section 4.2. Then HessEX(u1⊕w1, u2⊕w2) = =G(w1− G ∇u1X(q), w2− G ∇u2X(q)) + HessVX(u1, u2). Proof. This is a messy, but straightforward, proof in coordinates. ¤ With this as background, we make the following definition, recalling the notation uvq⊕wvqto denote a point in TvqTQ relative to the splitting defined by G ∇. Definition 6.3 (Linearized effective energy).Let Σ=(Q,G, V, F)be a forced simple mechanical system, let Xbe a complete infinitesimal isometry MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 39 for Σ, and let χbe a relative equilibrium. The linearized effective energy is the function on TTQ|image(χ0)defined by Eχ(uvq⊕wvq) = 1 2kwvq− G ∇uvqXk2 G+1 2HessVX(uvq, uvq), where vq=χ0(0).• Next we consider the linearized effective energy, but now for the reduced system. To do so, we assume that πB:Q→Bis a surjective submersion, as in Assumption 3.1. The effective energy EXis X-invariant and so drops to TQ/R≃TB ×R. We may further explicitly computed this “reduced effective energy,” denoted by Ered X, as Ered X(wb, v) = 1 2GB(wb, wb)+(VX)B(b) + 1 2(kXk2 G)B(b)(v−1)2. It then makes sense that the “reduced linearized effective energy” should be the Hessian of this function at a critical point, which corresponds, as we have seen, to a relative equilibrium. The following result records the form of the Hessian. Lemma 6.4. Let Σ = (Q,G, V, F )be a forced simple mechanical system, let Xbe a complete infinitesimal symmetry for Σsatisfying Assumption 3.1, and let (0b,1) ∈TB ×Rbe a critical point for (EX)TB×R. Then HessEred X(0b,1)(u1⊕v1⊕ν1, u2⊕v2⊕ν2) =GB(b)(v1, v2) + Hess(VX)B(b)(u1, u2)+(kXk2 G)B(b)ν1ν2. Proof. This is a straightforward computation. ¤ Based on this computation, let us make the following definition. Definition 6.5 (Reduced linearized effective energy).Let Σ = (Q,G, V, F) be a forced simple mechanical system, let Xbe a complete infinitesimal isometry for Σsatisfying Assumption 3.1, and let b0=πB(χ(0)). The reduced linearized effective energy is the function on Tb0B⊕Tb0B⊕R defined by Ered χ(ub0, vb0, ν) = 1 2kvb0k2 GB+1 2Hess(VX)B(b0)(ub0, ub0) + 1 2(kXk2 G)B(b0)ν2. • The preceding definition of the reduced linearized effective energy is obtained by “reducing” the effective energy, and then “linearizing” it. It should be possible to perform the operations in the opposite order to get to the same answer. To do this explicitly, we use the constructions of the preceding section. In particular, we use the vector bundle map ιB,X . As the following result indicates, one should think of this map as describing how 40 FRANCESCO BULLO AND ANDREW D. LEWIS the process of linearization is reduced when using the Ehresmann connection on πTTQ :TTQ →TQ associated with G ∇. Proposition 6.6 (Relating the linearized effective energies).Let Σ = (Q,G, V, F)be a forced simple mechanical system with Xa complete infinitesimal symmetry satisfying Assumption 3.1. If χis a relative equilibrium, then ιB,X (χ0(t))∗Ered χ=Eχfor all t∈R. Proof. Let q=χ(t) and let b=πB(q). We compute ιB,X (χ0(t))∗Ered χ(uvq⊕wvq) = =1 2kTqπB(wvq− G ∇X(uvq))k2 GB+1 2(kXk2 G)BνX(wvq− G ∇X(uq))2+ +1 2Hess(VX)B(b)(TqπB(ub), TqπB(ub)) =1 2khor(wvq− G ∇X(uvq))k2 G+kver(wvq− G ∇X(uvq))k2 G+ +1 2HessVX(q)(uvq, uvq), as desired. ¤ 7. Stability of relative equilibria In this section we apply the results of the preceding sections to study the stability of a relative equilibrium. We deal with stability in two ways: linear and nonlinear. Because we use the linearized effective energy as a Lyapunov function in our linear stability analysis, if the conditions we give for linear stability are satisfied, then one can conclude nonlinear stability as well. 7.1. Nonlinear stability definitions. Let us establish the basic setup for all the following definitions and results. We consider a forced simple mechanical system Σ = (Q,G, V, F) with a time-independent external force Fand with the following governing equation: G ∇γ0(t)γ0(t) = −gradV(γ(t)) + G](F(γ0(t))).(27) We assume for simplicity that all solutions to this equation can be defined on all of R. Let us define the notions of stability for relative equilibria that we consider in this paper. Definition 7.1 (Base and fiber stability).Let Xbe a complete infinitesimal symmetry for Σ, let χ:R→Qbe a regular relative equilibrium, and let b0=πB(χ(0)). MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 41 (i) The relative equilibrium χis base stable if, for all neighborhoods Uof 0b0, there exists a neighborhood Wof χ0(0) such that each solution γ:R→Qof equation (27), with initial condition γ0(0) ∈ W, satisfies TπB◦γ0(t)∈ U for all t∈R. (ii) The relative equilibrium χis locally asymptotically base stable if it is base stable, and if there exists a neighborhood Vof χ0(0) such that each solution γ:R→Qof equation (27), with initial condition γ0(0) ∈ V, satisfies limt→+∞TπB◦γ0(t) = 0b0. (iii) The relative equilibrium χis fiber stable if, for all neighborhoods URof 1, there exists a neighborhood Wof χ0(0) such that each solution γ:R→Qof equation (27), with initial condition γ0(0) ∈ W, satisfies JX(γ0(t)) kXk−2 G(γ(t)) ∈ URfor all t∈R. (iv) The relative equilibrium χis locally asymptotically fiber stable if it is fiber stable, and if there exists a neighborhood Vof χ0(0) such that each solution γ:R→Qof equation (27), with initial condition γ0(0) ∈ V, satisfies limt→+∞JX(γ0(t)) kXk−2 G(γ(t)) = 1.• To describe base stability, consider an invariant tube around the regular relative equilibrium χ. Base stability means that trajectories starting in the tube remain in the tube at subsequent times; see Figure 3. Fiber stability means that trajectories starting with nearby initial velocities will maintain nearby fiber velocities at subsequent times. Even if a system is base and fiber stable, trajectories with nearby initial conditions may diverge. What we call base stability of a relative equilibrium corresponds to the usual notion of stability for a relative equilibrium considered in the literature: stability of the equilibrium point in the reduced space. What we call fiber stability has to do, note, with only the velocity along the fiber. Thus fiber stability will generally not suffice to give other forms of stability along the relative equilibrium, such as the G-stability of Patrick [1992]. We also remark that the definitions of base and fiber stability are examples of what is sometimes called “partial stability” in more general contexts. With this sort of stability, one is only interested in the behavior of some of the states of the system. This is studied in the text [Vorotnikov 1998]. 7.2. Linear stability definitions. We let Σ = (Q,G, V, F) be a forced simple mechanical system where the force Fsatisfies Assumption 5.9, with Xa complete infinitesimal symmetry for the system, and with χ:R→Q a relative equilibrium. First we need a definition for linear stability of a relative equilibrium. The definition relies on the linearization along the relative equilibrium, which, from the developments of Section 5, satisfies an 48 FRANCESCO BULLO AND ANDREW D. LEWIS Proof. Note that combined linear (asymptotic) base and fiber stability of χ is equivalent to the linear (asymptotic) stability of the equilibrium point b0 for the reduced system on TB×R. Therefore, in the proof, we shall consider the stability in the reduced space, using the reduced linearized effective energy, Eref χ, as a candidate Lyapunov function. First note that, under the hypothesis that Hess(VX)B(b0) is positive-definite, it follows that Eref χis positive-definite about 0b⊕0b⊕0. Next, a straightforward computation, the details of which we omit, shows that dEred χ dt(x(t)⊕˙x(t)⊕ν(t)) = GB( ˙x(t), FB(b0)( ˙x(t))) + ν(t)GB( ˙x(t), fB(b0)) +ν(t)(kXk2 G)B(b0)αB(b0)( ˙x(t)) + ν(t)2hA[(X(q0)); X(q0)i, along a solution t7→ x(t)⊕˙x(t)⊕ν(t) to equations (29), where q0∈π−1 B(b0), and where Ais as in Assumption 5.9. One now can easily show that GB( ˙x(t), FB(b0)( ˙x(t))) = A(hlftq0( ˙x(t)),hlftq0( ˙x(t))), ν(t)GB( ˙x(t), fB(b0)) = A(ν(t)X(q0),hlftq0( ˙x(t))), ν(t)(kXk2 G)B(b0)αB(b0)( ˙x(t)) = A(hlftq0( ˙x(t)), ν(t)X(q0)). These computations allow us to conclude that dEred χ dt(x(t)⊕˙x(t)⊕ν(t)) = =A(hlftq0( ˙x(t)) + ν(t)X(q0),hlftq0( ˙x(t)) + ν(t)X(q0)). In part (i), A=−Rdiss is negative-semidefinite, and in part (ii), A=−Rdiss is negative-definite, and the result then follows directly. ¤ Alternatively, one can check the hypotheses of the theorem using the linearized effective energy. The following result contains the results of this transcription. Corollary 7.11 (Unreduced linear stability of relative equilibria).Let Σ = (Q,G, V, F)be a forced simple mechanical system, with Xa complete infinitesimal symmetry for Σsatisfying Assumption 3.1, and with χ:R→Q a regular relative equilibrium. Suppose that F(vq) = −R[ diss(vq−X(q)), where Rdiss is a Rayleigh dissipation function. The following statements hold: (i) χis linearly base and fiber stable if HessVX(χ(t)) is positivedefinite on any (and so every) complement to spanR{X(χ(t))} for some (and so for all) t∈R; (ii) χis linearly asymptotically base stable and linearly asymptotically fiber stable if HessVX(χ(t)) is positive-definite on any (and so MECHANICAL SYSTEMS ON RIEMANNIAN MANIFOLDS 49 every) complement to spanR{X(χ(t))}for some (and so for all) t∈R, and if Rdiss is positive-definite. Remarks 7.12. (1) Note that since positive-definiteness of the Hessian of (VX)Bat b0implies that b0is an isolated local minimum for (VX)B, the satisfaction of the hypotheses of Theorem 7.10 implies the satisfaction of the hypotheses of Theorem 7.7. (2) The presence of gyroscopic forces in the reduced linearization makes it difficult to draw the sharpest possible conclusions regarding linear stability. This is to be contrasted with the linear stability of equilibrium points, where it is possible to give much sharper stability conditions in the presence of only dissipative forces (see, for example, [Bullo and Lewis 2004, Theorem 6.42]). (3) In part (ii) of Theorem 7.10 we require that Rdiss be positive-definite. As is the case with stability of equilibria for mechanical systems, this hypothesis is stronger than required. • 8. Open problems and future directions We conclude with a discussion of some of the more obvious research directions suggested by our approach. (1) First and foremost, it would be important to analyze the setting of reduction by a general Lie group action. In such a setup, one might consider a principal fiber bundle π:Q→Bwith structure group being a Lie group G. One would then be interested in the Riemannian geometry of the principal bundle. For example, reduction should be describable in terms of ideas such as the reduced Riemannian metric on B, the mechanical connection and its curvature, the locked inertia tensor, the second fundamental form of the fibers, and the amended/effective potential. This picture is only partially understood at present (but see some initial results in [Cort´es and Mart´ınez 2003]), and the application of this picture to stability theory is described in [Marsden 1992]. (2) In a related direction, with regards to the stability analysis for relative equilibria, it would be interesting to describe the Energy- Momentum method of [Simo, Lewis, and Marsden 1991, Simo, Posbergh, and Marsden 1990] in Riemannian geometric language. (3) Generalizing from the Riemannian context, one could study the geometry of a G-invariant affine connection on a principal fiber bundle π:Q→B. From the point of view of mechanics, this is interesting since the dynamics of systems with nonholonomic constraints can be described using an affine connection that is generally 50 FRANCESCO BULLO AND ANDREW D. LEWIS different from the affine connection. This is an idea that goes back to Synge [1928]. (4) It would also be interesting to characterize the linearization of a mechanical control system along more general trajectories than relative equilibria. An interesting example of such trajectories are those arising in the theory of kinematic reduction for affine connection control systems [Bullo and Lewis 2003, Bullo and Lynch 2001]. (5) From a control theoretical viewpoint, very little is known about the linear and nonlinear controllability properties of mechanical control systems along relative equilibria. This is due to the complications arising from the gyroscopic forces present in the reduced equations. 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[1998] Partial Stability and Control, Birkh¨auser, Boston/Basel/Stuttgart, ISBN 0-8176-3917-9. Yano, K. and Ishihara, S. [1973] Tangent and Cotangent Bundles, number 16 in Pure and Applied Mathematics, Dekker Marcel Dekker, New York, ISBN 0-8247-6071-9. Francesco Bullo Associate Professor Mechanical & Environmental Eng. Univ. of California at Santa Barbara Engineering II Bldg. Santa Barbara, CA 93106-5070, U.S.A. [email protected] Andrew D. Lewis Associate Professor Dep. of Math. and Statistics Queen’s University Kingston, ON K7L 3N6, Canada [email protected]