Variational Structures for Hamel's Equations and Stabilization*

4th IFAC Workshop on Lagrangian and Hamiltonian Methods for Non Linear Control International Federation of Automatic Control August 29-31, 2012. Bertinoro, Italy

Variational Structures for Hamel’s Equations and Stabilization ? Kenneth R. Ball ∗ Dmitry V. Zenkov ∗∗ Anthony M. Bloch ∗∗∗ ∗

Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA (e-mail: [email protected]) ∗∗ Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA (e-mail: [email protected]) ∗∗∗ Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA (e-mail: [email protected]) Abstract: Hamel’s equations are an analogue of the Euler–Lagrange equations of Lagrangian mechanics when the velocity is measured relative to a frame which is not related to system’s local configuration coordinates. The use of this formalism often leads to a simpler representation of dynamics but introduces additional terms in the equations of motion. The paper elucidates the variational nature of Hamel’s equations and discusses their utility in control and stabilization. The latter is illustrated with the problem of stabilization of a falling disk. Keywords: Geometric mechanics, Hamilton–Pontryagin principle, bundle structure, feedback stabilization. 1. INTRODUCTION The Euler–Lagrange equations written in generalized coordinates, while universal, are not always the best tool for analyzing the dynamics of mechanical systems. For example, it is difficult to study the motion of the Euler top if the Euler–Lagrange equations (either intrinsically or in generalized coordinates) are used to represent the dynamics. On the other hand, the use of the angular velocity components relative to a body frame pioneered by Euler (1752) results in a much simpler representation of dynamics. Euler’s approach led to the development of the Euler–Poincar´e equations by Lagrange (1788) for reasonably general Lagrangians on the rotation group and by Poincar´e (1901) for arbitrary Lie groups (see Marsden and Ratiu (1999) for details and history). An extension of this formalism from Lie groups to arbitrary configuration manifolds was carried out by Hamel (1904). In Hamel’s formalism, the velocity components are measured relative to a set of independent (local) vector fields on the configuration space that are not associated with (local) configuration coordinates. See e.g. Neimark and Fufaev (1972) and Bloch et al. (2009a) for the history and contemporary exposition of Hamel’s formalism. Just as in the Euler–Poincar´e case, Hamel’s equations contain terms whose structure at first appears to be nonvariational. The presence of these terms is caused by nonvanishing Jacobi–Lie bracket of the vector fields that are used to measure the velocity components. We present two variational derivations of Hamel’s equations, with an emphasis on the nature of these bracket ? The research of AMB was partially supported by NSF grants DMS0806765 and DMS-0907949. The research of DVZ and KRB was partially supported by NSF grants DMS-0604108 and DMS-0908995.

978-3-902823-08-3/12/$20.00 © 2012 IFAC

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terms. The first approach is based on the formula for variations of velocity components that generalizes the variation formula for the Euler–Poincar´e reduction (see Marsden (1992), Marsden and Ratiu (1999) and Bloch et al. (2009a) for details). The second approach utilizes the Hamilton– Pontryagin principle and produces these bracket terms using unconstrained variations, albeit taken in a different, larger-dimensional space. The origins of the Hamilton–Pontryagin principle may be traced back to Livens (1919); see also Pars (1965). The recent results of Yoshimura and Marsden (2006a,b, 2007) reveal the links between this principle, implicit Lagrangian systems, and Dirac structures. The latter are important in interconnected mechanical systems, electric circuits, electromechanical systems, and control, as discussed in e.g. van der Schaft and Maschke (1994, 1995), van der Schaft (1998), and Bloch and Crouch (1997). As shown in Yoshimura and Marsden (2006b), the dynamics and the Legendre transform are the outcomes of a variational procedure when the Hamilton–Pontryagin principle is used. If a regular approach, either Lagrangian or Hamiltonian, is used to set up the dynamics, the interconnections induced by the constraints lead to systems of differential-algebraic equations. The latter are known to be difficult to model numerically. The use of frames and Hamel’s formalism eliminates the Lagrange multipliers in a natural fashion and represents the dynamics in the form of differential equations. Thus, pairing Hamel’s formalism with suitable integrators may result in good numerical techniques. We are particularly interested in merging the Hamel and variational integrator formalisms. Thus our interest in the nature of the bracket terms, as is it currently unclear what the discrete analogue of the said terms is. A general analysis of this approach will be a subject of future

10.3182/20120829-3-IT-4022.00010

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publications. Some early results are quite promising, see Zenkov et al. (2012).

(ii) The curve q(t) satisfies the Euler–Lagrange equations (1).

In this paper we demonstrate the utility of Hamel’s equations in feedback stabilization problems, using the slow upright uniform motions of a falling disk as an example. Specifically, we use a fusion of a momentum shaping technique and the nonholonomic energy-momentum stability analysis developed in Zenkov et al. (1998) to construct a stabilizing controller for the disk.

We point out here that this principle assumes that a variation of the curve q(t) induces the variation δ q(t) ˙ of its velocity vector according to the formula d δ q(t) ˙ := δq(t). dt For more details and a proof, see e.g. Bloch (2003) and Marsden and Ratiu (1999).

The paper is organized as follows. Lagrangian mechanics and the Hamilton–Pontryagin principle are reviewed in Section 2. Hamel’s formalism along with associated variational principles are the subject of Section 3. Hamel’s equations in redundant coordinates are then used to represent the dynamics of a spherical pendulum in Section 4. An example of a control application is given in Section 5. 2. LAGRANGIAN MECHANICS 2.1 The Euler–Lagrange Equations A Lagrangian mechanical system is specified by a smooth manifold Q called the configuration space and a function L : T Q → R called the Lagrangian. In many cases, the Lagrangian is the kinetic minus potential energy of the system, with the kinetic energy defined by a Riemannian metric and the potential energy being a smooth function on the configuration manifold Q. If necessary, nonconservative forces can be introduced (e.g., gyroscopic forces that are represented by terms in L that are linear in the velocity), but this is not discussed in detail in this paper. In local coordinates q = (q 1 , . . . , q n ) on the configuration space Q we write L = L(q, q). ˙ The dynamics is given by the Euler–Lagrange equations d ∂L ∂L = i , i = 1, . . . , n. (1) dt ∂ q˙i ∂q These equations were originally derived by Lagrange in 1788 by requiring that simple force balance F = ma be covariant, i.e. expressible in arbitrary generalized coordinates. A variational derivation of the Euler–Lagrange equations, namely Hamilton’s principle (see Theorem 2.1 below), came later in the work of Hamilton in 1834/35. Let q(t), a ≤ t ≤ b, be a smooth curve in Q. A variation of the curve q(t) is a smooth map β : [a, b] × [−ε, ε] → Q that satisfies the condition β(t, 0) = q(t). This variation defines the vector field ∂β(t, s) δq(t) = ∂s s=0 along the curve q(t). Theorem 2.1. The following statements are equivalent: (i) The curve q(t), where a ≤ t ≤ b, is a critical point of the action functional Z b L(q, q) ˙ dt (2) a

on the space of curves in Q connecting qa to qb on the interval [a, b], where we choose variations of the curve q(t) that satisfy δq(a) = δq(b) = 0. 179

2.2 The Hamilton–Pontryagin Principle Let Q be a manifold, T Q be its tangent, and T ∗ Q be its cotangent bundles. Let q, (q, v), and (q, p) be local coordinates on Q, T Q, and T ∗ Q, respectively. Let t 7→ (q(t), v(t), p(t)), t ∈ [a, b], be a curve in the Pontryagin bundle T Q ⊕ T ∗ Q. Following Yoshimura and Marsden (2006a,b, 2007), define the action functional on T Q ⊕ T ∗ Q by the formula Z bh

i (3) L(q(t), v(t)) + p(t), q(t) ˙ − v(t) dt. S= a

Consider the space of curves in T Q ⊕ T ∗ Q that satisfy the conditions q(a) = qa , q(b) = qb , with a ≤ t ≤ b, where qa and qb are two points in the configuration space Q. The variational derivative of action (3) on this space of curves is computed to be     Z b  ∂L ∂L δS = − p˙ δq + − p δv + (q˙ − v)δp dt. ∂q ∂v a Theorem 2.2. The following statements are equivalent: (i) The curve (q(t), v(t), p(t)) is a critical point of the action functional (3) on the space of curves in T Q ⊕ T ∗ Q connecting qa ∈ Q to qb ∈ Q on the interval [a, b], with variations satisfying δq(a) = δq(b) = 0. (ii) The curve (q(t), v(t), p(t)) satisfies the implicit Euler–Lagrange equations ∂L ∂L − p˙ = 0, p = , q˙ = v. (4) ∂q ∂v Equations (4) include the Euler–Lagrange equations, the Legendre transform p = ∂v L, and the second order condition q˙ = v. We emphasize that variations δv and δp are not induced by variations δq. 3. LAGRANGIAN MECHANICS IN NON-COORDINATE FRAMES 3.1 The Hamel Equations In this paragraph we briefly discuss the Hamel equations, following the exposition of Bloch et al. (2009a). In many cases the Lagrangian and the equations of motion have a simpler structure when written using velocity components measured against a frame that is unrelated to the system’s local configuration coordinates. An example of such a system is the rigid body. Let q = (q 1 , . . . , q n ) be local coordinates on the configuration space Q and ui ∈ T Q, i = 1, . . . , n, be smooth independent local vector fields on Q defined in the same

LHMNLC 2012 August 29-31, 2012. Bertinoro, Italy

coordinate neighborhood. In certain cases, some or all of ui can be chosen to be global vector fields on Q. The components of ui relative to the basis ∂/∂q j will be denoted ψij ; that is, ∂ ui (q) = ψij (q) j , ∂q where i, j = 1, . . . , n and where summation on j is understood. Let ξ = (ξ 1 , . . . , ξ n ) ∈ Rn be the components of the velocity vector q˙ ∈ T Q relative to the basis u1 , . . . , un , i.e., q˙ = ξ i ui (q); (5) then l(q, ξ) := L(q, ξ i ui (q)) (6) is the Lagrangian of the system written in the local coordinates (q, ξ) on the tangent bundle T Q. The coordinates (q, ξ) are Lagrangian analogues of non-canonical variables in Hamiltonian dynamics. Define the quantities ckij (q) by the equations ckij (q)uk (q),

[ui (q), uj (q)] = (7) i, j, k = 1, . . . , n. These quantities vanish if and only if the vector fields ui (q), i = 1, . . . , n, commute. Here and elsewhere, [ · , · ] is the Jacobi–Lie bracket of vector fields on Q. Viewing ui as vector fields on T Q whose fiber components equal 0 (that is, taking the vertical lift of these vector fields), one defines the directional derivatives ui [l] for a function l : T Q → R by the formula ∂l ui [l] = ψij j . ∂q The evolution of the variables (q, ξ) is governed by the Hamel equations d ∂l ∂l = ckij k ξ i + uj [l], (8) dt ∂ξ j ∂ξ coupled with equations (5). If ui = ∂/∂q i , equations (8) become the Euler–Lagrange equations (1). Equations (8) were introduced in Hamel (1904) (see also Neimark and Fufaev (1972) and Bloch et al. (2009a) for details and some history). 3.2 Hamilton’s Principle for Hamel’s Equations Theorem 3.1. Let L : T Q → R be a Lagrangian and l be its representation in local coordinates (q, ξ). Then, the following statements are equivalent: (i) The curve (q(t), ξ(t)) is a critical point of the functional Z b

l(q, ξ) dt

(9)

a

with respect to variations δξ, induced by the variations δq = ζ i ui (q), ζ(a) = ζ(b) = 0, and given by δξ k = ζ˙ k + ck (q)ξ i ζ j . (10) ij

(ii) The curve (q(t), ξ(t)) satisfies the Hamel equations (8) coupled with the equations q˙ = ξ i ui (q). For the proof of Theorem 3.1 and the early development and history of these equations see Poincar´e (1901), Hamel (1904), and Bloch et al. (2009a). 180

3.3 The Hamilton–Pontryagin Principle We start by rewriting action (3) using the frame ui (q), i = 1, . . . , n. Denote the components of q, ˙ v, and p relative to the frame ui (q) and its dual by ξ, η, and µ, respectively: ∂ q˙ = ξ j uj = ξ j ψji i , (11) ∂q ∂ v = η j uj = η j ψji i , (12) ∂q p = µj uj = µj φji dq i . (13) Here and below, φji are the elements of the inverse of ψ. The action functional (3) becomes Z bh i l(q(t), η(t)) + hp(t), q(t) ˙ − v(t)i dt, S=

(14)

a

where q, ˙ v, and p are given by formulae (11)–(13). Theorem 3.2. The following statements are equivalent: (i) The curve (q(t), η(t), µ(t)), a ≤ t ≤ b, is a critical point of the action functional (14) on the space of curves in T Q ⊕ T ∗ Q connecting qa and qb on the interval [a, b], where we choose variations of the curve (q(t), η(t), µ(t)) that satisfy δq(a) = δq(b) = 0. (ii) The implicit Hamel equations ∂φkm r m uj [l] − µ˙ j − ψ ψ µk ξ i ∂q r i j ∂ψ m (15) − ir φkm ψjr µk η i = 0, ∂q ∂l µ= , ξ=η (16) ∂η hold. Coupled with (11), the implicit Hamel equations capture the dynamics for the Lagrangian l(q, ξ). Proof. Taking the variation of (14) gives Z bh i δS = δl(q, η) + δ(hp, q˙ − vi) dt a Z b ∂l(q, η) i ∂l(q, η) i = δq + δη ∂q i ∂η i a  + hδp, q˙ − vi + hp, δ qi ˙ − hp, δvi dt. Next, we evaluate δv and obtain ∂ψim ∂ i s δv = δη i ui + η i δui = δη i ui + η δq ∂q s ∂q m ∂ψim k i = δη i ui + φ η uk δq s . (17) ∂q s m Therefore, the term hp, δvi becomes ∂ψim k hp, δvi = µi δη i + φ µk η i δq s . ∂q s m Integration by parts replaces the term hp, δ qi ˙ with −hp, ˙ δqi, as the term dhp, δqi/dt vanishes after integration. Evaluating p, ˙ we obtain p˙ = µ˙ j uj + µi

dui ∂φk = µ˙ j uj + rs ψir µk ξ i dq s dt ∂q ∂φk = µ˙ j φjs dq s + rs ψir µk ξ i dq s . ∂q

LHMNLC 2012 August 29-31, 2012. Bertinoro, Italy

Therefore, −hp, ˙ δqi = −µ˙ j φjs δq s −

∂φks r ψ µk ξ i δq s . ∂q r i

Using these formulae, the variation of action (14) becomes  Z b  ∂l ∂φks r ∂ψim k j i i δS = − µ ˙ φ ψ φ − µ ξ − µ η δq s j s s r i k s m k ∂q ∂q ∂q a    ∂l i − µ δη + hδp, q ˙ − vi dt + i ∂η i Z b  ∂l m ∂φkm r m = ψj − µ˙ j − ψi ψj µk ξ i m r ∂q ∂q a  ∂ψim k r i − φ ψ µk η φjs δq s ∂q r m j    ∂l i + − µi δη + hδp, q˙ − vi dt. ∂η i Recall that the variations δv and δp are not induced by δq. By the independence of the variation δq, δη, and δp, vanishing of the variation of the action functional (14) is equivalent to the implicit Hamel equations (15) and (16).  The inverse matrix differentiation rule implies ∂ψjm k ∂φkm m ψ φ , = − ∂q r j ∂q r m and therefore ∂φkm r m ∂ψim k r ψ ψ − φ ψ − ∂q r i j ∂q r m j   m ∂ψj r ∂ψim r k ψ − ψ = ckij . (18) = φm ∂q r i ∂q r j Thus, substituting (16) in (15) and utilizing (18) produces Hamel’s equations (8). We emphasize that equations (15) and (16) include Hamel’s equations, the Legendre transform µ = ∂η l, and the second order condition ξ = η. 3.4 Remarks on the Frame Selection As discussed in Bloch (2003), Bloch et al. (1996), and Bloch et al. (2009a), (see also Fernandez and Bloch (2011) and Maruskin et al. (2012)), the presence of constraints and symmetry naturally defines subbundles of the system’s velocity phase space T Q. For underactuated mechanical systems, the controlled directions define a subbundle of the system’s momentum phase space T ∗ Q. Selecting a frame then should be carried out in such a way that the suitable subframes of the frame and its dual span the mentioned subbundles. Such selections lead to a simpler representation of dynamics and highlight the structure of the mechanical system under consideration (subsystems, interconnections, etc.). 4. THE SPHERICAL PENDULUM Here we discuss a representation of dynamics of a spherical pendulum that allows one to develop effective and structure-preserving integrators, as discussed in Bobenko and Suris (1999) and Zenkov et al. (2012). 181

Consider a spherical pendulum whose length is r and mass is m. While the pendulum is usually viewed as a point mass moving on the sphere, the development here is based on the representation µ˙ = µ × ξ + τ, (19) γ˙ = γ × ξ (20) of pendulum’s dynamics. Here ξ is the angular velocity of the pendulum, µ is its angular momentum, γ is the unit vertical vector (and thus the constraint kγk = 1 is assumed), and τ is the torque produced by gravity, all written relative to a body frame. The latter is assumed orthonormal, thus allowing one to interpret the dual vectors as regular vectors, if necessary. Note that the projection of τ on γ is zero. Equations (19) and (20) may be interpreted as the dynamics of a degenerate rigid body. Indeed, select an orthonormal body frame with the third vector aligned along the rod of the pendulum. The inertia tensor of the pendulum relative to such a frame is I = diag{mr2 , mr2 , 0}, and the Lagrangian reads (21) l(ξ, γ) = 21 hI ξ, ξi − mgha, γi, where a is the vector from the fixed point to the bob of the pendulum. Equations (19) and (20) are the dynamics associated with Lagrangian (21). With our frame selection, the third component of the angular momentum of the body vanishes, ∂l µ3 = 3 = I3 ξ 3 = 0, ∂ξ and thus there are only two nontrivial equations in (19). Thus, one needs five equations to capture the dynamics of the pendulum. This reflects the fact that rotations about the direction of the pendulum have no influence on its motion. The dynamics then can be simplified by setting ξ 3 = 0. With this assumption, the aforementioned five first-order differential equations read µ˙ 1 = mgrγ 2 , µ˙ 2 = −mgrγ 1 , (22) 1 2 3 2 1 3 3 2 1 1 2 γ˙ = −ξ γ , γ˙ = ξ γ , γ˙ = ξ γ − ξ γ . (23) Equations (22) and (23) are in fact Hamel’s equations written in the redundant configuration coordinates (γ 1 , γ 2 , γ 3 ) relative to the frame ∂ ∂ u1 = γ 3 2 − γ 2 3 , ∂γ ∂γ 1 ∂ 3 ∂ u2 = γ −γ . ∂γ 3 ∂γ 1 We emphasize that the representation of pendulum’s dynamics discussed here, though redundant, eliminates the use of local coordinates on the sphere, such as spherical coordinates. Spherical coordinates, while being a nice theoretical tool, introduce artificial singularities at the north and south poles. That is, the equations of motion written in spherical coordinates have denominators vanishing at the poles, but this has nothing to do with the physics of the problem and is solely caused by the geometry of the spherical coordinates. Another important remark is that the length of the vector γ is a conservation law of equations (19) and (20), kγk = const, (24)

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and thus adding the constraint kγk = 1 does not result in a system of differential-algebraic equations. 5. STABILIZATION OF SLOW MOTIONS OF A FALLING DISK We illustrate the usefulness of Hamel’s formalism with the problem of stabilization of slow vertical motions of a falling disk. See Bloch et al. (2009b) for more details. We use the nonholonomic energy-momentum method which employs a variational approach to stability in this context. Consider a uniform disk rolling without sliding on a horizontal plane. It is well-known that some of the steady state motions are the uniform motions of a disk along a straight line. Such motions are unstable if the angular velocity of the disk is small. Stability is observed if the angular velocity of the disk exceeds a certain critical value, see Neimark and Fufaev (1972) and Bloch et al. (2009b) for details. Below we use a steering torque for stabilization of slow unstable motions of the disk. We assume that the disk has a unit mass and a unit radius. The moments of inertia of the disk relative to its diameter and to the line orthogonal to the disk and through its center are A and B = 2A, respectively. The configuration coordinates for the disk are (θ, φ, ψ, x, y) as in Figure 1. Following Neimark and Fufaev (1972), we select u1 to be the vector in the xy-plane and tangent to the rim of the disk, u2 to be the vector from the contact point to the center of the disk, and u3 to be u1 × u2 , as shown in Figure 1. In agreement with our general frame

here and below the parameters a and b label the levels of these conservation laws. These conservation laws are obtained by integrating the equations dξ dη A = Aξ tan θ − Bη, (B + 1) = −ξ. dθ dθ Formulae (28) may be interpreted as momentum conservation laws (see Bloch et al. (2009a) and Zenkov (2003)). Now consider a steady state motion θ = 0, ξ = 0, η = ηe . This motion is unstable if ηe is small. Set ˙ u = −f (θ)η θ, (29) where f (θ) is a differentiable function. The motivation for the definition (29) of the steering torque u is that it preserves the structure of equations (26) and (27), and thus the controlled system will have conservation laws whose structure is similar to that of the uncontrolled system. Viewing θ as an independent variable, we replace equations (26) and (27) with the linear system dξ dη A = Aξ tan θ − (B + f (θ))η, (B + 1) = −ξ. (30) dθ dθ The general solution of (30), ξ = Fc (θ, a, b), η = Gc (θ, a, b), (31) is interpreted as the controlled conservation laws. The functions that define these conservation laws are typically difficult or impossible to find explicitly. Next, we reduce the dynamics to the common level set of the controlled conservation laws (31). From (25), this defines a family of one degree of freedom Lagrangian (or Hamiltonian) systems (A + 1)θ¨ + AFc2 (θ, a, b) tan θ − (B + 1)Fc (θ, a, b)Gc (θ, a, b) − g sin θ = 0. The stability of the relative equilibrium θ = 0, ξ = 0, η = ηe is tested using the nonholonomic energy-momentum method of Zenkov et al. (1998). This method requires that d AFc2 (θ, ae , be ) tan θ dθ
− (B + 1)Fc (θ, ae , be )Gc (θ, ae , be ) − g sin θ

Fig. 1. The geometry of the rolling disk. selection process, the fields u1 , u2 , and u3 span the fibers of the constraint distribution, the fields u2 and u3 span the constrained symmetry directions, and the dual of u2 spans the control subbundle. The component of disk’s angular ˙ the u2 and u3 components are velocity along u1 equals θ, denoted by ξ and η. Using this frame, the Hamel equations for the disk are computed to be (A + 1)θ¨ + Aξ 2 tan θ − (B + 1)ξη − g sin θ = 0, (25) Aξ˙ − Aξ θ˙ tan θ + Bη θ˙ = u, (26) (B + 1)η˙ + ξ θ˙ = 0, (27) where u is the steering torque and g is the acceleration of gravity. In the absence of the steering torque, the last two equations can be written as the conservation laws ξ = F (θ, a, b), η = G(θ, a, b); (28) 182

> 0, θ=0

where ae and be are defined by the equations Fc (0, ae , be ) = 0, Gc (0, ae , be ) = ηe . This stability condition is obtained by constructing a suitable Lyapunov function, see Zenkov et al. (1998) for details. Using (30), the stability condition becomes Ag f (0) > − B. (32) (B + 1)ηe2 That is, any function f (θ) whose value at θ = 0 satisfies inequality (32) defines a stabilizing steering torque. Observe that in the settings considered in the present paper the energy-momentum method gives conditions for nonlinear Lyapunov (nonasymptotic) stability. Hence stabilization by the torque (29) is nonlinear and nonasymptotic. Asymptotic stabilization can be achieved by adding dissipation-emulating terms to the control input.

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6. CONCLUSION This paper demonstrates the utility of Hamel’s equations and in particular their variational nature for formulating certain problems in dynamics and analyzing their stability. Among our future goals are developing variational integrators for Hamel’s equations to aid in simulation, and developing their use in both open loop control and stabilization of complex coupled systems. ACKNOWLEDGEMENTS We would like to thank Jerry Marsden for his inspiration and helpful discussions in the beginning of this work, and the reviewers for useful suggestions. AMB and DVZ would like to acknowledge support from Mathematisches Forschungsinstitut Oberwolfach, where a part of this work was carried out. REFERENCES Bloch, A.M. (2003). Nonholonomic Mechanics and Control, volume 24 of Interdisciplinary Appl. Math. Springer-Verlag, New York. Bloch, A.M. and Crouch, P.E. (1997). Representations of Dirac structures on vector spaces and nonlinear LC circuits. In Differential Geometry and Control, volume 64 of Proc. Sympos. Pure Math., 103–117. AMS, Providence, Rhode Island. Bloch, A.M., Krishnaprasad, P.S., Marsden, J.E., and Murray, R. (1996). Nonholonomic mechanical systems with symmetry. Arch. Rational Mech. Anal., 136, 21–99. Bloch, A.M., Marsden, J.E., and Zenkov, D.V. (2009a). Quasivelocities and symmetries in nonholonomic systems. Dynamical Systems: An International Journal, 24(2), 187–222. Bloch, A.M., Marsden, J.E., and Zenkov, D.V. (2009b). Quasivelocities and stabilization of relative equilibria of underactuated nonholonomic systems. Proc. CDC, 48, 3335–3340. Bobenko, A.I. and Suris, Yu.B. (1999). Discrete time Lagrangian mechanics on Lie groups, with an application to the Lagrange top. Communications in Mathematical Physics, 204, 147–188. Euler, L. (1752). Decouverte d’un nouveau principe de Mecanique. M´emoires de l’acad´emie des sciences de Berlin, 6, 185–217. Fernandez, O. and Bloch, A.M. (2011). The Weitzenbock connection and time reparameterization in nonholonomic mechanics. Journal of Mathematical Physics, 52, 012901. Hamel, G. (1904). Die Lagrange–Eulersche Gleichungen der Mechanik. Z. Math. Phys., 50, 1–57. Hamilton, W.R. (1834). On a general method in dynamics, part I. Phil. Trans. Roy. Soc. Lond., 247–308. Hamilton, W.R. (1835). On a general method in dynamics, part II. Phil. Trans. Roy. Soc. Lond., 95–144. Lagrange, J.L. (1788). M´ecanique Analytique. Chez la Veuve Desaint. Livens, G.H. (1919). On Hamilton’s principle and the modified function in analytical dynamics. Proc. Roy. Soc. Edinburgh, 39, 113–119. Marsden, J.E. (1992). Lectures on Mechanics, volume 174 of London Mathematical Society Lecture Note Series. Cambridge University Press. 183

Marsden, J.E. and Ratiu, T.S. (1999). Introduction to Mechanics and Symmetry, volume 17 of Texts in Appl. Math. Springer-Verlag, New York, 2nd edition. Maruskin, J., Bloch, A.M., Marsden, J.E., and Zenkov, D.V. (2011). A fiber bundle approach to the transpositional relations in nonholonomic mechanics. Preprint. Neimark, Ju.I. and Fufaev, N.A. (1972). Dynamics of Nonholonomic Systems, volume 33 of Translations of Mathematical Monographs. AMS, Providence, Rhode Island. Pars, L.A. (1965). A Treatise on Analytical Dynamics. Wiley, New York. Poincar´e, H. (1901). Sur une forme nouvelle des ´equations de la m´ecanique. CR Acad. Sci., 132, 369–371. van der Schaft, A.J. (1998). Implicit Hamiltonian systems with symmetry. Reports on Mathematical Physics, 41, 203–221. van der Schaft, A.J. and Maschke, B.M. (1994). On the Hamiltonian formulation of nonholonomic mechanical systems. Reports on Mathematical Physics, 34, 225–233. van der Schaft, A.J. and Maschke, B.M. (1995). The Hamiltonian formulation of energy conserving physical systems with external ports. Archiv f¨ ur Elektronik und ¨ Ubertragungstechnik, 49, 362–371. Yoshimura, H. and Marsden, J.E. (2006a). Dirac structures in Lagrangian mechanics. Part I: Implicit Lagrangian systems. Journal of Geometry and Physics, 57, 133–156. Yoshimura, H. and Marsden, J.E. (2006b). Dirac structures in Lagrangian mechanics. Part II: Variational structures. Journal of Geometry and Physics, 57, 209– 250. Yoshimura, H. and Marsden, J.E. (2007). Reduction of Dirac structures and the Hamilton–Pontryagin principle. Reports on Math. Phys., 60, 381–426. Zenkov, D.V. (2003). Linear conservation laws of nonholonomic systems with symmetry. Disc. Cont. Dyn. Syst., (extended volume), 963–972. Zenkov, D.V., Bloch, A.M., and Marsden, J.E. (1998). The energy-momentum method for the stability of nonholonomic systems. Dynamics and Stability of Systems, 13, 128–166. Zenkov, D.V., Leok, M., and Bloch, A.M. (2012). Hamel’s formalism and variational integrators on a sphere. Preprint.