On the dynamics of a heavy symmetric ball that rolls without sliding on a uniformly rotating surface of revolution

We study the class of nonholonomic mechanical systems formed by a heavy symmetric ball that rolls without sliding on a surface of revolution, which is either at rest or rotates about its (vertical) figure axis with uniform angular velocity. The first studies of these systems go back over a century, but a comprehensive understanding of their dynamics is still missing. The system has an $SO(3) \times SO(2)$ symmetry and reduces to four dimensions. We extend in various directions, particularly from the case $\Omega = 0$ to the case $\Omega \ne 0$, a number of previous results and give new results. In particular, we prove that the reduced system is Hamiltonizable even if $\Omega \ne 0$ and, exploiting the recently introduced `moving energy', we give sufficient conditions on the profile of the surface that ensure the periodicity of the reduced dynamics and hence the quasi-periodicity of the unreduced dynamics on tori of dimension up to three. Furthermore, we determine all the equilibria of the reduced system, which are classified in three distinct families, and determine their stability properties. In addition to this, we give a new form of the equations of motion of nonholonomic systems in quasi-velocities which, at variance from the well known Hamel equations, use any set of quasi-velocities and explicitly contain the reaction forces.


Introduction
1.1 Motivations. This paper is devoted to the class of nonholonomic mechanical systems formed by a ball that rolls without sliding on a surface of revolution, under the action of gravity, which is assumed to be directed as the surface figure axis. The ball is assumed to be dynamically symmetric, namely, its center of mass coincides with its center and its three moments of inertia on which the ball rolls. The smoothness ofΣ puts some conditions on the curvature of Σ, which are clarified in Proposition 1.
The equations of motion of the system are derived in the Appendix, as an instance of a novel form of the equations of motion of nonholonomic systems in quasi-velocities which we derive there. At variance from Hamel equations, that choose the quasi-velocities so as to "hide" the reaction forces [34,37,11], our equations use any set of quasi-velocities and include the explicit expression of the reaction forces as a function on the phase space (Proposition 16). From a general perspective, this might be useful in the study of a number of questions in nonholonomic mechanics in which the reaction forces play a dominant role, such as the existence of first integrals, invariant measures etc.
Since the SO(2)-action given by spatial rotations of the system around the surface figure axis has isotropy, the quotient space M 4 = M 8 /SO(3) × SO(2) is a stratified space. It consists of a singular, one-dimensional stratum M sing 4 that contains all reduced kinematical states in which the center of the ball is at the 'vertex' of the surface (the point of Σ that belongs to the figure axis) with zero velocity, and of a regular four-dimensional stratum M reg 4 . Following [35,22] we will embed M 4 in R 5 through the use of a set of 5 invariant polynomials. This will allows us to give some results on the entire reduced space M 4 . Subsequently, we will specialize the analysis to M reg 4 or even to its subset M • 4 obtained by removing all states in which the center of the ball passes (with any velocity) through the vertex. In so doing, when this will make the description more transparent, we will reverse to polar coordinates.
In Section 3 we study some general properties of the reduced and unreduced systems. After giving the expressions of the two Routh integrals and of the moving energy, extending a similar analysis in [22] we study their independence (Proposition 3). Next, we show that the motions of the reduced system (including those that transit through the vertex) are of four possible types (equilibria, periodic motions, motions asymptotic to equilibria, motions which go to infinity; Proposition 4) and we discuss their reconstruction to the full system (Proposition 5). In particular, the already mentioned results on the reconstruction under compact symmetry groups [36,30] imply that motions of the full system in relative equilibria and relative periodic orbits are quasi-periodic on tori of dimensions up to, respectively, two and three. Lastly, we prove that the level sets of the moving energy in M 4 are all compact-so that the reduced dynamics is generically periodic and the unreduced one is generically quasi-periodic-in two cases: if Ω = 0 and the surface goes to +∞ at infinity, and if Ω = 0 and the surface goes to +∞ at infinity sufficiently fast, more than quadratically in the distance (Proposition 8). We stress that it is only the behaviour at infinity of the surface-and no other details of it-that plays a role in these two results. The first was in fact proven in [35,43,22], but was there stated only for either convex or compact surfaces. The case Ω = 0 is new. (A very weak version of it was proven in [28], with a continuation argument from the case Ω = 0, for convex surfaces and sufficiently small Ω's).
In Section 4 we restrict our analysis to the subset M • 4 (all states with the ball at vertex removed) and first prove the existence in M • 4 of a rank-two Poisson tensor that makes the system Hamiltonian, with the moving energy as Hamilton function (Proposition 9) and the two Routh integrals as Casimirs. This tensor reduces to the ones of [39,22] and (up to a factor related to a time reparameterization) of [15] for Ω = 0. The interest of this Hamiltonization result resides also in the fact that while the Hamiltonizability of nonholonomic systems has been so far extensively studied in the case of linear constraints, very little is known in the case of affine constraints (the only other result we are aware of concerns the Veselova system [31]). Next, we show that the restriction of the dynamics to the level sets of the two Routh integrals can be seen as a natural Lagrangian system with one degree of freedom, namely with a Lagrangian which is the difference between the kinetic energy of a point holonomically constrained to the surface Σ and of an 'effective' potential energy which depends on the value of the two Routh integrals (Proposition 10).
In Section 5 we determine the equilibria of the reduced system in M • 4 , thus excluding those at the vertex (Proposition 11). An equilibrium of the reduced system corresponds to motions of the unreduced system in which the center of the ball moves (or stands still in space) on a parallel of the surface Σ, namely on a horizontal circle, and the component of the angular velocity of the ball normal to the surface is constant. We prove that there are reduced equilibria on any parallel of Σ, which are different if the parallel is critical (a local maximum or minimum or a saddle point of the radial height) or regular. On each critical parallel there are two families of reduced equilibria, the first for all Ω's and the second only for Ω = 0, both parametrized by the vertical component ω z ∈ R of the ball's angular velocity. In the first family the center of the ball stands still in space; this happens also if the surface Σ rotates, with any Ω. In the reduced equilibria of the second family, instead, the center of the ball rotates uniformly on the parallel with nonzero angular velocity cΩ with a certain 0 < c < 1 which depends on the moment of inertia of the ball. On regular parallels there is, for each Ω ∈ R, a family of reduced equilibria parametrized by the (nonzero) angular velocity of the center of the ball.
In Section 6 we study the stability of the reduced equilibria, regarding them as equilibria of the restriction of the reduced system to a level set of the two Routh integrals, namely, to a symplectic leaf of the rank-two Poisson structure. In order to avoid ambiguities, we thus speak of 'leafwisestability'. This study reduces to the study of the critical points of the effective potential. We first give analytical conditions for the leafwise-(in)stability of the reduced equilibria of the three families (Proposition 12) and then we study these conditions, with particular attention to the effect of the surface rotation. The resulting bifurcation scenario, which is somehow rich, is described in , and a number of situations are considered. Overall, we reach a fairly complete understanding of the reduced equilibria's leafwise-stability.
In Section 7 we study in some detail, and partly numerically, the particular case in which the surface is a paraboloid. This is has two motivations. First, since the behaviour at infinity of the surface is exaclty quadratic in the distance from the center, our result about the compactness of the level sets of the moving energy does not apply when Ω = 0. Nevertheless, using the fact that in this case the two Routh integrals can be explicitly determined, we can prove that the common level sets of the three first integrals are compact, so that the dynamics of the reduced system is generically periodic. This suggests that our integrability results can be improved. Second, we investigate numerically the existence and number of reduced equilibria on the level sets of the two Routh integrals, finding that on each of them there are between one (leafwise-stable) and three (one of which leafwise-unstable) reduced equilibria.
In the very short Conclusions we point out some open problems and some future research directions.
2 The system and its reduction 2.1 The system. We start with the holonomic system formed by a homogeneous ball of mass m and radius a, the center C of which is constrained to belong to a surface of revolution Σ embedded in R 3 (x, y, z) and produced by the rotation, about the z-axis, of the graph Γ of an even smooth function f : R → R. More precisely, in view of a later rescaling of the coordinates, we assume that Σ is described by the equation We call f the 'profile function' and its graph Γ the 'profile curve'. Note that f has either a minimum or a maximum at r = 0. The configuration manifold of this holonomic system can be identified with are the a-rescaled (x, y)-coordinates of C, so that OC = (ax 1 , ax 2 , af (|x|)), and the matrix R fixes the attitude of the ball. After (right) trivialization of the tangent bundle of SO (3), the phase space of the system is the 10-dimensional manifold where ω = (ω x , ω y , ω z ) is the angular velocity of the ball relative to, and written in, the spatial frame.
We assume that the only active force that acts on the system is weight, directed as the downward z-axis. We denote by g the gravity acceleration and by mka 2 the moment of inertia of the ball with respect to C; thus 0 < k < 1 (k = 2 5 for a homogeneous ball). Then, up to an overall factor ma 2 , the Lagrangian of the system is L(x, R,Ṙ, ω) = 1 2 |ẋ| 2 + 1 2 x ·ẋ |x| f (|x|) withĝ = g/a. Next, we introduce the nonholonomic constraint that the ball rolls without sliding on a surface Σ which lies below Σ and rotates with constant angular velocity Ωe z about the z-axis. In the rescaled coordinates, the points ofΣ have unit normal distance from those of Σ. The surfaceΣ is produced by the rotation of the curveΓ which is parallel to the graph Γ of f , with unit normal distance to it, and lies below it. It is necessary to assume thatΓ is a regular curve and that, at each point of contact withΣ, the ball touchesΣ in only that point. The latter condition requires that, at each point at which it is not concave (namely, its signed curvature is nonnegative), the curveΓ has radius of curvature > 1.
As it turns out, the latter condition follows from the former, which also ensures thatΓ is diffeomorphic to Γ: In such a case,Γ is diffeomorphic to Γ and has curvature radius > 1 at each point at which it is not concave.
Proof. Γ is the image of the immersion ι : . Thus,Γ is the image of the mapι : R → R 2 given byι The fact that f is defined in all of R rules out the possibility that f (x) < −(1 + f (x) 2 ) 3/2 for all x ∈ R 2 (by a standard comparison theorem for ODEs, since the solution of y = −(1 + y 2 ) 3/2 , y(0) = 0, blows up to −∞ in finite time, if f would satisfy such a condition then its derivative could not be defined in all of R). Thus,ι is an immersion if and only if f satisfies (2).
Finally, if f satisfies (2) then the map C : We will assume that (2) is satisfied. This excludes cases such that of a conical Σ. However, many of our results can be applied to such cases as well after removing the vertex or deforming the surface in a suitable neighbourhood of the vertex. Cases in which the profile function is defined only in an open bounded interval, and possibly diverges at its boundary, could be easily treated as well. However we note that in such cases it might happen that condition (2) is satisifed with the opposite sign, and this might affect the stability analysis of Section 6.3.
The nonholonomic constraint forces the velocity v P of the point P of the ball in contact with the surfaceΣ to be equal to Ω e z × OP . Since v P = v C + ω × CP and OP = OC + CP , the nonholonomic constraint is Equation (3) defines an eight-dimensional submanifold M 8 of M 10 which is diffeomorphic to R 2 × SO(3) × R 3 and can be globally parametrized with (x, R,ẋ, ω z ). Indeed, since CP = a n(x) with 1 where the (downward) normal unit vector to Σ at its point ax 1 , ax 2 , af , the first two entries of (3) can be written as (The third equation in (3) is obviously not independent of the first two). We thus identify Clearly, the functions x·ẋ |x| f and xi |x| f , i = 1, 2, that enter expressions (1) and (6) are not defined at x = 0 but extend smoothly to 0 at x = 0. In order to make smoothness at x = 0 transparent, following [22] we substitute the profile function f with a smooth function ψ : R → R such that The existence of such a function is granted by a result of Whitney [42] (see also [33], pages 103, 108) on account of the fact that f is even. Note that f (r) = rψ r 2 2 and ψ r 2 However, since f (r) = ψ r 2 2 + r 2 ψ r 2 2 and ψ r 2 we will use f when we need to stress the dependence on the convexity properties of the profile. The equations of motion of this nonholonomic system are derived in the Appendix. We need them only as a tool to deduce those of the reduced system.  (2)). Since the actions of SO(3) and SO(2) commute, the reduction can be performed in stages.
Since the Lagrangian and the constraint are independent of the attitude R of the ball, the SO(3)-reduction consists in simply cutting off the SO(3) factor of M 8 , and the SO(3)-reduced space is the five-dimensional manifold The SO(2)-action on M 8 induces an action on M 5 given by P.(x,ẋ, ω z ) = (P x, Pẋ, ω z ), which is free at all points of M 5 except at those with x =ẋ = 0 (the kinematical states in which the ball is at the vertex of the surface and the velocity of its center of mass is zero-hence, its angular velocity is vertical). 2 The reduction under this action is well known. In fact, SO(2) does not act on the R-factor of M 5 , while its action on the factor R 2 × R 2 is nothing but the familiar SO(2)-action of the 1:1 oscillator [35,22]. Therefore, the reduced space M 5 /SO(2) = M 8 /SO(3) × SO(2) can be identified with the semialgebraic variety M 4 = (p 0 , p 1 , p 2 , p 3 , p 4 ) ∈ R 5 : 4p 0 p 1 = p 2 2 + p 2 3 , p 0 ≥ 0 , p 1 ≥ 0 immersed in R 5 (p 0 , p 1 , p 2 , p 3 , p 4 ) =: p, with quotient map M 5 → M 4 given by (a set of generators of the invariant polynomials of the SO(2)-action, see [35,17]; see also [15]). The last coordinate p 4 for R 5 has been chosen as ω · n, instead of ω z , because this will somehow simplify the expression, and the analysis, of the moving energy. It also simplifies the equations that define the other two first integrals of the system, J 1 and J 2 below, but this is actually not that important.
The semialgebraic variety M 4 consists of two strata: a "singular" one-dimensional stratum which is the quotient of the one-dimensional submanifold M sing  where p 1 > 0, which is diffeomorphic to R + × R 3 and can be globally parametrized with either (p 1 , p 2 , p 3 , p 4 ) or (r,ṙ,θ, ω n ) (or, for that matter, with (r,ṙ,θ, ω z ) as well). In fact, we will switch between these two parametrizations depending on the needs: the former is closely linked to the theory in M 4 and M reg 4 , the latter has a more direct physical interpretation.
However, we will prefer using its embedding in R 5 .

2.3
The equations of motion of the reduced system. Following [35,22], we write the equations of motion of the SO(3) × SO(2)-reduced system in M 4 (from now on, 'reduced system') as the restriction to M 4 of a set of equations in R 5 . The deduction of these equations is done in the Appendix, on the basis of a new form of the equations of motion of nonholonomic systems.
The equations of motion of the reduced system are the restriction to M 4 of the equatioṅ where X = (X 0 , X 1 , X 2 , X 3 , X 4 ) is the vector field in R 5 with components and Note that 1 2 < µ < 1 and that ψ, F, G 3 , G 4 , g 3 and g 4 are functions of p 1 alone and are independent of Ω. Instead, f and F are functions of r, and F (r) = 1/F(r 2 /2).
For consistency, we note that M 4 is invariant under the flow of the vector field X in R 5 : X vanishes at the points of M sing 4 and is tangent to M reg 4 given that L X (p 2 2 + p 2 3 − 4p 0 p 1 ) = 0. From (9) it follows that the equilibria of the reduced system are the points where p 2 = 0 and : p 2 = 0 , X 2 (p 0 , p 1 , 0, p 3 , p 4 ) = 0 .
The reduced equilibria forming the singular stratum M sing 4 are the projection of relative equilibria in M 8 which consist of motions in which the ball stands at the vertex of the surface and uniformly spins with constant, vertical angular velocity. Relative equilibria that project onto reduced equilibria in E reg 4 consist instead of motions of the nonholonomic system in M 8 in which the ball uniformly rolls along a horizontal circle inΣ. We will study reduced equilibria in E reg 4 and their stability in Section 5. Instead, we will not study in this work the stability of the reduced equilibria in M sing 4 , and the related existence of motions asymptotic to/from them, because that would require the analysis of the system in the SO(3)-reduced space M 5 , which is extraneous to the approach taken here and is left for a separate work.
Finally, we note that the dynamics of the reduced system relative to a certain Ω = 0 is conjugate by the reflection to that of the reduced system relative to −Ω. In fact, if we make momentarily explicit the dependence of the vector field X on the surface's angular velocity Ω by denoting it X Ω , it follows from (9) that In particular, the dynamics at Ω = 0 is invariant under the reflection C.

Reduced and unreduced dynamics
In this Section we first describe some general features of the dynamics of the reduced and unreduced systems and then particularize to the case of coercive profile functions.
3.1 The first integrals. The reduced system (and hence the unreduced one) is known to have three integrals of motion: the moving energy discovered in [27] and two other integrals, whose existence was proven by Routh for Ω = 0 (and for the special case of a spherical profile also for Ω = 0, [40], section 224) and by Borisov, Mamaev and Kilin for Ω = 0 [15]. In order to express the latter two integrals we note that the equations for p 3 and p 4 are where with G 3 , G 4 , g 3 and g 4 as in (11). Let R x → U (x) ∈ GL(2) be the solution of the matrix differential equation and R x → u(x) ∈ R 2 the solution of the differential equation (recall that linear (non)homogeneous equations have global existence of the solutions).
Proposition 2. The restrictions to M 4 of the function E : R 5 → R given by and of the two components J 1 , J 2 of the map J : R 5 → R 2 given by are first integrals of the reduced system (8).
Proof. We show that E, J 1 , J 2 are first integrals of system (8) in the entire R 5 . That L X E = 0 is checked with a computation. If we denote with a dot the derivative with respect to time and with a prime the derivative with respect to p 1 , then, along a solution of (8) The fundamental matrix U satisfies the equation U = GU , which implies (U −1 ) = −U −1 G. Using this equality, u = Gu + g and (15) one verifies that d dt J = 0. We will refer to E| M4 as to the 'reduced moving energy' of and to J 1 | M4 and J 2 | M4 as to 'reduced Routh integrals' of the system. The pull-backs of these functions to M 8 give three SO(3) × SO(2)invariant first integrals of the unreduced system.
We also note that, if we momentarily make explicit the dependence of the first integrals on Ω by denoting them E Ω and J Ω = (J Ω,1 , J Ω,2 ), then where C is the reflection (13). We now prove that the three first integrals are everywhere functionally independent at all points of M 4 which are not equilibria. Specifically, we neglect the singular stratum M sing 4 (which consists of equilibria) and prove that E, J 1 , J 2 are functionally independent at all points of the regular stratum M reg 4 but the equilibria. For Ω = 0 this was proven in [22] with a direct computation. For Ω = 0 a direct computations is somewhat cumbersome and we use a somehow different argument. This argument makes explicitly appear in the proof the component X 2 of the reduced vector field and in this way sheds some light on why, in M reg 4 , the independence is lost exactly at the reduced equilibria.
Remarks. (i) The pull-back of E| M4 differs by a factor k + 1 from the reduced moving energy of the (unreduced) system as defined in [27]. The existence of this first integral was proven in [27] and its expression was then computed in [16].
(ii) With reference to the theory developed in [27,20], we note that the reduced moving energy of the (unreduced) system is the difference between the energy E 0 = L+2ĝf and the 'momentum' of the vector field Y = − Ω x2 |x| , Ω x1 |x| , 0, 0, Ω on the configuration manifold R 2 × SO(3) of the system. This is a 'kinematically interpretable' moving energy in the sense of [20] and its conservation follows from Proposition 8 of [20].
(iii) As shown in [24], when Ω = 0 the Routh integrals are "gauge momenta" [23]. In the case of the rotating cylinder the two Routh integrals are gauge momenta as well [27]. In analogy with the case of linear constraints [25], the fact that, being SO(3) × S 1 -invariant, the Routh integrals are "weakly-Noetherian" (in the sense of [23]) might suggest that they are always gauge momenta.
3.2 Some results on the reduced and unreduced dynamics. The existence of three independent integrals of motion makes the reduced dynamics in M 4 very simple.
Proposition 4. Assume that p ∈ M 4 is not an equilibrium point of X and let η p be the connected component of the fiber of (E, J)| M4 that contains p.
i. If η p does not contain any equilibrium, then the integral curve of X through p either is periodic or leaves any compact subset of M 4 for both positive and negative times. ii. If η p contains an equilibrium, then for positive times the integral curve of X through p either leaves any compact subset of M 4 or is asymptotic to an equilibrium. The same happens for negative times.
and, by Proposition 3, is a component of a regular fiber of (E, J)| M4 . As such, η p is a closed embedded one-dimensional submanifold of M reg 4 , which is moreover invariant under the flow of X and does not contain any equilibrium. Thus, η p is the image of the maximal integral curve of X through p. If η p is diffeomorphic to S 1 , then the integral curve of X through p is periodic. If η p is diffeomorphic to R, then it is parametrized by the maximal integral curve of X through p, say ϕ : (T − , T + ) → M 4 with ϕ(0) = p and some −∞ ≤ T − < 0 < T + ≤ +∞. Assume now, by contradiction, that η + := ϕ([0, T + )) is contained in a compact subset K of M 4 . Then T + = +∞ and, since η p is an embedded submanifold, lim t→+∞ ϕ(t) =: p + exists in K. Elementary facts about ODEs imply that then X(p + ) = 0. But this is impossible because p + ∈ η p , given that η p is closed, and η p does not contain equilibria. Similarly for η − := ϕ((T − , 0]).
(ii.) Let η eq be the set of points of η p at which X vanishes. Thus η eq = η p ∩ (M sing is the image of the maximal integral curve of X through p. At variance from case i., however, now η * p is not closed. Thus, the integral curve through p either leaves every compact set or tends to an equilibrium point. We note that reduced motions may leave any compact set in M 4 in two ways: either the center of the ball goes to infinity or some components of the velocity go to infinity. The conservation of the moving energy, together with the 'Hamiltonization' of the reduced system which shows that it is a family of one-degree-of-freedom Hamiltonian (or Lagrangian) systems of mechanical type, (Proposition 9) will imply that the latter possibility can only take place with motions that tend to the vertex. Because of the singularity of the reduced space at the vertex, it seems to us that an investigation of motions asymptotic to them is more naturally performed on the SO(3)-reduced system in M 5 , and we leave it for a future work.
The knowledge of the reduced dynamics in M 4 gives some information on the properties of the motions of the unreduced system in M 8 . In particular, a rather complete description can be given for motions that project over equilibria and periodic orbits of the reduced system. Assume that a compact Lie group G acts freely on a manifoldM and thatX is a G-invariant vector field onM . Let π :M → M :=M /G be the quotient map and X the reduced vector field, which is π-related toX. The preimage under π of an equilibrium of X is called relative equilibrium of X and the preimage of a periodic orbit of X is called relative periodic orbit ofX. The work of [30,36] proves that for each relative equilibrium (resp. the relative periodic orbit) there exist an integer 0 ≤ k ≤ rank G (resp. 1 ≤ k ≤ 1 + rank G) and a vector ω ∈ R k such that the relative equilibrium (resp. relative periodic orbit) is fibered by X-invariant submanifolds diffeomorphic to T k , and the restriction of the flow ofX to each of these submanifolds is conjugate to the linear flow α → α + tω mod(2π) on T k . We say that the flow in the relative equilibrium or relative periodic orbit is quasi-periodic with k frequencies.
) is a union of relative equilibria in each of which the flow of the unreduced system is periodic (unless p 4 = 0 in which case the relative equilibrium consists of equilibria). ii. π −1 (E reg 4 ) is a union of relative equilibria in each of which the flow of the unreduced system is quasi-periodic with 0 ≤ k ≤ 2 frequencies.
iii. In every relative periodic orbit, the flow of the unreduced system is quasi-periodic with 1 ≤ k ≤ 3 frequencies.
Proof. In view of Propositions 4 and 5, in order to reach a complete picture of the dynamics of the (reduced or unreduced) system it is necessary to determine the reduced equilibria in E reg 4 , and the motions asymptotic to them, and the regions of the reduced space M reg 4 \ E reg 4 in which the (connected components of the) level sets of (E, J) are compact and those in which they are not. In the next section we make a first step in this direction, looking for situations in which all the level sets of (E, J) are compact and hence the reduced dynamics in the complement of the set of the reduced equilibria and of their stable and unstable sets is periodic, and the unreduced dynamics in the complement of the set of relative equilibria and of their stable and unstable sets is quasi-periodic.
Remarks: (i) The integrability by quadratures of the reduced system was proved in [15] by exploiting the existence of an invariant measure and of the two Routh integrals and applying the Euler-Jacobi theorem. However, this method cannot prove the periodicity of the reduced dynamics. (At best, after replacing one of the Routh integrals with the moving energy, it gives the weaker result that the reduced dynamics is, after a time reparametrization, linear on tori of dimension two).
(ii) For the dynamics in relative equilibria and relative periodic orbits in presence of a non compact symmetry group, which also is of interest in nonholonomic mechanics, see [3,26].
3.3 Coercive profiles and quasi-periodicity of the unreduced dynamics. The simplest case in which all the level sets of (E, J)| M4 are compact is when those of E| M4 are compact. Extending a result in [22] for the case Ω = 0 and for a convex profile, we give some conditions that ensure this fact. (Equivalently, lim p1→+∞ ψ(p 1 ) = +∞ in the first case and lim p1→+∞ ψ(p1) p1 = +∞ in the second). Proposition 7. The reduced moving energy E| M4 has all its level sets compact in any one of the following two cases: (H1) Ω = 0 and f is coercive.
Hence, in any level set L E of E, both P and Q are bounded from below and from above. It easily follows from this that, in L E , both p 1 and p 4 are bounded, so that 0 ≤ p 1 ≤ c 1 (E) and |p 4 | ≤ c 4 (E) for some positive c 1 (E) and c 4 (E). Since ( bounded as well in L E . Finally, from p 2 2 + p 2 3 = 4p 0 p 1 it follows that, in L E , p 2 and p 3 are bounded as well. Since the map J is continuous, under either of the two hypotheses of Proposition 7 the level sets of the map (E, J)| M4 are compact and, as already pointed out, the reduced dynamics is generically periodic and the unreduced dynamics is generically quasi-periodic on tori of dimensions up to three.
Remarks: (i) For Ω = 0, Proposition 7 was stated in [22] for convex profile functions, but a simple inspection to the proof shows that what is there used is only the coercivity of f , not its convexity.
(ii) When Ω = 0, the asymptotic superquadraticity of the profile function is likely to be not only sufficient but also necessary for the compactness of the level sets of E| M4 . Indeed, for p 2 = p 4 = 0 and large p 1 , E| M4 is approximately equal to γψ + p 2 3 4p1 − Ωp 3 + µΩ 2 p 1 and hence, if ψ goes to +∞ not faster than p 1 , to p 2 3 4p1 − Ωp 3 + µΩ 2 p 1 whose level sets are hyperbolas (recall that µ < 1). The level sets of the map (E, J)| M4 might nevertheless be compact. In fact, in Section 7 we will show that this happens for the parabolic profile f (r) = br 2 with b > 0; the same argument could be easily applied to the case of the conic profile f (r) = br with b > 0. A study of the compactness of the map (E, J)| M4 for a generic profile is difficult because the functions J 1 and J 2 are not explicitly known.
4 Hamiltonization of the reduced system 4.1 A rank-two Poisson structure. The system formed by a sphere that rolls without sliding on a surface of revolution which is at rest, namely our system for Ω = 0 and a convex profile, has been one of the first-if not even the very first-nonholonomic system with linear constraints and a symmetry group for which it has been shown that the reduced system is Hamiltonian with respect to a Poisson structure of rank two, with the reduced energy as Hamiltonian [15,39,22,7].
We show here that the same remains true when Ω = 0, but with the reduced moving energy, instead of the reduced energy, as Hamiltonian. This is of interest for two reasons: From a geometrical perspective, the very existence of Poisson structures for systems with affine (rather than linear) constraints was so far unknown, except in the very special case of the Veselova system [31]. And from a dynamical perspective, it helps enlightening some aspects of the dynamics of the reduced system, which turns that of a (family of) Hamiltonian systems with one degree of freedom which are of mechanical type (hence, also Lagrangian).
We limit ourselves to consider \ {p 1 = 0} of the vector field X = (X 0 , X 1 , . . . , X 4 ) in R 5 given by (9) becomes the vector field X • in M • 4 with components Similarly, the reduced moving energy (19) becomes the function E • : M • 4 → R given by The representative J • : M • 4 → R 2 of J| M reg 4 \{p1=0} has the same expression (20) as J, but we prefer using the symbol J • to stress that we are working in a subset of M reg 4 , and with a different parametrization.

Proposition 8. Consider the bivector
The characteristic distribution of the bivector Λ is spanned by the two vector fields ∂ p2 and ∂ p1 + (G 3 p 4 + Ωg 3 )∂ p3 + (G 4 p 3 + Ωg 4 )∂ p4 , which are everywhere linearly independent. Thus Λ has everywhere rank two and the associated Poisson brackets trivially satisfy the Jacobi identity, so that it is Poisson.
where e 1 = 1 0 and e 2 = 0 1 . Moreover, and, similarly, We point out that, for Ω = 0, the origin of the rank-two Poisson structure Λ is not clear. There are two possible approaches: 1. There exists an almost-Poisson formulation of nonholonomic mechanical systems with linear constraints and Lagrangian without gyrostatic terms [8,41]. In presence of symmetry-and under suitable hypotheses-this almost-Poisson structure induces a Poisson structure on the reduced space, that makes the reduced system Hamiltonian with the energy as Hamiltonian [6,4,32,5,7]. A similar theory for the case of affine constraints (or, equivalently, for Lagrangians with gyrostatic terms) does not exist yet. We speculate that such an extension might exist, particularly if the reduced moving energy is 'kinematically interpretable' in the sense of [20].
2. In [22], it is shown that every dynamical system with periodic flow possesses (infinitely many) rank-2 Poisson formulations, suggesting a dynamical origin of these structures. This point of view may account for the existence of Λ in the case of coercive profiles, but not in general. It is possible that the approach of [22] could be extended by using the existence of three first integrals, even if their level sets are not compact.
withp 3 andp 4 defined by (22), and is a submanifold of M • 4 diffeomorphic to R + ×R (p 1 , p 2 ). The Poisson structure Λ induces a symplectic form ω j on each symplectic leaf M j 2 , and the restriction of X • to M j 2 equals the vector field ω j dE • | M j 2 , namely, the ω j -Hamiltonian vector field whose Hamiltonian is the restriction of the reduced moving energy E • to M j 2 . If we use (p 1 , p 2 ) as coordinates on M j 2 , then ω j (p 1 , p 2 ) = 1 2p 1 F 2 dp 2 ∧ dp 1 and E • | M j 2 (p 1 , p 2 ) = 1 2 p 2 2 2p1F(p1) 2 + W j (p 1 ) with "effective potential" wherep 3,j andp 4,j stand forp 3 (·, j) andp 4 (·, j). If we pass to the (Darboux) coordinates (Q, P ) = p 1 , p2 2p1F 2 ∈ R + × R on M j 2 , then the symplectic 2-form ω j becomes dP ∧ dQ and E • | M j 2 becomes 1 2 2p 1 F 2 p 2 2 + W j (p 1 ). Thus, the restriction of the reduced system to each symplectic leaf can be regarded as a Hamiltonian system that describes a one-degree-of-freedom mechanical (holonomic) system on the cotangent bundle T * R + (Q, P ) of the configuration space R + Q = p 1 = r 2 /2. Equivalently, this can be regarded as a Lagrangian system on T R + (Q,Q) = (p 1 ,ṗ 1 ) with 'natural' Lagrangian 1 2ṗ 2 1 2p1F 2 − W j (p 1 ). To allow for easier interpretation, we prefer switching to the coordinates (r,ṙ). Correspondingly, we reverse to the original profile function f (r) and we use the two functions Proposition 9. The restriction of the reduced equations (8) to any level set M j 2 of the two reduced Routh integrals, written in coordinates (r,ṙ) ∈ T R + , is the Lagrangian system with Lagrangian with the effective potential 5 Reduced equilibria in E reg 4 .
Thus, for each r > 0 and Ω ∈ R there are one or two 1-parameter families of reduced equilibria with those r and Ω. Families RE1 and RE2 are parametrized by ω n ∈ R, while family RE3 is parametrized by v θ = 0. For fixed r and Ω, the curve ω n =ω n (r, v θ , Ω) in the plane (v θ , ω n ) has two branches, one in the half plane v θ > 0 and one in the half plane v θ < 0. When Ω = 0 these two branches are symmetrical with respect to the origin. The qualitative properties of these curves depend on the sign of f (r), and are shown in Figure 2 for Ω = 0; a nonzero Ω shifts both branches up or down, depending on the signs of Ω and of the quantity r f + 1 F and has no effect on them if the latter quantity vanishes. Curiously, if f (r) > 0 then there are exactly two reduced equilibria with ω n = 0. A more difficult question is which reduced equilibria are present for any given value of . This depends in a non obvious way on the profile of the surface Σ and on Ω, given that the map J • depends on them, and can be investigated, numerically if not analytically, on a case by case basis. The case of an upward half-cone was studied in [14]. The case of an upward paraboloid is studied in Section 7.
Remarks: (i) The reason why, when Ω = 0, we consider the reduced equilibria (r, 0, 0, ω n ) as part of the family RE1, instead of RE2, is because of their stability properties.
It follows from (14) that, if (r, 0, v θ , ω n ) is an equilibrium of the reduced system for a certain value of Ω, then (r, 0, −v θ , −ω n ) is an equilibrium of the reduced system for −Ω, and they have the same stability properties. (This can also be checked with (30) and with the formulas of Proposition 11). We may therefore restrict our study of the reduced equilibria to the case Ω ≥ 0.
When Ω = 0, the invariance of X under the reflection C as in (13) implies that if (r, 0, v θ , ω n ) is a reduced equilibrium then so is (r, 0, −v θ , −ω n ) and they have the same stability properties. Note that, by (21), if one of them belongs to M j 2 , then the other belongs to M −j 2 .
When Ω = 0 we may thus restrict ourselves to study reduced equilibria for j 1 ∈ R, j 2 ≥ 0.

Motions in relative equilibria.
Motions in all relative equilibria in M 8 consist of a uniform rotation of the center of mass of the ball on a parallel (hence, a horizontal circle) of the surface Σ, and of a uniform rotation of the ball around the axis normal to Σ (which changes periodically with the same frequency as the center of mass). See also Proposition 5.
By Proposition 10, there are three families of relative equilibria, which we call with the same names of the reduced equilibria onto which they project, and there is at least one such family on any parallel of Σ. For each Ω ∈ R: • Relative equilibria of type RE1 consist of motions in which the center of mass of the ball uniformly moves (if Ω = 0) or stands (if Ω = 0) on a horizontal 'critical' parallel of the surface Σ. At these points the normal vector n is vertical. Note that, since 1 2 < µ < 1, the angular velocity v θ = Ωµ of the center of mass is smaller than that of the surface. Thus, the ball either rolls (if Ω = 0) or stands (if Ω = 0) on the corresponding critical parallel of the surfaceΣ, and at the same time rotates around its vertical axis with any constant angular velocity ω z = ω n .
• In relative equilibria RE2, v θ = 0 and the center of mass of the ball stands still in space.
Correspondingly, the ball rolls uniformly on a critical parallel of the surfaceΣ. Here too, the ball may rotate with any constant angular velocity ω n = ω z around its vertical axis. • In relative equilibria of type RE3 the ball rolls along a non-critical parallel of the surfaceΣ, with any nonzero v θ .
Example. The case of a ball on a plane (ψ = 0) is well known and elementary [19,37]. The equations of motion for the SO(3)-reduced system in M 5 (x,ẋ, ω z ) areẍ = −µΩẏ,ÿ = µΩẋ, ω z = 0 (Equations (5.44) in [37]). ω z = ω n is constant. If Ω = 0 the center of mass moves on a straight line or stands still. For Ω = 0 the solution with initial conditions (x 0 , y 0 ,ẋ 0 ,ẏ 0 ) is and the center of mass moves along a circle. According to Proposition 10 the S 1 -reduction to M 4 of this system in M 5 has two families of reduced equilibria at any distance r from the origin, one of type RE1 and one of type RE2. The lift to M 5 of the reduced equilibria of type RE1 are motions withẋ 0 = −µΩy 0 ,ẏ 0 = µΩx 0 with nonzero (x 0 , y 0 ): the ball spins with any ω z around its center of mass, that moves along a circle centered at the origin. The lift to M 5 of the reduced equilibria of type RE2 are motions with initial conditionsẋ 0 =ẏ 0 = 0: the ball spins with any ω z around its center of mass, that stands still in space.
Remarks: (i) Relative equilibria of type RE2 resemble certain motions of a ball on a rotating umbrella produced in the Japanese 'turning umbrella' (kasamawashi) art. In some of these performances, an umbrella is kept in uniform rotation about its inclined axis, and a ball rolls on its surface in such a way to remain fixed in space. At each instant, the ball touches a point of the umbrella whose tangent plane is horizontal. The difference with our treatment is that, due to the inclination of the umbrella, that system is not invariant under rotation about the vertical. We will come back on this system in a future work.
(ii) In view of the example of the ball on the rotating plane, the existence of the reduced equilibria of types RE1 and RE2 can be regarded as obvious. However, the stability of these equilibria depends on the surface profile, see next Section. 6 (Leafwise) stability of the reduced equilibria 6.1 Leafwise-stability. We study now the stability of the reduced equilibria-where 'stability' is relative to the restriction of the reduced system to a level set M j 2 of the map J • . In order to avoid ambiguities on this point, we introduce the following terminology: We say that an equilibrium of the reduced system is leafwise-stable (leafwise-unstable) if it is a Lyapunov-stable (Lyapunov unstable) equilibrium of the restriction of the reduced system to the level set M j 2 of the map J • to which it belongs. ('Leafwise' refers, of course, to the symplectic leaves of the Poisson structure of M reg 4 ). Leafwise-stability of a reduced equilibrium does not imply its stability as equilibrium of the reduced system in M reg 4 , because motions nearby might run away with small but nonzero v θ . However, it implies the SO(3) × SO(2)-orbital stability of the motion in the corresponding relative equilibria of the unreduced system.
6.2 Leafwise-stability of RE1 reduced equilibria. The properties of leafwise-stability of the reduced equilibria of type RE1 are read without any difficulty from the expression of the function S 1 as in (32). Assume f (r) = 0.
Note that the properties of leafwise-stability of reduced equilibria of type RE1 are independent of the angular velocity ω n = ω z of the ball.
Thus, the rotation of the surface has a stabilizing effect also on the reduced equilibria of type RE2: they all become leafwise-stable for Ω → +∞.
The regions of leafwise-stability and leafwise-instability of these reduced equilibria in the halfplane (Ω, ω n ) ∈ R + × R are depicted in Figure 3 for the cases in which f (r) = 0. Note that, in these cases, the stability properties depend also on the angular velocity ω n = ω z with which the ball rotates about its vertical axis.  6.4 Leafwise-stability of RE3 reduced equilibria. Reduced equilibria of type RE3 exhibit more complex bifurcation scenarios than those of types RE1 and RE2. As above, we may assume Ω ≥ 0. First we note that, for large Ω, the surface rotation may have either a stabilizing or a destabilizing effect on these reduced equilibria, depending on the direction in which the ball moves along the surface's parallel, or even (in non-generic but nontrivial cases) no effect at all: Proposition 13. Consider r > 0 such that f (r) = 0 and v θ = 0.
Next, we investigate the leafwise-stability and instability of the reduced equilibria of type RE3 with given r, as a function of Ω and v θ . Recall that for given r and Ω there are two branches of these equilibria in the plane (v θ , ω n ), one with v θ > 0 and one with v θ < 0, which are given by (29) and are shown in Figure 2.
(In some of the computations below we prefer using (36) and (37)  Proof. (i.) Fix Ω and r. If ∆ 11 (r) = 0 then S 3 (r, v θ , Ω) = ∆ 0 (r, v θ ) is an even polynomial in v θ and we may study it only for v θ > 0. Since it has degree four, it has at most two positive roots.
And if it has two positive roots, none of them is an extremal point. If ∆ 11 (r) = 0, then the zeroes of S 3 (r, v θ , Ω) are the values of v θ at whichΩ(r, v θ ) = Ω. This is an odd function of v θ , and again we may study it only for v θ > 0. The positive zeroes ofΩ(r, v θ ) are the positive roots of the even polynomial of degree four v θΩ (r, v θ ). Hence, they are at most two and v θ →Ω(r, v θ ) can have at most one extremal point on the positive axis. It follows that, for v θ > 0, its graph intersects in at most two points any horizontal line. And if there are two intersections, none of them is at an extrmal of v θ →Ω(r, v θ , Ω). (ii.) For small |v θ |, the sign of S 3 (r, v θ , Ω) is the same as that of ∆ 00 (r). (iii.) This follows from items ii. and iii. of Proposition 13.
We detail now a few situations, not with the purpose of being exhaustive (which would require too many cases and subcases, and can be done on a case by case basis) but with that of covering a few typical situations and disclosing some general patterns. In particular, we neglect almost all nongeneric cases. We defineΩ Case 1: f (r) > 0, f (r) > 0. The three coefficients of the polynomial ∆ 0 (r, v θ ) are all positive (recall that µ < 1). Thus ∆ 0 (r, v θ ) > 0 for all v θ = 0 and it follows that all RE3 reduced equilibria with this r are leafwise-stable when Ω = 0. This has already been proved by [35].
If ∆ 11 (r) < 0, thenΩ has the opposite sign of that of the case ∆ 11 (r) > 0; the resulting situation is depicted in Figures 4.b and 4.d. We now study a few other cases. The analysis is similar to that of Case 1, and we may limit ourselves to a few comments-mostly, to draw the graph of the functionΩ. Instead of plotting the bifurcation diagrams in the plane (ω n , v θ ) we may describe them by specifying the type and the order (left to right) of the components of leafwise-stability ("S") and of leafwise-instability ("U ") in each branch v θ < 0 and v θ > 0. We write the resulting strings between brackets, with a comma that separates the branch v θ < 0 (first) from the branch v θ > 0. Thus, for instance, the bifurcation diagrams of Case 2: f (r) > 0, f (r) < 0. In this case ∆ 00 (r) and ∆ 11 (r) are negative, ∆ 02 (r) is positive and ∆ 04 (r) may have any sign. If ∆ 04 (r) > 0 then the graph ofΩ(r, v θ ) is as in Figure 5.a and the bifurcation diagram is of type (SU, U S). If ∆ 04 (r) < 0 there are two (generic) cases, depending on the sign of the discriminant D(r) := ∆ 02 (r) 2 − 4|∆ 00 (r)∆ 04 (r)| .
If D(r) > 0 then the graph ofΩ(r, v θ ) is as in Figure 5.b, withΩ m (r) finite and positive, and the (generic) bifurcation diagrams are of type (U SU, U SU ) if 0 ≤ Ω <Ω m (r) and of type (U SU, U ) if 0 >Ω m (r). The graph ofΩ(r, v θ ) when D(r) < 0 is as in Figure 5.c and the (generic) bifurcation diagrams are of type (U, U ) if 0 ≤ Ω <Ω m (r) and of type ( Case 3: f (r) > 0, f (r) = 0. This case is nongeneric, but it is worth mentioning because it is the case of a cone, for which the existence and stability of reduced equilibria has been investigated in [14].
The bifurcation diagram is of type (S, S) for Ω = 0 and of type (S, SU ) for Ω > 0.
• If ∆ 11 (r) > 0 and D(r) > 0 then the graph ofΩ is as in Figure 5. Other cases can be studied similarly.
7 Example: the ball on an upward paraboloid 7.1 The parabolic surface. We investigate now some aspects of the dynamics for the parabolic profile with a constant b > 0. This has two purposes. One is to prove that, even if the profile is not superquadratic, all motions which do not pass through the vertex are bounded, and hence generically quasi-periodic, even for Ω = 0 (Proposition 15; this had been previously proven only for small values of |Ω|, see [27]). The other is to investigate, numerically, the presence and number of (particulalry leafwise-unstable) reduced equilibria on each level set of the map J. We give all expressions in polar coordinates. Note that 7.2 Reduced equilibria. The system has only reduced equilibria of type RE3, with The two branches they form are independent of r if Ω = 0, but for Ω > 0 they are shifted below by an amount which decreases with r and varies between Ω(1 + 1 b ) and Ω b . All these reduced equilibria pertain to case 1 of Section 6.4, with ∆ 11 = −γµb 2 r 3 < 0. For Ω = 0 they are all leafwise-stable and for Ω > 0 all those with negative v θ are leafwise-stable. We thus focus on the reduced equilibria with Ω > 0 and v θ > 0.
For v θ > 0, the possible situations are those of Figures 4.b and 4.d. The functionΩ is given bỹ and some of its level curves (with values increasing from top to bottom) are shown in Figure 7 for three different values of the parameter b. For eachr > 0, the level curveΩ(r, v θ ) =Ω m (r) is the one tangent to the (horizontal) line r =r. Therefore, the function r →Ω m (r) is a strictly decreasing function which tends to +∞ for r → 0 and to 0 for r → +∞ and its graph resembles that of a branch of a hyperbola. Its inverse Ω →r m (Ω), which gives the r-coordinate of the minimum of the level curves ofΩ, has these same properties. We stress that, for each Ω > 0,r m (Ω) > 0. This provides the following picture for the stability of the reduced equilibria P 3 (r, v θ , Ω) with v θ > 0. For each Ω > 0, they are all leafwise-stable if r <r m (Ω). For r >r m (Ω) there are the three intervals S-U-S of values of v θ as in Figure 4.d. As r increases, the first "S" interval, the one closest to v θ = 0, becomes extremely narrow while the amplitude of the middle "U" interval reaches a maximum and then goes (slowly) to zero as r → +∞. We stress that all reduced equilibria become stable for v θ large enough.
Concerning the dependence on the parameters, Figure 6 indicates that as b increases, namely, as the paraboloid becomes steeper, the amplitude of the U-interval decreases at small r but increases at large r. We mention that increasing γ expands the instability region at all r, while increasing µ expands it at small r and seems to have little effect at large r.

7.3
The J-restricted reduced systems. In order to understand the dynamics we investigate now the J • -restricted systems. For the parabolic profile the integration of the differential equations that give the two first integrals J 1 and J 2 can be done explicitly. Expressed as functions of r instead of p 1 , the solutions of equations (17) and (18) are The effective potential V j is given by (28) with see (22).
Proof. Since p 3 (0, j) = j 1 and p 4 (0, j) = j 2 , for r → 0 + the function V j is asymptotic to For r → +∞, c(r) and s(r) are both asymptotic to r √ µ/2 and the same is true for the matrix U (r). Instead, u(r) is asymptotic to r 2 . Thus, both p 3j and p 4j are asymptotic to r 2 . This implies that, if Ω > 0, then V j is asymptotic for r → +∞ to 1 2 µp 4j (r) 2 and hence to r 4 . This implies that, for all j 1 = 0, the dynamics of the reduced system is periodic (and hence that of the unreduced one is quasi-periodic) except for the equilibria and the motions asymptotic to and from the unstable ones. We do not investigate here motions in the level set j 1 = 0 because it contains the vertex.

7.4
The equilibria of the J-restricted reduced systems. Proposition 15 implies that, for any j 1 = 0 and Ω ≥ 0, the effective potential has at least one minimum, and hence the restriction of the reduced system to M j 2 has at least one stable equilibrium. In fact, since V j is a real analytic nonconstant function, its minima are all isolated and, since the system has one degree of freedom, they are the only stable equilibrium configurations. Generically, there is obviously an odd number of equilibria on each M 2 j , but their exact number-and the numbers of the stable and unstable ones-is of special interest because gives global information on the dynamics in M j 2 . We already know that, when Ω = 0, all reduced equilibria are leafwise-stable. This implies that, for each j = 0, V j has a single critical point, which is a minimum, and the reduced system in M j 2 has only one equilibrium. Figure 8.a shows, for a typical choice of the values of the parameters µ, b, j, the value of the r-coordinate of the reduced equilibrium on M 2 j as a function of j = (j 1 , j 2 ). This is a single valued surface. At fixed j 2 , the r-coordinate of the reduced equilibrium tends to a constant value when |j 1 | → ∞ and there is a single maximum of r, which goes to +∞ when |j 2 | → ∞. Not surprisingly, when j 1 → 0 the coordinate the reduced equilibrium tends to the vertex (r → 0). Note the symmetry of the surface S under reflections of (j 1 , j 2 ).
In order to determine the number of unstable equilibria for Ω = 0 we resorted to a numerical analysis, whose results are illustrated by Figures 7.b-d. As soon as Ω = 0, two (or exceptionally, at the bifurcations, one) other reduced equilibria are created for j = (j 1 , j 2 ) in about half of the (j 1 , j 2 )-plane, one of which is leafwise-unstable and the other (if present) is leafwise-stable. The figures show the equilibria surface for different values of Ω and in different ranges of j 1 , j 2 . Even though the figures cannot show it clearly, the shape of the surface is similar for all values of Ω but, as one sees observing that the figures have different scales, as Ω → 0 the two additional equilibria go to infinity in r and/or v θ . We have provided a general analysis of the dynamics of a heavy dynamically symmetric ball that rolls without sliding on a uniformly rotating surface of revolution. Even though this study has clarified a number of aspects of this class of systems, some questions remain open.
1. The possibility and the properties of motions through-or asymptotic to-the vertex have not been studied. The possibility of motions in which the point of contact tends to the vertex and (some component of) the angular velocity grows unbounded is not ruled out by our analysis and should be invetigated. One natural possibility is to analyse these motions in the five-dimensional SO(3)-reduced system.
2. When Ω = 0, we have only proven the boundedness of motions under the hypothesis that the profile of the surface goes superquadratically to +∞ at infinity. We have proven this fact using the compactness of the level sets of the moving energy. However, as pointed out in section 3.3, what is necessary is the compactness of all the level sets of the map (E, J) which, as the example of the (upward) paraboloid of section 7 shows, might be satisfied under the assumption alone of coercitivity of the profile. A general study of this question might require a careful analysis of the asymptotic properties of the functions J 1 and J 2 defined by the differential equations (15). 3.
When Ω = 0, if the profile goes asymptotically to −∞, or to a constant, then there are certainly unbounded motions, in which the ball goes to infinity. Even though some particular statements are made by Routh in [40], a characterization of the initial conditions which lead to bounded or unbounded motions is essentially missing.
4. In connection with point 3., we remark that the example of the ball that rolls on a horizontal plane suggests that the rotation of the surface may have a 'stabilizing' effect on the dynamics. In fact, in all motions but the equilibria the ball runs away to infinity if Ω = 0, but as soon as Ω = 0 the ball moves on circles! Preliminary investigations show that such a stabilizing effect of the rotation is present in other profiles, e.g. in the downward paraboloid and cone, and we conjecture that, as soon as Ω = 0, all motions in any profile are bounded (with the possible exception of those asymptotic to the vertex).
5. Also the (local and global) structure of the foliation by the invariant tori (in integrable cases) is still not studied. This study would require some comprehension of the frequencies of motions. Some results on this, for the case of a corcive profile and Ω = 0 are given by [35].
9 Appendix: The equations of motion 9.1 The nonholonomic equations of motion in quasi-velocities with the reaction forces. The equations of motion of mechanical systems subject to nonholonomic constraints can be written in several ways. Particularly when the configuration space involves a Lie group it is customary to employ a technique originally developed by Poincaré for holonomic systems [38], which is based on the use of coordinates and quasi-velocities-namely linear combinations of the velocities. For instance, for rigid bodies this allows to use the components of the angular velocity (with respect to a fixed or moving frame) instead of the velocities of the Euler angles or other local coordinates on SO(3). The nonholonomic case was first considered by Hamel [34].
However, in Hamel's approach the quasi-velocities are chosen so that the nonholonomic constraint is given as zero of some of them. This leads to a set of equations on the constraint manifold-Hamel equations-in which the reaction forces are not explicitly identified. In our opinion, instead, the explicit consideration of the reaction forces is under several respects important, e.g. in determining the conservation of momenta and energy [27,28,20].
We thus derive here a form of the equations of motion of nonholonomic systems that employs quasi-velocities and contains, in an explicit way, the reaction forces. Specifically, we write these equations as the restriction to the nonholonomic constraint manifold M ⊂ T Q of a set of equations in the tangent bundle of the configuration manifold Q that leave M invariant (namely, as a vector field which is tangent to M ). This is a generalization of an analogous form of the equations that uses Lagrangian coordinates and velocities, which is our starting point and for which we refer to [27].
We consider a nonholonomic system (Q, L, M) with an n-dimensional configuration manifold Q, a mechanical Lagrangian L : T Q → R, and an affine distribution M on Q with constant rank that describes the nonholonomic constraint. More specifically: i. By a mechanical Lagrangian we mean a function of the form L = L 2 + L 1 + L 0 , where L 2 is a Riemannian metric on Q, L 1 is a function whose restriction to each fiber of T Q is linear, and L 0 is a basic function, hence constant on the fibers of T Q. ii. We write the affine distribution as M = ξ + D, with D a non-integrable distribution on Q of constant rank r, 1 < r < n, and ξ a vector field on Q. Clearly, the vector field ξ is defined up to a section of D. We denote by M the (n + r)-dimensional subbundle of T Q formed by the fibers of M. iii. Lastly, we assume that the nonholonomic constraint is ideal, namely, that the reaction forces it exerts satisfy d'Alembert principle, see [2,27] for details. It is well known that, under these hypotheses, there is a unique function R L,M : M → D • with the property that the restriction to M of Lagrange equations with the reaction forces, defines a vector field on M , and hence a dynamical system on M [2,27]. Here, [L] is the usual Euler-Lagrange operator. The expression of these equations using lifted coordinates (q,q) in T Q is given in [27], and can be recovered as a particular case of the present treatment. The local representative L of the Lagrangian L has the form L = L 2 + L 1 + L 0 with L 0 independent of the v's, L 1 linear in the v's, and L 2 (q, v) = 1 2 v · A(q)v with A = ∂ 2 L ∂v∂v a positive definite matrix that depends only on q. The fibers of the distribution D based in Q U can be represented as the kernel of a q-dependent (n − r) × n matrix S(q) of rank n − r: the fiber of M based at the point of Q U of coordinates q is given by the equation where q → s(q) ∈ R n is a a smooth map that depends on the vector field ξ (specifically, s(q) = −S(q)ξ loc (q) if ξ| Q U = i ξ loc i ∂ qi ). The image of M ∩ T Q U under the coordinate map (q, v) is the (n + r)-dimensional submanifold M U := (q, v) ∈ U × R n : S(q)v + s(q) = 0 of U × R n .
Define now maps : U × R n → R n , σ : U × R n → R n−r and R : U × R n → R n as follows: (q, v) has components 4 i = where γ ijh = B −T ik ∂B T kj ∂q l − ∂B T lj ∂q k B −1 lh are the so-called "transpositional symbols", σ(q, v) ∈ R k has components σ a = ∂S ai ∂q j v i + ∂s a ∂q j B −1 jh v h , a = 1, . . . , n − r , and Proposition 16. The representative of equation (40) in the coordinates (q, v) is the restriction to M U of the equationq = B(q) −1 v , in U × R n .

9.2
The (reduced) equations of motion of our system. We write now the equations of motion of the ball on the rotating surface of revolution considered in this paper and of its SO(3) × SO(2)-reduction.