The interconnection of sliding and rolling in the problem of the motion of a homogeneous sphere on a rough horizontal plane

The motion of a homogeneous sphere on a rough horizontal plane when the angular velocities of the twisting and spinning of the sphere are equal to zero at the initial instant is considered. It is proved that, for any initial conditions, the angular velocity of the rolling of the sphere and the sliding velocity vanish after the same finite time. It is shown that the sliding and rolling are interconnected and, in particular, that the rolling of a sphere without sliding is impossible.     

Hypographs satisfying an external sphere condition and the regularity of the minimum time function

We prove that if the hypograph of a continuous function f admits at every boundary point a supporting ball then it has “essentially” positive reach, i.e. the hypograph of the restriction of f outside a closed set of zero measure has (locally) positive reach. Hence such a function enjoys some properties of a concave function, in particular a.e. twice differentiability. We apply this result to a minimum time problem in the case of a nonlinear smooth dynamics and a target satisfying internal sphere condition.     

Time-optimal steering of a point mass onto the surface of a sphere at zero velocity

The problem of the time-optimal steering of a point mass onto the surface of a sphere at zero velocity, by a control force of bounded magnitude is investigated. It is assumed that the surface is penetrable and that the point may “land” on the sphere either from the outside or from the inside. An optimal control, in the open-loop and feedback form of trajectories the optimal time and the Bellman function are constructed using Pontrya'gin's maximum principle. The multidimensional boundary-value problem is reduced, by introducing self-similar variables, to the numerical solution of an algebraic equation of degree four and a transcendental equation. It is shown that the boundary-value problem degenerates when the optimal trajectory is nearly linear; a solution of the synthesis problem is constructed in the degenerate case. The efficacy of the approach proposed here is illustrated by specific examples in which families of trajectories are computed, and by an analysis of control regimes.     

An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.     

A Nonholonomic Model of the Paul Trap

In this paper, equations of motion for the problem of a ball rolling without slipping on a rotating hyperbolic paraboloid are obtained. Integrals of motions and an invariant measure are found. A detailed linear stability analysis of the ball’s rotations at the saddle point of the hyperbolic paraboloid is made. A three-dimensional Poincaré map generated by the phase flow of the problem is numerically investigated and the existence of a region of bounded trajectories in a neighborhood of the saddle point of the paraboloid is demonstrated. It is shown that a similar problem of a ball rolling on a rotating paraboloid, considered within the framework of the rubber model, can be reduced to a Hamiltonian system which includes the Brower problem as a particular case.     

Asymptotics of Maxwell Time in the Plate-Ball Problem

The problem on rolling of a sphere on a plane without slipping or twisting is considered. One should roll the sphere from one contact configuration to another so that the length of the curve traced by the contact point in the plane is the shortest possible. The asymptotics of Maxwell time for rolling of the sphere along small amplitude sinusoids is studied. A two-sided estimate for this asymptotics is obtained.     

The problem of drift and recurrence for the rolling Chaplygin ball

We investigate the motion of the point of contact (absolute dynamics) in the integrable problem of the Chaplygin ball rolling on a plane. Although the velocity of the point of contact is a given vector function of variables of the reduced system, it is impossible to apply standard methods of the theory of integrable Hamiltonian systems due to the absence of an appropriate conformally Hamiltonian representation for an unreduced system. For a complete analysis we apply the standard analytical approach, due to Bohl and Weyl, and develop topological methods of investigation. In this way we obtain conditions for boundedness and unboundedness of the trajectories of the contact point.     

Closed-Loop Control of the Motion of a Sphere Rolling on a Moving Horizontal Plane

A uniform sphere is rolling without slipping on a horizontal plane. The motion of the sphere is controlled via the control of the acceleration of the plane. At the time t=0, the sphere and the plane are stationary and the center of the sphere is located at a point A in the plane. Given a time interval [0, t _f], the problem dealt with here is: Find a closed-loop strategy for the acceleration of the moving plane such that, at the time t=t _f, the plane and the sphere will be nearly at rest and the center of the sphere will be in a given neighborhood of the origin. By introducing the concept of path controllability, a closed-loop strategy for the solution of the above-mentioned problem is proposed and its efficiency is demonstrated by solving numerically some examples.     

Topological analysis of an integrable system related to the rolling of a ball on a sphere

A new integrable system describing the rolling of a rigid body with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velocity) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid body dynamics (for which no real separation of variables has been found either). In particular, a fixed point of focus type is possible for this system, which can serve as a topological obstacle to the real separation of variables.     

On slow motion of a sphere in a viscous liquid

Stokes’ and Seth’s solutions for the slow motion of a sphere in a viscous, incompressible liquid have been discussed from the viewpoint of the structure of the velocity field and its relation to the drag of the sphere. The problem is analysed from a different angle in this paper. It is believed that it throws more light on thephysics of the problem.     

Global existence and decay estimates of the Boltzmann equation with frictional force: The general results

In this paper, we give the existence theory and the optimal time convergence rates of the solutions to the Boltzmann equation with frictional force near a global Maxwellian. We generalize our previous results on the same problem for hard sphere model into both hard potential and soft potential case. The main method used in this paper is the classic energy method combined with some new time–velocity weight functions to control the large velocity growth in the nonlinear term for the case of interactions with hard potentials and to deal with the singularity of the cross-section at zero relative velocity for the soft potential case.     

The dynamics of the chaplygin ball with a fluid-filled cavity

We consider the problem of rolling of a ball with an ellipsoidal cavity filled with an ideal fluid, which executes a uniform vortex motion, on an absolutely rough plane. We point out the case of existence of an invariant measure and show that there is a particular case of integrability under conditions of axial symmetry.     

带有微弱章动的圆球在粗糙水平面上的运动

对于圆球在粗糙水平面上的运动,在文[1]中,作者忽略了章动,得到了近似解析解。本文在此基础上给出了有章动情况下的控制方程。通过求解这些方程,证明文[1]关于接触点速度的结论在有章动时仍然正确。还得到其它一些有趣的结果,例如:球心和接触点的速度与球的自转角速度和章动角速度有一定联系;球心和接触点的速度的方向具有不变性。在进一步假设微弱章动的情况下,文中得到近似解析解,从而证明文[1]结果的正确性。     

How to control the Chaplygin ball using rotors. II

In our earlier paper [3] we examined the problem of control of a balanced dynamically nonsymmetric sphere with rotors with no-slip condition at the point of contact. In this paper we investigate the controllability of a ball in the presence of friction. We also study the problem of the existence and stability of singular dissipation-free periodic solutions for a free ball in the presence of friction forces. The issues of constructive realization of the proposed algorithms are discussed.     

Extensions de fonctions d'un voisinage de la sphère à la boule

In this note, we are interested in the problem of extending a germ defined along the standard sphere and whose restriction to the sphere is Morse to a function F defined on the ball bounded by the sphere, without critical point. We give an algebraic necessary condition dealing with the Morse complexes of f with $\mathbb{Z}$ coefficients.     

Invariant measure in the problem of a symmetrical ball rolling on a surface

The problem of a symmetrical ball, rolling without slip on a fixed surface, is considered, along with some generalizations. It is proved that, under fairly general assumptions as to the nature of the external forces. the system admits of a smooth invariant measure.     

Rubber rolling over a sphere

“Rubber” coated bodies rolling over a surface satisfy a no-twist condition in addition to the no slip condition satisfied by “marble” coated bodies [1]. Rubber rolling has an interesting differential geometric appeal because the geodesic curvatures of the curves on the surfaces at corresponding points are equal. The associated distribution in the 5 dimensional configuration space has 2–3–5 growth (these distributions were first studied by Cartan; he showed that the maximal symmetries occurs for rubber rolling of spheres with 3:1 diameters ratio and materialize the exceptional group G ₂). The 2–3–5 nonholonomic geometries are classified in a companion paper [2] via Cartan’s equivalence method [3]. Rubber rolling of a convex body over a sphere defines a generalized Chaplygin system [4–8] with SO(3) symmetry group, total space Q = SO(3) × S ² and base S ², that can be reduced to an almost Hamiltonian system in T*S ² with a non-closed 2-form ω_NH. In this paper we present some basic results on the sphere-sphere problem: a dynamically asymmetric but balanced sphere of radius b (unequal moments of inertia I _j but with center of gravity at the geometric center), rubber rolling over another sphere of radius a. In this example ω_NH is conformally symplectic [9]: the reduced system becomes Hamiltonian after a coordinate dependent change of time. In particular there is an invariant measure, whose density is the determinant of the reduced Legendre transform, to the power p = 1/2(b/a − 1). Using sphero-conical coordinates we verify the result by Borisov and Mamaev [10] that the system is integrable for p = −1/2 (ball over a plane). They have found another integrable case [11] corresponding to p = −3/2 (rolling ball with twice the radius of a fixed internal ball). Strikingly, a different set of sphero-conical coordinates separates the Hamiltonian in this case. No other integrable cases with different I _j are known.      

The brachistochrone problem for a disc

The motion of a vertical disc along a curve under the influence of gravity is investigated. On the assumption of regular rolling without slip and separation of contact points, the problem of plotting the curve of most rapid motion of the disc centre from the origin of coordinates to an arbitrary fixed point of the lower half-plane is solved. As usual, the velocity at the initial instant of time is zero, and at the final instant of time it is not fixed. In explicit parametric form, the classical brachistochrone for contact points of the disc is plotted and investigated. The response time, trajectory and kinematic and dynamic characteristics of motion are calculated analytically. Previously unknown qualitative properties of regular rolling are established. In particular, it is shown that the disc centre moves along a cycloid connecting specified points. The envelopes of the boundary points of the disc, produced as its centre moves along the cycloid, are brachistochrones. The feasibility of mechanical coupling of the disc and the curve by reaction forces at the contact point (the normal pressure and dry friction) is investigated.     

Local Poincaré inequalities in non-negative curvature and finite dimension

In this paper, we derive a new set of Poincaré inequalities on the sphere, with respect to some Markov kernels parameterized by a point in the ball. When this point goes to the boundary, those Poincaré inequalities are shown to give the curvature-dimension inequality of the sphere, and when it is at the center they reduce to the usual Poincaré inequality. We then extend them to Riemannian manifolds, giving a sequence of inequalities which are equivalent to the curvature-dimension inequality, and interpolate between this inequality and the Poincaré inequality for the invariant measure. This inequality is optimal in the case of the spheres.     

A generalization of the Poisson integral formula for the functions harmonic and biharmonic in a ball

We construct an analytic solution to the problem of extension to the unit N-dimensional ball of the potential on its values on an interior sphere. The formula generalizes the conventional Poisson formula. Bavrin’s results obtained for the two-dimensional case by methods of function theory are transferred to the N-dimensional case (N ≥ 3). We also exhibit a solution to a similar extension problem for some operator expressions depending on a potential known on an interior sphere. A connection is established between solutions to the moment problem on a segment and on a semiaxis.