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3Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 3-17. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221205
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70E18, 70E40
The Integrable Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint
A. A.Kilin, T. B.Ivanova
This paper addresses the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that there is no slipping of the sphere as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. It is also assumed that, in the direction perpendicular to the above-mentioned one, the sphere can slip relative to the plane. Examples of realization of the above-mentioned nonholonomic constraint are given. Equations of motion are obtained and their first integrals are found. It is shown that the system under consideration admits a redundant set of first integrals, which makes it possible to perform reduction to a system with one degree of freedom.
Keywords: nonholonomic constraint, first integral, integrability, reduction
Introduction
This paper investigates the problem of a sphere with axisymmetric mass distribution (a spherical top) rolling with partial slipping on a horizontal plane.
The problem of the rolling motion of a spherical body is one of the basic problems of the dynamics of a rigid body and goes back to L.Euler [1], J. L. R. D'Alembert [2], S.Poisson [3], A. A. Cournot [4], E. J. Routh [5] and others. Various versions of this problem have been studied in many classical and modern works. Therefore, without claiming to give a complete review of the literature, we mention only the main (predominantly classical) works concerned with the
Received October 25, 2022 Accepted December 07, 2022
This work is supported by the Russian Science Foundation under grant 22-21-00797.
Alexander A. Kilin [email protected] Tatiana B. Ivanova [email protected]
Ural Mathematical Center, Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia
dynamics of spherical tops. A more detailed review of the relevant literature can be found, for example, in [6-11].
The main results concerning the motion of a rigid body on a smooth plane are linked to the names of S. D. Poisson [3], A. A. Cournot [4], V. Puiseux [12, 13], F. Klein [14], E. J. Routh [15] and others. As an arbitrary convex body moves on a smooth plane, two projections of the linear momentum onto the supporting plane, as well as the area integral — the projection of the angular momentum onto the vertical axis, are preserved. After reduction to the level set of these integrals, this problem generally reduces to a nonintegrable system with two degrees of freedom [16]. In the case of a spherical top, to these integrals one adds a Lagrange integral corresponding to the axial symmetry of the problem, and Jellett's integral [17] (which in this case linearly depends on the area integral and the Lagrange integral). In this case, Jellett's integral is preserved if an arbitrary force applied to the sphere at the point of contact is added. A generalization of Jellett's integral to the case of an additional torque can be found in [18, 19].
No less attention was given in the literature to the motion of spherical bodies rolling without slipping on an absolutely rough plane. Seminal contributions on this subject include the works of H. Hertz [20], E. Routh [5], D.K.Bobylev [21], S. A. Chaplygin [22], G.Hamel [23], J.Hadamard [24], and P. Appell [25]. In fact, these works laid down the foundations for development of the modern dynamics of systems with nonholonomic constraints.
In the case of a rigid body rolling without slipping on a plane, the equations describing the rotation of a body about its center of mass decouple, as in the case of a smooth plane, from the complete system. However, in the general case these equations admit no first integrals except for the energy integral. Due to this fact (and the absence of an invariant measure) such systems exhibit "nonconservative" behavior and even strange attractors and mixed dynamics [26, 27].
In the case of spherical bodies with an arbitrary mass distribution, the equations of motion admit an additional first integral quadratic in velocities [28]. However, even in this case the system shows dissipative effects which are a consequence of nonexistence of an invariant measure. In the case of dynamical symmetry the problem of a spherical body rolling on an absolutely rough plane (as in the case of a smooth plane) is integrable [5, 29]. In this case, the equations of motion admit, in addition to energy, two integrals linear in velocities: Jellett's integral and a generalization of the Lagrange integral, which is sometimes called the Chaplygin integral.
Another model used in describing the rolling motion of a rigid body on a surface is the rubber-rolling model. In this model, the condition of no slipping relative to the plane at the point of contact is supplemented with the condition that the projection of the angular velocity onto the normal to the surface be equal to zero. This model was first proposed by J. Hadamard [24] and developed further by H. Beghin [30]. For a modern account of these results, see [31]. Later, J.Koiller and K.Ehlers [32] suggested calling this model the rubber-rolling model as the one capturing most clearly the resistance to spinning. In recent publications, especially applied ones, concerned with robotics, this model is sometimes called the pure-rolling model.
In [26] it is shown that in the rubber-rolling model the dynamics of a spherical body can exhibit effects such as a strange attractor. A systematic account of results on the rubber rolling of a rigid body on a plane and a sphere can be found in [33-35]. As shown in [33], the problem of a spherical top (and, generally speaking, that of an arbitrary body of revolution) considered within the framework of the rubber-rolling model is integrable. For this problem, an invariant measure and an additional integral that is a generalization of the Lagrange integral have been found.
When the dynamics of a rigid body rolling on a rough plane is considered or when the rubber-rolling model is used, two or three nonholonomic constraints, respectively, are imposed
on the system. In this paper, we examine another model in which we impose on the system only one nonholonomic constraint which corresponds to the absence of slipping in the direction of the projection of the symmetry axis of the top onto the plane. A constraint of such kind is usually imposed in the case where bodies with a sharp edge move on ice. The best-known system with such a constraint is the Chaplygin sleigh [36, 37] and a disk with a sharp edge on ice [38]. We note that similar restrictions (the no-slip constraint in some specified direction) also arise in problems concerning the rolling motion of wheeled vehicles, both those having usual wheels [39-42] and those having omniwheels [43, 44].
Another interesting example in which only one nonholonomic constraint takes place is a system of two coupled bodies, one of which is a spherical shell and the other is (partially) fastened inside the first body. One of the first works in this direction was evidently that of S. A. Chaplygin [45]. In this work S. A. Chaplygin proved integrability of two problems of a spherical shell rolling without slipping and having a second body inside it. In the first problem, a homogeneous sphere rolls inside the shell, and in the second, a spherical pendulum is fastened at its center. Possible generalizations and special cases of the latter problem arise in considering the dynamics of spherical robots controlled by an internal pendulum mechanism [46-53]. Related problems concerning the dynamics of a rigid body with a spherical support can be found in [54-57].
An interesting fact pointed out in [58] is that the dynamics of a spherical pendulum fastened inside a spherical shell rolling on an absolutely rough plane is described by equations of motion of a rigid spherical body on a smooth plane. Similarly to [58], it can be shown that, if an axisymmetric body is fastened inside a weightless spherical shell in such a way that it can rotate freely about its symmetry axis, then the motion of the internal body is subject to one nonholonomic constraint described above.
This paper is structured as follows. In Section 1 we formulate the model under consideration and give examples of its implementation. In Section 2 we derive equations of motion of the system considered. In Section 3 we find all first integrals of the system and successively perform reduction to a system with one degree of freedom.
1. Formulation of the problem and realization of a nonholonomic constraint
In this paper we address the problem of the rolling motion of a heavy unbalanced sphere of radius R and mass m with axisymmetric mass distribution on a plane.
We formulate the main assumptions under which we will examine the problem at hand:
- the contact between the plane and the sphere is a point contact, and the interaction between them results only in forces applied to the point of contact.
- as the sphere rolls, it does not lose contact with the plane.
- the sphere does not slip as it rolls in the direction of the projection of its symmetry axis onto the plane. Also, the sphere can slip in the direction perpendicular to the above-mentioned one.
The restrictions formulated above imply that only one nonholonomic constraint is imposed on the sphere. This distinguishes it from the model of rolling without slipping in which two nonholonomic constraints are simultaneously imposed on the body (the horizontal projection of the velocity of the point of contact is zero).
Although the condition of partial slipping is seemingly artificial, it turns out that it can be fulfilled (with some accuracy) in a real mechanical system.
One way to fulfil this condition is to use a roller-bearing sphere (see Fig. 1). Such a sphere is a generalization of the omniwheel for which also only one nonholonomic constraint [43] is imposed on the system. Indeed, let us arrange weightless rollers on the surface of the sphere as follows:
- the axis of each roller is directed along a tangent to the large circle (meridian) passing through the axis of dynamical symmetry of the sphere and the point where the roller is located;
- the profiles of the rollers have such a form that the external edge of the section formed by the intersection of the rollers lying on the same meridian with the meridian plane forms a circle of radius R. In other words, each meridian of the sphere is a classical omniwheel (see Fig. 1b).
(a) (b)
Fig. 1. The roller-bearing sphere (a) and the section formed by its intersection with the meridian plane (b)
Such a sphere rolls along the meridian without slipping. In the direction perpendicular to the axis of the roller, the sphere can slide freely, which is made possible by rotation of the roller.
Another more realistic way to fulfil the above-mentioned constraint is the motion of a weightless spherical shell rolling without slipping on a plane and containing a heavy axisymmetric body (rotor) that rotates freely about its symmetry axis rigidly attached to the shell (see Fig. 2). It is easy to show that in this case the required restriction is imposed on the motion of the internal body and is interpreted as the absence of slipping during rolling along the meridian.
Similar systems of coupled bodies are considered in the paper [59], which is concerned with the dynamics of two coupled bodies: a dynamically symmetric shell and a body that is fastened inside it and rotates about some axis fixed inside the shell with a constant angular velocity.
We also mention the paper [57], which addresses the problem of a system of two coupled bodies that rolls without slipping on a plane and consists of a spherical shell and a rigid body fastened at its center to a spherical joint. In that paper it was shown that the dynamics of an internal body is similar to that of a spherical top with the same mass distribution which rolls on an absolutely smooth plane. The constraint in the form of two coupled bodies fulfilled in the way presented by us is a generalization of this analogy to the case of one nonholonomic constraint.
A third possible way to fulfil the constraint is the motion of a sphere rolling on ice and consisting of sharp skates in the form of disks threaded onto the axis of dynamical symmetry (see Fig. 3). In the classical approach, the skate on ice cannot move in the direction perpendicular
slip/ roll
Fig. 2. An example of fulfilment of a constraint Fig. 3. A sphere consisting of disks
in the form of a spherical shell with an axisym- (skates) moving on an icy plane
metric rotor inside
to its plane. Therefore, the sphere consisting of disks (skates) can slide along the plane of the disk in contact with the supporting plane at the current instant. In this case, a roll-over of the sphere from one disk to another without slipping occurs in the direction of the meridian.
2. Equations of motion
2.1. Configuration space and kinematic relations
To describe the motion of the sphere, we introduce three coordinate systems:
- a fixed (inertial) coordinate system Oxyz with origin at some point of the supporting plane and with the axis Oz directed vertically upward. Let us denote the unit vectors of this coordinate system by the vectors a, 3 and 7 (see Fig. 4);
- a moving coordinate system ox1 x2x3 rigidly attached to the sphere, with origin at the center of mass of the system and with the axes directed along the principal axes of inertia of the body. We denote the unit vectors of this coordinate system by the vectors e1, e2 and e3 (see Fig. 4);
- a half-moving coordinate system £2x3 with origin at the geometrical center of the sphere. The axis C^2 is parallel to the projection of the unit vector of the vertical 7 onto the plane Ox1x2, and the axis Csupplements the axes C^2 and Cx3 to the right triplet. We denote the unit vectors of this coordinate system by the vectors t, n and e3.
In the case of axisymmetric mass distribution under consideration, one of the axes (for example, Ox3, see Fig. 4) coincides with the symmetry axis of the sphere. Then the vector of displacement of the center of mass has the form a = (0, 0, a3), and the matrix of the principal moments of inertia is I = diag(J1, I1, I3).
Remark 1. Here and in what follows, we denote the vectors by the letters written in bold italics, a, 3, F, M, ..., and write their scalar and vector product, respectively, in the form (a, 3) and a x b. The sign 0 denotes the tensor product, i.e., (a 0 b) = aibj. The letters written in boldface Roman font denote the matrices: A, B, ....
Let a, 3 and 7 be the unit vectors of the fixed coordinate system Oxyz that are referred to the axes of the moving coordinate system ox1x2x3, and let rc = (xc, yc) be the coordinates of the point of contact on the plane in the fixed coordinate system Oxyz.
Fig. 4. Spherical top on a plane
Then the orthogonal matrix Q, whose columns are projections of the vectors a, 3 and 7 onto the axes of the moving coordinate system ox1x2x3,
Q
(a-i ft- Yi\ a2 Y2
Vas A3 Y3)
e SO(3),
and the vector rc form a pair (rc, Q) e R2 x SO(3) which uniquely defines the position of the body.
Remark 2. By definition, the vectors a, ( and 7 satisfy the relations
(a, a) = 1, (3, 3) = 1, (y, y) = 1,
(a, 3) = 0, (3, Y) = 0, (Y, a) = 0.
Thus, the set of vectors a, 3 and 7 is a redundant set of coordinates.
Let u = (w1, u2, w3) and v = (v1, v2, v3) be the angular velocity and the velocity of the center of mass of the body, respectively, referred to the axes of the moving coordinate system ox1 x2x3.1 Then the evolution of the orientation and position of the sphere is given by the kinematic relations
ex = a x u,
x = (v, a),
33 = 3 x u,
y = (v, 3),
77 = 7 x u, z = (v, 7),
(2.1)
where x, y and z are the coordinates of the center of mass o of the sphere in the fixed coordinate system Oxyz.
2.2. Constraint equations
The condition of motion without loss of contact with the plane and the condition of partial nonslipping relative to the plane lead to imposition of two constraints on the system.
The first constraint corresponds to the condition that the motion occurs without loss of contact of the sphere with the plane; also, it is holonomic and has the form
Z + (r, 7) = 0,
(2.2)
1 Here and in what follows (unless otherwise specified) all vectors are referred to the moving axes.
where r is the radius vector of the point of contact, which can be represented as
r = -Rj - a. (2.3)
The time derivative of the constraint (2.2), with the last equation of (2.1) taken into account, leads to the following restriction on the velocity:
¡1 = (%, Y) = (v + w x r, y) = 0, (2.4)
where vp = v + w x r is the velocity of the sphere at the point of contact with the plane.
The condition that there be no slipping in the direction of the projection of the axis of dynamical symmetry of the sphere onto the plane imposes the following nonholonomic constraint on the system:
¡2 = (vp, v) = (v + w x r, v) = 0, (2.5)
where v is a unit vector in the direction of the projection of the symmetry axis of the body onto the plane (see Fig. 4) which is given by the equation
Y x (e3 x y) . .
V1 - 73
2.3. Dynamic equations
The Lagrangian function of the system in the moving coordinate system ox1x2x3 has the
form
L = -rnv2 + -(a;, la;) + mg(r, 7),
where g is the free-fall acceleration.
The equations of motion of the system can be obtained from the D'Alembert-Lagrange principle and written in quasi-velocities (for a detailed derivation in general form, see, e.g., [60-62]) taking into account the constraints (2.4) and (2.5) in the form
mV = mv x w + X, y + A3v,
(2.7)
Iw = Iw x w + (mg + X^(r x y) + X2(r x v),
where X1 and X2 are undetermined multipliers which are calculated from a joint solution of Eqs. (2.7) and the time derivatives of the constraints (2.4) and (2.5).
To completely describe the motion of the sphere in absolute space, the system of equations (2.7) must be supplemented with the quadratures (2.1) describing the orientation of the sphere in space and the trajectory of the point of contact.
In order to obtain the equations of motion in explicit form, we multiply the first equation of (2.7) by y and express X1 from the resulting equation. Taking into account the time derivative of the constraint (2.5), we obtain X1 in the following form:
Xx = -m(w, r x y)'. (2.8)
In the same way we express the multiplier X2 from the first equation of (2.7), multiplied by v, and the time derivatives of the constraints (2.4) and (2.5):
X2 = —m ((v, v) + (w x r, v)' + (v x w, v)). (2.9)
Substituting X1 and X2 into Eqs. (2.7), we obtain a closed system of differential equations (2.1) and (2.7) which, by construction, preserves the integrals of motion f1 and f2 corresponding to the imposed constraints (2.4) and (2.5). The phase flow on the zero level set of these integrals will describe the dynamics of the system.
As can be seen from (2.7), (2.8) and (2.9), the multipliers X1 and X2 do not explicitly depend on a, 3, x and y. Thus, after substituting X1 and X2, the equations for u, v and 7 decouple from the complete system and form a closed reduced system of nine differential first-order equations:
V + (u, r x 7)7 + (u, r x V)v = = v x u — (u, (r x 7)') 7 — (u, (r x V)■) V — (v x u, V)v — V(v, V), Ju = Iu x u + m (g — (u, (r x 7)')) (r x 7) — m ((u, (r x v)■) + (v, V + u x v)) (r x v),
77 = 7 x u,
(2.10)
where J = I + m(r x 7) ® (r x 7) + m(r x v) ® (r x v) and v is defined by (2.6).
3. First integrals and reduction
3.1. Reduction by the symmetry field
The resulting system (2.10) admits four first integrals:
- a geometric integral:
Fo = 72 = 1;
- integrals following from the constraint equations (2.4), (2.5):
F1 = (v + u x r, 7) = 0, F2 = (v + u x r, v) = 0;
- an energy integral
E = im{v, v) + — mg(r, 7).
We note that the projection of the second equation of the system (2.10) onto e3 has the trivial form
UJ3 =
Hence, the system (2.10) admits another first integral of motion, the Lagrange integral:
F3 = I3w3 = const. (3.1)
The integral (3.1) is a consequence of the invariance of the system under rotations about the axis Ce3. By virtue of this invariance, Eqs. (2.10) possess also a symmetry field defined by the differential operator
d d d d d d , .
Let us write Eqs. (2.7), taking the constraints (2.4) and (2.5) into account, and simultaneously perform a reduction to the level set of the integral (3.1) and by the symmetry field (3.2).
To do so, following [28, 63], we transform to new variables which are integrals of the vector field (3.2). As such variables we choose projections of the velocity v and the angular velocity tv onto the axes of the coordinate system
t \ Y1v3 - Y3P1 t ^ Ylvl + Y2v2 t ^
vT = (V, r) = 3-'jL1, vn = (V, n) = - \-V3 = (v, e3),
V1 - 73 V1 - 73
Y1w3 - y3w1 Yi^i + Y3^3 , \
wr = (w, r) = i-13 \ uin = (w, n) = ~ ^-V> w3 = (w, e3),
V1 - Y3 V1 - Y3
and the quantity y3. Here the unit vectors t and n of the coordinate system C£1£3x3 are expressed in terms of the unit vectors 7 and e3 as follows:
e3 X 7 n=e3x(e3x 7)
In the new variables the equations for vT, uT, un and y3 decouple from the complete system and form a reduced system of equations:
(R + a3Y3) (M3 Y3 \
VT = (jJT(jJn r—^1, un = WTM+ rr^Un , V1 - Y33 V 71 V 1 - Y3 /
Y3 = UtJ 1 - Y3>
(3.3)
. ma3 (1 - Y3) {g + R^T) + iiw3Y3 + mVTWn(R + a3Y3)
yr^l (I, + m (a| + -ñ2 + 2a3Ü73))
M3 wra
(Ii + m (a3 + R3 + 2a3RY3))'
where M3 is a fixed value of the Lagrange integral. In this case, the projections vn and v3 are defined from the constraint equations which take the form
v3 = -u)TR\Jl -7|, vn = -ujt(R"/3 + a3).
The evolution of the remaining configuration variables is defined by the quadratures (2.1).
Equations (3.3) admit an energy integral which in the new variables up to a constant term takes the form
^ 1 3 Ii 3 I1 + m (a'3 + R3 + 2a3Ry3) 3 .
E = -mv3 + -juj3 + -i--"—^uj3 + mga313. (3.4)
In addition, it is easy to show that Eqs. (3.3) possess an invariant measure with density
p(Y3) = I1 + m (a3 + R3 + 2a:iRY3) ■ (3.5)
Thus, for integrability of the system we need another integral. Next, we turn to analysis of the reduced system (3.3).
3.2. Additional integrals of motion
As was shown by J.H. Jellett [17], if a rolling sphere is acted upon, from the side of the plane, only by the force applied to the point of contact, then the equations of motion admit an additional integral that was named after the person who found it. The system at hand satisfies the above requirement and hence admits Jellett's integral which in the chosen notation takes the form
F4 = (la;, r) = R\Jl — 73 un - (Rl3 + a3)M3.
As shown in [19], the existence of the Lagrange integral F3 and Jellett's integral F4 implies the existence of the area integral
F5 = (la;, 7) = - - I, (3.6)
which is linearly dependent on F3 and F4:
F4 + a3F3 + RF5 = 0.
Remark 3. The area integral can also be obtained from the condition of invariance of the system under rotations about the vertical.
Next, we show that the system (3.3) admits another independent integral linear in velocities.
We note that the first three equations of the system (3.3) are linear in wT. Let us divide the first and the second equation of (3.3) by the third. As a result, we obtain a closed system of two differential first-order equations in vT (y3), w„(y3):
<K = u (R + Q-373)
1 - 73 (3 7)
dcon= M3 73
d% hs/l^l 1-71' n"
The solution of the system (3.7) can be represented as
„, _ R(M3 - cil3) + a3(M3-/3 -C\) + C2 + \I:,a:, arccos y3
hV^ri h
2
(3.8)
M,r, c,
Wn ~ T /1 2 '
hV1" 73
where C1 and C3 are constants of integration.
Expressing C1 and C3 from (3.8), we obtain two integrals of motion linear in velocities:
F5 = 73M3 - /iV 1 - y33 = Ci,
( !--\ (3-9)
F6 = h (vt - (ry3 + a3)^n) - m3i r\j1 - 73 + a3 arccos 7 J = C2.
As can be seen from (3.6) and (3.9), the first of the linear integrals is an area integral and F6 is a new independent integral. Thus, the reduced .system (3.3) admits three independent first integrals and is superintegrable, and all nonsingular trajectories are periodic functions of time.
The integral similar to F6, but with a logarithmic dependence on y3, was found in the problem of a disk rolling on ice in [38] by generalizing the main theorems of dynamics to systems with nonholonomic constraints.
3.3. Reduction to one degree of freedom
Using the integrals (3.9), we can reduce the system under consideration to a system with one degree of freedom. To do so, we substitute vn and wT from (3.8) into the last two equations of (3.3). This yields
Y3 = 1 - Y2,
pujT = -maJl-Yl (g + Ruj2t) - ~ ^ i/,i.\/:; - Cl73) + m(R + a2%)B), {3'W)
I2 (1 - Y3)
where C1 and C3 now denote constant values of the integrals F5 and F6, respectively, p is defined by relation (3.5), and the quantity B is introduced to abbreviate the notation and has the form
5 = C2 + a3M3 arccos 73 + - CQ
V1 - Y33
These equations admit an energy integral that can be obtained from (3.4) by substituting into it the dependences wn and vT (3.8)
puj2 (C.-M^s)2 . mB2
E = T + 2I1 (1 - 732) + ~W - ( }
Fixing the level set of the energy integral, E = h, and solving the equation E(wT) = h for wT, taking the first equation of (3.10) into account, we obtain a quadrature for y3 in the following form:
3 2 (1 - y2) / (Ci - M3y3)3 mB3 \
7s = —-— h - —^----x- - mgatfs .
P V 2/i(1" 732) 2J?
An example of the phase portrait of the system (3.10) with the parameters
R = 1, m = 1, g = 1, a3 = 0.1, I1 = 0.3, I3 = 0.5 (3.12)
on the level set of the first integrals
M3 = 1, C1 = 0.5, C3 = -2
is given in Fig. 5.
In this paper, we will not examine in detail the system (3.10). We will only consider briefly the singularities of Eqs. (3.10) with y3 = ±1. To these singularities there correspond instants when either of the poles of the rolling sphere comes into contact with the supporting plane and the radius vector of the contact point has the form r = (0, 0, -a3 ± R). In this case, we will call the point with coordinates (0, 0, -a3 + R) the northern pole and the point with coordinates (0, 0, -a3 - R) the southern pole.
The analysis of Eqs. (3.10) and the asymptotics of the energy integral (3.11) as y3 ^ ±1 shows that, when IC-^ = |M3|, the function y3 is a bounded and, in the general case, periodic function of time. The maximal and the minimal absolute values of Y3(t) are strictly less than unity: |Y3(t)| < 1. Thus, y3 cannot reach the values ±1, and hence, on these level sets of the
. 73
Fig. 5. Phase portrait of the system (3.10) with parameters (3.12) on the level set of the first integrals M3 = 1, C1 = 0.5, C2 = -2
integrals, the poles of the sphere cannot come in contact with the supporting plane when the sphere rolls (or rotates).
In the case M3 = C1, the sphere can roll with its southern pole contacting the supporting plane (with y3 = 1), and in the case M3 = —C1, it is the northern pole of the sphere that contacts the plane (with y3 = —1). Obviously, it is more convenient to investigate the dynamics of the system on these level sets if one uses other variables that have no singularities at the poles of the unit sphere (71 = 1.
The question of classification of motions of the system depending on the values of first integrals and parameter values still remains open.
Another possible avenue of research into the system considered is to analyze the quadratures (2.1), which describe the motion of the sphere in absolute space. This can be done by using the approaches that were applied to investigate similar systems (see, e.g., [7, 38]).
Acknowledgments
The authors thank E. N. Pivovarova for useful discussions.
Conflict of interest
The authors declare that they have no conflicts of interest.
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