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4Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 4, pp. 533-543. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231201
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 70E18, 70K25
The Problem of the Rolling Motion of a Dynamically Symmetric Spherical Top with One Nonholonomic Constraint
A. A.Kilin, T. B.Ivanova
This paper investigates the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that the sphere can slip in the direction of the projection of the symmetry axis onto the supporting plane. Equations of motion are obtained and their first integrals are found. It is shown that in the general case the system considered is nonintegrable and does not admit an invariant measure with smooth positive density. Some particular cases of the existence of an additional integral of motion are found and analyzed. In addition, the limiting case in which the system is integrable by the Euler-Jacobi theorem is established.
Keywords: nonholonomic constraint, first integral, nonintegrability, Poincaré map
Introduction
In this paper we address the problem of a sphere with axisymmetric mass distribution (spherical top) rolling with partial slipping on a horizontal plane. We assume that the contact interaction of the sphere with the plane is anisotropic, so that the sphere can slip as it moves in the direction of the projection of the symmetry axis onto the supporting plane.
Constraints of this type arise, for example, in considering the problem of the motion of a body with a sharp edge rolling on an ice surface [1-4]. In this case, the slipping at the point of contact is allowed only in the direction tangent to the sharp edge.
A similar constraint arises also in considering the dynamics of a rigid body freely rotating about some axis fixed inside a weightless spherical shell which rolls without slipping on
Received November 02, 2023 Accepted December 10, 2023
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
a plane [5-7]. In this problem, slipping is possible in the direction perpendicular to the projection of the symmetry axis on the plane. In [5] it is shown that in this case the system is integrable. Moreover, it admits a redundant set of first integrals that allows a reduction to a system with one degree of freedom.
The system under study has no such simple physical realization. However, it can be regarded as a model of the rolling motion of an omnisphere which has rollers fastened on the entire surface, with their axes directed along the tangents to the parallels of this sphere (see Fig. 1). A similar realization is possible for the problem treated in [5], but in that case the rollers' axes are directed along the tangents to the meridians of the sphere.
Fig. 1. A model of the omnisphere which has rollers fastened on the entire surface, with their axes directed along the tangents to the parallels of this sphere
We note that the recent paper [8] examines the dynamics of an inhomogeneous disk with a similar model of anisotropic contact interaction. The presence of one nonholonomic constraint in this problem has given rise to interesting dynamics. For example, it was shown that chaotic dynamics in this problem can coexist with a two-parameter family of periodic solutions.
As we will see below, the dynamics of the system considered in this paper is also fairly diverse. In particular cases it can be integrable. However, in the general case the dynamics is nonintegrable and, moreover, it exhibits behavior typical of dissipative systems. It is interesting that, within the framework of the model considered, the flip-over of the top similar to tippe top inversion [9-12] can be observed even without adding dissipation to the system.
This paper is structured as follows. In Section 1 we formulate the model under study. In Section 2 we derive equations of motion for the system. In Section 3 we find first integrals of the system and perform reduction. In Section 4 we examine particular cases where an additional integral of motion exists. In Section 5 we present results of numerical simulation, and prove the absence of an additional analytic integral in the general case.
1. Formulation of the problem
In this paper we address the problem of a spherical top (a heavy unbalanced sphere of radius R and mass m with axisymmetric mass distribution) rolling on a plane. We formulate the main assumptions under which we 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 perpendicular to the projection of its symmetry axis onto the plane. But it does slip along the projection of the symmetry axis of the sphere on the plane.
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).
An example of physical realization of the system in the form of an omnisphere is shown in Fig. 1.
2. Equations of motion
2.1. Configuration space and kinematic relations
To describe the motion of the sphere, we introduce two 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, / and 7 (see Fig. 2);
- a moving coordinate system ox1x2x3 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. 2).
Fig. 2. Spherical top on a plane
In the case of axisymmetric mass distribution under consideration, one of the axes, for example, ox3 (see Fig. 2) coincides with the symmetry axis of the sphere. Then e3 = (0, 0, 1), the displacement vector of the center of mass is directed along e3 and has the form a = (0, 0, a3), and the matrix of principal momenta of inertia I = diag(/1, I1, I3)12.
1 Here and in what follows (unless otherwise specified) all vectors are referred to the moving axes.
2Here and in what follows, we denote the vectors by the letters written in bold italics, a, /, F, M, ..., and write their scalar and vector product, respectively, in the form (a, 3) and a x b. The sign ®
We will specify the position of the system by the coordinates of the radius vector of the center of mass ro = (x, y, z) and by the orthogonal matrix Q of transition from the fixed coordinate system to the moving coordinate system. The columns of this matrix are the projections of the vectors a, (3, 7 onto the axes of the moving coordinate system ox1x2x3.
Let u = (wl, u2, w3) and v = (v1, v2, v3) be the angular velocity and the velocity of the center of mass of the body, respectively. Then the evolution of the orientation and position of the sphere is given by the kinematic relations
a = a x u, (3 = (3 x u, Y = Y x u,
' H H >77 > (2.1)
x = (v, a), y = (v, ¡3), Z = (v, y).
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:
fi = (vp, Y) = 0, (2.2)
/2 = (vp, t) = 0, (2.3)
where vp = v + u x r is the velocity of the sphere at the point of contact with the plane, r = -Ry — a is the radius vector of the point of contact, and t is the unit vector lying in the horizontal plane and perpendicular to the projection of the symmetry axis of the top onto the plane (see Fig. 2):
V1 — Y32
The constraint equation (2.2) is a consequence of the condition of motion without loss of contact with the plane, which can be written as a holonomic constraint:
z + (r, y) = 0.
The constraint equation (2.3) corresponds to the condition of partial nonslipping in the direction of the vector t. In a sense, this constraint is complementary to the constraint considered in [5], where the sphere did not slip in the direction of the vector v.
2.3. Dynamical equations
The Lagrangian function of the system in the moving coordinate system ox1x2x3 has the
form
L = -rnv2 + -(u>, Iu>) + 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., [13-15]) taking into account the constraints (2.2) and (2.3) in the form
mi} = mv x u + Ai y + A2t,
(2.5)
Iu = Iu x u + (mg + A1)(r x y) + A2(r x t),
denotes the tensor product, i. e., (a ® b) = a^bj. The letters written in boldface Roman font denote the matrices: A, B, ...
where X1 and A2 are undetermined multipliers. These are calculated from a joint solution of Eqs. (2.5) and the time derivatives of the constraints (2.2) and (2.3):
Ai = —m(u, r x 7)', 1 (2.6) A2 = —m ((v, t) + (u x r, t) + (v x u, t)).
Substituting A1 and A2 into Eqs. (2.5), we obtain a closed system of differential equations (2.1) and (2.5) which, by construction, preserves the integrals of motion f1 and f2 corresponding to the imposed constraints (2.2) and (2.3). The phase flow on the zero level set of these integrals will describe the dynamics of the system.
As can be seen from (2.5) and (2.6), the multipliers A1 and A2 do not explicitly depend on a, 3, x and y. Thus, after substituting A1 and A2, 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)y + (u, r x T)t = v x u — (v x u, T)t—
— (u, (r x 7)')y — (u, (r x t)')t — (v, t)t,
Ju = Iu x u + m(r x 7)(g — (u, (r x 7)')) — (2.7)
—m(r x t) ((u, (r x t)') + (v, t + u x t)),
7 = 7 x u,
where J = I + m(r x 7) ® (r x 7) + m(r x t) ® (r x t), and t is defined by (2.4).
3. Invariants and reduction
3.1. Invariants
The resulting system (2.7) admits five first integrals:
a geometric integral:
72 = 1;
- integrals following from the constraint equations (2.2) and (2.3):
F1 = (v + u x r, 7) = 0, F2 = (v + u x r, t) = 0;
- an energy integral:
E = -m(v, v) + — mg(r, 7);
- Jellett's integral:
G = (Ju, r).
The existence of this integral is due to the fact that, as the sphere is rolling, the only force acting on it from the plane is the force applied to the point of contact of the sphere with the plane [16].
Since the system is invariant under rotations about the axis Ce3, Eqs. (2.7) also have a symmetry field defined by the differential operator
^ d d d d d d , .
As will be shown below (see Section 4), the system under study does generally not admit an invariant measure with smooth positive density.
3.2. Reduction
Let us write Eqs. (2.5), taking the constraints (2.2) and (2.3) into account, and simultaneously perform a reduction by the symmetry field (3.1).
For this, we define the vectors v and n, which are expressed in terms of the vectors y and e3 as follows:
= 7 x (e3 x 7) n=e3x{e3x 7)
V^yf ' V^yf
We note that the vectors (t, v, 7) and (t, n, e3) form right triples of orthogonal unit vectors (see also Fig. 2), and the corresponding projections of the velocity v and the angular velocity tv onto these vectors are integrals of the vector field (3.1).
Thus, following [17, 18], as new variables we choose the quantity y3, the projections of the velocity v onto the vectors t, v, 7 and the projections of the angular velocity t onto the vectors t, n, e3 :
/ \ Y1v2 - Y2v1 / \ 7iw2 - 72W1
vT = (V, r) = i-12 \ Wr = (w, r) = 2-,2\
V1 - y3 V1 - %
= (v, v) = v3 (1 - 7I) - 73(71% + I2V2), = (u>, n) = -llUJJ\+1'2tf-,
V1 - YÎ
VY = (v, 7) = vi7i + v272 + w3 = e3).
In the new variables the equations for vv, wr, w3 and y3 decouple from the complete system and form a reduced system of five differential first-order equations:
Y3 = wt\I1 - Y2,
' w„(a3 + RY3) + Ruj3 \J I-73) >
v = —, ^
ma3 (1 - 7|) {a3Y3to2 ~ g) - (i^w,, + /3w3 ^l - 732)
^ = Vl - 73 (/i+mai (1-7^)) ' ^^
mh^niRls + «3)K(^ + «-373) + O , h^Ah + + a373))
UJ — - — - ->
" yr^Tf Q(ls)V^ÏI e(7s)
^ mfi^nt^ + ^r(«-373 + R)) , mfly/1 - y'^S^A^A1! ~ h) ~ aiLi) 3 f?(73) f?(73)
where é?(y3) = I1I3 + I1mR2 (1 - y3) + I3m(a3 + Ry3)2. The projections v7 and vT are defined from the constraint equations which take the form
Vf = a>3ujt\/l -71, vT = u)n(a,3 + Ry3) + Ew3\Ji~yI-
The evolution of the remaining configuration variables is defined by the quadratures (2.1).
Remark 1. We note that Eqs. (3.2) cannot be applied to the analysis of singular configurations IY31 = 1. This is due to the fact that these configurations correspond to singular orbits (zero-dimensional in this case) of the action of rotation groups, whereas Eqs. (3.2) are defined on the space of regular (one-dimensional) orbits (see, e.g., [15]). Thus, in Eqs. (3.2) one needs to set I73I < 1.
Equations (3.2) admit an energy integral and Jellett's integral, which in the new variables (up to a constant term) take the form
^ m o h + ma3 i1 ~ li) 2 2 A 2 ni (. „ . ,, r ~ \2
E = jv2 + -i--^uj2t + + + j Í (E73 + a3)un + RsJ 1 -73^3) +
G = 11R \Jl — 72 ujn - I3(Rl3 + a3)u3. (3.3)
Thus, by the Euler-Jacobi theorem, for the system (3.2) to be integrable, an integral and an invariant measure are required.
Next, we consider cases where an additional integral of motion exists.
E0 = ^v2V + {uj2 + + -fw32 + ( JI - 72u3 + l3ujn ) , (4.1)
4. Cases of the existence of an additional integral of motion
4.1. A balanced sphere
Let us consider the case of an inhomogeneous, but balanced sphere, i.e., let us assume that the center of mass is at the geometric center of the sphere (a3 = 0), and that I1 = I3. In this case, the energy integral and the Jellett integral take the form
G0 = hRsJl-^ujn - /,//-,.•,. (4.2)
In addition to the energy integral and the Jellett integral, the system (3.2) admits, for a3 = 0, an additional (quadratic) integral of motion
F0 = mI2(Vv + RuT)2 + I - I3)№2. (4.3)
Fixing the values of the first integrals (4.1), (4.2) and (4.3), we can reduce the analysis of the system (3.2) to investigating the flow on the two-dimensional integral manifold
M2 = {(73> Vv > ^t > ^3) | E0 = E, G0 = g, Fo = F}.
The projection of the surfaces of the fixed level set of the integrals (4.1), (4.2) and (4.3) into the space of velocities {(vv, wT, wn, w3)} is
- an ellipsoid for the energy integral;
- a hyperplane for the Jellett integral;
- a cylinder (elliptic or hyperbolic, depending on the values of I1 and I3) or two planes (with I1 = I3) for the integral F0.
In nonsingular cases, the intersection of these surfaces is one or two closed curves without self-intersections. Hence, the nonsingular integral manifolds M2 are one or two two-dimensional cylinders.
For the system (3.2) to be integrable by the Euler-Jacobi theorem, it is necessary that it admit not only an additional integral, but also an invariant measure. However, in the case at hand this is not so, and the following proposition holds.
Proposition 1. When a3 = 0, the .system (3.2) does not admit an invariant measure with smooth positive density.
Proof. When a3 = 0, the system (3.2) admits a three-parameter family of fixed points
Y3 = y0 = const, vV = v0 = const, wT = 0, wn = 0, w3 = w0 = const. (4.4)
On the integral manifold M2, this family corresponds to several isolated fixed points. In absolute space, the family (4.4) corresponds to rotations of the sphere about its symmetry axis, which can be directed in an arbitrary manner, with simultaneous sliding along the vector v. The equations of motion linearized near the solutions (4.4) have the form
q = Aq, A
^0 0 0 0 0 0 0 0
^0 0
0 0
I3oj°3(l3+mR2)
e
0
I1
0„,0 v
mRI3 JO'v
e
1-(y0)
mR2(Il-^V1-^)^ mRhv
0 0 0
0 0
(4.5)
where q is the vector of deviations from the equilibrium point (4.4).
The characteristic polynomial of the system (4.5) has the form (E is a unit matrix)
det(A - AE) = A3 (liesJ 1-(y3°)2A2 + hhmRv0^A + w2/2 sjl-^)2^ + mE2)) = 0.
Three zero eigenvalues of the linearization matrix correspond to the integrals of motion, and the trace of the linearization matrix A is
tr A = -
mRI3Y0 V
00 V
(4.6)
It follows from (4.6) that there exist integral manifolds M2 on which the trace of the linearization matrix near the equilibrium points (4.4) is nonzero. Consequently, in accordance with the theorem of the existence of an integral invariant [19] (see also [20]), the system (3.2) has no invariant measure with smooth positive density in a neighborhood of the solutions (4.4) for a3 = 0. □
Thus, in the case a3 = 0 the analysis of the system (3.2) reduces to the study of the non-area-preserving phase flow on a two-dimensional cylinder. The investigation of this flow depending on the system's parameters and the constants of first integrals is an interesting problem that requires a separate study, and so we will not address it here.
4.2. The limiting case I3 = 0
Consider the case of mass distribution in which the displacement vector of the center of mass has the form a = (0, 0, a3), and the matrix of principal moments of inertia is I = diag(I1, I1, 0). An example of such a mass distribution is a weightless shell with an inhomogeneous rod or a point mass inside.
In this case the energy integral is divided into two independent quadratic integrals:
Ei = jvl + y ( (E73 + a3K + R V:1 " 732W3
/1 2 , Ji + (1 ~ 73) 2 .
E2 = +---uT + mga373.
In this case, the system also possesses an invariant measure with density
P(73) = ^1" 732 (A + ma! (1 " 73))•
Thus, when I3 = 0 the system (3.2) is integrable by the Euler-Jacobi theorem.
Remark 2. The positive definiteness of the integral E1 implies that the following relations are satisfied on its level set:
Vv = ( (rY3 + «3 Vn + Ry 1 - 75^3 =
In this case, the nonsingular common level sets of the integrals E1, E2 and G are completely filled with periodic orbits.
5. Numerical simulation
In this section we present the results of numerical simulation of the system's motion for arbitrary parameters.
For this we perform a reduction to the level set of the Jellett integral G = G by expressing from (3.3) and by substituting the resulting expression into Eqs. (3.2). As a result, in the four-dimensional space {(y3, vv, wr, w3)} we obtain a phase flow that preserves the energy integral. For this flow we can construct a Poincare map in a standard way.
For this we choose the (hyper)plane wr = 0 as a secant. The section formed by the intersection of the isoenergetic level set E = E with this plane forms a two-dimensional surface. When the phase flow intersects this surface on the fixed level set of the energy integral, it forms a Poincare map on it. As local coordinates on this surface we choose (n, Y3), where3
V = arctan ^mRsJl - s]q{q ~ hh) + ) •
It follows from the first equation of (3.2) that all periodic trajectories of the system intersect the chosen section wr = 0 or lie completely in it.
For ease of numerical simulation, we choose the sphere's radius R as a unit of length, the
sphere's mass m as a unit of mass, and the quantity ^ as a unit of time.
Figure 3a shows the Poincare map of the system (3.2) on the plane (n, Y3) for the parameter values
a3 = 0.51, I1 = I3 = l E = 2.15, Q = -0.56. (5.1)
3
Figure 3b shows an enlarged fragment of Fig. 3a and depicts a stable and an unstable branch of the separatrices of an unstable fixed point. Their intersection is indicative of the existence of chaotic dynamics in the system and of the absence of an additional analytic integral in the general case.
3The function arctan(y, x) = argument(x + iy), where i is an imaginary unit, returns the angle of the radius vector in the interval from —n to n with the coordinates of the end of the radius vector being x and y.
Fig. 3. (a) Poincare map of the system (3.2) with the parameters (5.1) on the plane (n, Y3). (b) Transverse intersection of the separatrices near an unstable fixed point (the enlarged region is framed in the left panel)
Conclusion
In this paper we investigate the dynamics of a sphere with axisymmetric mass distribution (spherical top) on a horizontal plane. It is assumed that, as the sphere is moving, it can slip in the direction of the projection of its symmetry axis onto the plane of motion.
It is shown that in the general case the system is nonintegrable. In particular, using numerical simulation it is shown that in the general case there is no additional integral. Also, it is proved that the system under study does not admit an invariant measure with smooth density.
Some particular cases of the existence of an additional integral of motion are found and analyzed. Also, the limiting case in which the system is integrable by the Euler-Jacobi theorem is established.
Further analysis of the system may involve the study of the phase flow on an integral manifold in the case of a balanced sphere (with a3 = 0) depending on the system's parameters and the constants of first integrals. Another problem of interest is to investigate possible flip-overs of the spherical top similar to tippe top inversion. Preliminary numerical investigations show that in the system under study such a flip-over can be observed even without adding friction forces.
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