Dynamics of the Tippe Top via Routhian Reduction
Dynamics of the Tippe Top via Routhian Reduction
M.C. Ciocci1 and B. Langerock2
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Sint-Lucas school of Architecture, Hogeschool voor Wetenschap & Kunst, B-9000
Ghent, Belgium
October 22, 2018
Abstract
We consider a tippe top modeled as an eccentric sphere, spinning on a
horizontal table and subject to a sliding friction. Ignoring translational effects,
we show that the system is reducible using a Routhian reduction technique.
The reduced system is a two dimensional system of second order differential
equations, that allows an elegant and compact way to retrieve the classification
of tippe tops in six groups according to the existence and stability type of the
steady states.
1 Introduction
The tippe top is a spinning top, consisting of a section of a sphere fitted with a
short, cylindrical rod (the stem). One typically sets off the top by making it spin
with the stem upwards, which we call, from now on the initial spin of the top. When
the top is spun on a table, it will turn the stem down towards the table. When
the stem touches the table, the top overturns and starts spinning on the stem.
The overturning motion as we shall see is a transition from an unstable (relative)
equilibrium to a stable one. Experimentally, it is known that such a transition only
occurs when the spin speed exceeds a certain critical value.
Let us make things more precise and first describe in some detail the model
used throughout this paper. The tippe top is assumed to be a spherical rigid body
with its center of mass ǫ off the geometrical center of the sphere with radius R.
In addition, we assume that this eccentric sphere has a mass distribution which is
axially symmetric about the axis through the center of mass O and the geometrical
center C of the sphere. The tippe top is subject to a holonomic constraint since
we only consider motions of the tippe top on a horizontal plane. An important
observation is that the system has an integral of motion, regardless of the model of
the friction force that is used. This integral is called the Jellet J and is proportional
to the initial spin n0 (equation (2.7)). For the sake of completeness we mention
here that a nonholonomic model (rolling without slipping) for the tippe top is not
appropriate, since the equations of motion then allow two more integrals of motion
(the energy and the Routh-integral) that prohibit the typical turning motion, cf. eg.
[1, 5]. We will choose the friction force to be linearly dependent on the slip velocity
of the contact point Q of the top and the horizontal plane.
In earlier works [7, 15] conditions of stability for asymptotic states of the tippe
top were retrieved by the Lyapunov function method starting with the Newton
equations of motion. We take a different approach to the problem. Our goal is to
describe the behaviour of the tippe top in terms of Lagrangian variables and thereby
classify its asymptotic motions as function of non-dimensional (physical) parameters
http://arxiv.org/abs/0704.1221v3
A/C, the inertia ratio and ǫ/R, the eccentricity of the sphere and of the value the
Jellet J . As main novelty, we show that by ignoring the translational velocity of the
center of mass in the friction law, the Lagrangian formulation of the dynamics of the
tippe top allows a restriction to a system on SO(3) which is amenable to Routhian
reduction [22]. The reduced system allows a rather simple stability analysis of the
relative equilibria based on the relationship of the value of the Jellet’s integral J
and the tumbling angle θ, see below. Through the Routhian reduction we retrieve
a simple stability criterium which leads to the same conclusions as in [7, 15].
The hypothesis of neglecting translational effects is similar to the one made
by Bou-Rabee et al. [16]. They motivated it by reasoning that for all possible
asymptotic states (the relative equilibria of the system) the velocity of the center
of mass is zero [7], and, for this reason, in a neighborhood of these solutions the
translational friction can be neglected. In [15] the dynamics of the spherical tippe
top with small friction has been studied without such an approximation and a full
analysis of the asymptotic long term dynamics of the system is given in terms of the
Jellet and the eccentricity of the sphere and inertia ratio. Remarkable is the fact
that the stability results obtained by means of the Routhian reduction procedure
fully coincides with the results of [15]. This justifies a-posteriori the approximation
assumption in the friction force and shows that it accurately describes the behavior
of the tippe top.
The main result of this paper is summarized in the following theorem, also
compare with [1, 15].
Theorem 1 In the approximation of negligible translational effects, a spinning ec-
centric sphere on an horizontal (perfectly hard) surface subject to a sliding friction
is reducible with a Routhian reduction procedure [22]. The relative equilibria of the
reduced system are precisely the steady states of the original system. They are purely
rolling solutions and except for the trivial state of rest, they are of three types:
(i) (non-inverted) vertically spinning top with center of mass straight below the
geometric center;
(ii) (inverted) vertically spinning top with center of mass straight above the geo-
metric center;
(iii) intermediate spinning top, the top precesses about a vertical while spinning
about its axle and rolling over the plane without gliding.
The existence and stability type of these relative equilibria only depend on the inertia
ratio A
, the eccentricity of the sphere ǫ
and the Jellet invariant J . In particular,
six regimes are identified in terms of the Jellet invariant and only three exhibit the
‘tipping’ behavior.
It turns out that the vertical states always exist, and intermediate states may branch
off from them. Qualitative bifurcation diagrams corresponding to the possible dif-
ferent regimes are sketched in Fig. 1.
The reader is referred to Sec. 3 – Sec. 4 for the details and specification of the
parameter ranges. There are three main groups Group I, II and III determined by
existence properties of intermediate equilibrium states (similar to the subdivision
from [20]). Tippe tops of Group I may admit intermediate states for θ > θc, where
θ denotes the angle between the vertical and the symmetry axis of the top. Two
subgroups are distinguished according to change in stability type of the intermediate
states. Relevant is that for tippe tops belonging to this class the non-inverted
position is always stable, so they never flip however large the initial spin when
launched under an angle θ close to 0. Tippe tops of Group II may admit intermediate
states for all θ, and they show complete inversion when the initial spin is large
Group Ia
Group Ib
Group IIa
Group IIb
Group IIc
Group III
Figure 1: Bifurcation diagrams of relative equilibria in function of the Jellet invariant J . Solid
black branches correspond to stable relative equilibria, while dashed black branches correspond to
unstable ones. The vertical states θ = 0 and θ = π always exist, from which intermediate states
(with 0 < θ < π) may branch off.
enough. Tops of Group III tend to flip over up to a certain angle θc < π when
spun rapidly enough. Since stability results are often in the literature expressed
using the ‘initial’ spin n0 of the tippe top, one can read the J
2 in the figures as
n20. We anticipate to further results by noting that the instability inequalities are
independent of the friction coefficient unless it is zero.
The structure of the paper is as follows. In Section 2 our model is described and
the equation of motion are given according to the Lagrangian formalism. After
introducing the Jellet integral of motion, the Routhian reduction is performed in
Section 2.2. The steady states of the system are then calculated and their sta-
bility type is determined, yielding a tippe top classification in six groups which is
summarized in Section 4.1.
2 Equations of motion
As is mentioned in the introduction, we consider the eccentric sphere model of
such a top, see Fig. 2. That is, we consider a sphere with radius R whose mass
distribution is axially symmetric but not spherically symmetric, so that the center
of mass and the geometric center do not coincide. The line joining the center of
mass and the geometrical center is an axis of inertial symmetry, that is, in the plane
perpendicular to this axis the inertia tensor of the sphere has two equal principal
moments of inertia A = B. The inertia moment along the axis of symmetry is
denoted by C and the total mass of the sphere is m.
The eccentricity ǫ is the distance between the center of mass O and the geometric
center C of the sphere, with 0 < ǫ < R. The point Q is the point of contact with the
plane of support. We assume that an inertial (laboratory) frame Mxyz is chosen,
where M is some point on the table and the z-axis is the vertical. Let us denote
the unit vectors along the axis of the reference frame Ox, Oy, Oz fixed to the body
by respectively ex, ey, ez. The coordinates of the center of mass are denoted by
PSfrag replacements
h(θ) = R − ǫ cos(θ)
Figure 2: Eccentric sphere version of the tippe top. R is the radius of the sphere, the center
of mass O is off center by ǫ. The top spins on a horizontal table with point of contact Q. The
axis of symmetry is Oz and the vertical axis is Oz, they define a plane Π (containing ~OQ) which
precesses about Oz with angular velocity ϕ̇. The height of O above the table is h(θ).
rO = (x, y, z)Mxyz . A second reference frame is denoted by Oxyz, and is defined
in such a way that its third axis is precisely the symmetry axis of the top and the
y-axis is perpendicular to the plane Π through the z- and z-axes (see Fig. 2). Again
we denote the unit vectors along the axis of this reference frame by ex, ey, ez
Let (θ, ϕ, ψ) be the Euler angles of the body with respect to the inertial frame,
Fig. 2, chosen in such a way that (i) the vertical plane Π is inclined at ϕ to the
fixed vertical plane xz and precesses with angular velocity ϕ̇ around the vertical Oz;
(ii) the angle θ is the angle between the vertical Oz and the axle Oz of the top; θ̇
causes the nodding (nutation) of the axle in the vertical plane Π; and (iii) the angle
ψ orients the body with respect to the fixed-body frame, ψ̇ is the spin about the
axle.
As it was pointed out before, the tippe top is constrained to move on a horizontal
plane. This holonomic constraint is expressed by z = R − ǫ cos θ = h(θ). We
assume throughout the paper that the only forces acting on the sphere are gravity
G = −mgez and a friction force F exerted at the point of contact Q of the sphere
with the plane. It is now immediate to write down the Lagrangian for the tippe
L = 1
m(ẋ2 + ẏ2) + (ǫ2m sin2 θ +A)θ̇2 +A sin2 θϕ̇2 + C(ψ̇ + ϕ̇ cos θ)2
−mg(R− ǫ cos θ), (2.1)
where g is earth acceleration.
This function is defined on the tangent space of the configuration manifold
M = IR2 × SO(3). In order to obtain the equations of motion, it only remains
to define a suitable friction force F. The Lagrangian equations of motion for the
tippe top then read:
= QFi , (2.2)
where qi represents one of the coordinates (x, y, ϕ, θ, ψ), and QF = QFi dq
i is a one-
form on M representing the generalized force moment of the friction force at the
point of contact. It is defined by, with F = Rf+Rnez the orthogonal decomposition
of F:
QF = Rf · exdx+Rf · eydy + (q×Rf ) · (eydθ + ezdϕ+ ezdψ).
1In [16] the origin of the reference system attached to the body is in the center of the sphere
C, and not in the center of mass O.
Modeling the friction force One typically models the friction force F = Rf +
Rnez to be proportional to the slip velocity of the point of contact vQ. We denote
by Rn = Rnez the normal reaction of the floor at Q, which is of order mg, and
Rf = FXex+FY ey is the (sliding) friction which opposes the slipping motion of the
body. The fact that the sliding friction opposes the slipping motion is expressed by
Rf · vQ ≤ 0. We adopt a viscous friction law [1, 8, 9, 20] and assume that
Rf = −µRnvQ. (2.3)
Here µ is a coefficient of friction with the dimension of (velocity)−1. It now takes
a few tedious computations to arrive to the coordinate expressions for the force
moments of the friction force. The coordinates of the point of contact ~OQ := q
are Q = (xQ, 0, zQ)Oxyz = (R sin θ, 0, ǫ − R cos θ)Oxyz . The velocity of the point of
contact Q equals
vQ = vO + ω × q, (2.4)
vO = (ẋ, ẏ, h
′(θ)θ̇)Mxyz is the velocity of the center of mass. A coordinate expression
for the angular velocity is given by
ω = −ϕ̇ sin(θ)ex + θ̇ey + nez, where n := ψ̇ + ϕ̇ cos(θ) (2.5)
n is the spin (that is, the component of ω about Oz). The generalized force moments
now read:
Qx = −µRn(ẋ− sinϕ θ̇(R − ǫ cos θ) + cosϕ sin θ(Rψ̇ + ǫϕ̇))
Qy = −µRn(ẏ + cosϕ θ̇(R− ǫ cos θ) + sinϕ sin θ(Rψ̇ + ǫϕ̇))
Qθ = −µRn(R − ǫ cos θ)(cosϕẏ − sinϕẋ+ (R − ǫ cos θ)θ̇)
Qϕ = −µRnǫ sin θ(cosϕẋ+ sinϕẏ + sin θ(ǫϕ̇+Rψ̇))
Qψ = −µRnR sin θ(cosϕẋ+ sinϕẏ + sin θ(Rψ̇ + ǫϕ̇)).
For the sake of completeness we write an explicit expression for the normal compo-
nent of the reaction force Rn, which can be determined from Newton’s law for the
center of mass of the sphere:
Rn(θ, ϕ, θ̇, ϕ̇, ψ̇, ẋ, ẏ) =
g + θ̇2h′′ + h′ϕ̇ sin(θ)(ϕ̇ cos(θ)− C(ψ̇ + ϕ̇ cos θ)/A)
1/m+ h′/A[−hµ(sinϕẋ− cosϕẏ − θ̇h) + h′]
(2.6)
To conclude this section, we briefly discuss other models for the tippe top. The
eccentric sphere model does not accurately model the contact effects of the tippe
top stem. However, it does describe the fundamental phenomenon of the over-
turning. For the sake of completeness we remark that as soon as the top rises to
spin on its stem, it behaves as a ‘normal’ spinning top with rounded peg, we refer
to [12, Chapter 6] for a satisfactory introduction to this topic. Note that our model
also does not describe the peculiar ‘Hycaro’ tippe tops by Prof. T. Tokieda [18],
which need a non-axisymmetric asymmetric mass distribution. These tippe tops
have a ‘preferred direction’ meaning that the top would flip over only if spun in the
preferred direction with a certain initial spin, and, no matter what the initial spin
is, it would just continue rotating around the rest position when spun the other
way round. We refer to [4] for a detailed analysis when elasticity properties of the
horizontal surface and tippe top are taken into account. Their model allows for
jumps of the tippe top on the horizontal surface (that we assumed to be rigid).
Finally, we mention that we do not debate over the issue of whether transitions
sliding-rolling and rolling-sliding occur in the motion of an eccentric sphere on a
flat surface. We chose to concentrate on the sliding model only, because we were
interested in capturing the ‘overturning’ phenomenon which cannot occur under the
non-holonomic constraint of pure rolling. We refer the interested reader to [1, 5, 8]
for a discussion of the topic and to [3, 13, 19] for an analysis of the motion of the
rolling eccentric sphere also called the Routh’s sphere.
2.1 Constants of motion: the Jellet invariant
It was first shown by Jellet [6] by an approximate argument, and later proved by
Routh [17] that the system, even if dissipative, has a conserved quantity:
J = −L · q = const, (2.7)
where L is the angular momentum of the tippe top about the center of mass.
We prove this by using Euler equations which govern the evolution of the angular
momentum L̇ = q× F. The total time derivative of J then becomes:
J̇ = −L̇ · q− L · q̇
= 0− (Aω − (A− C)(ω · ez)ez) · (ǫω × ez) = 0 .
Straightforward calculations show that Jellet’s constant can be written as
J = Cn(R cos(θ)− ǫ) +Aϕ̇R sin2(θ) . (2.8)
We emphasize once more that the Jellet’s constant is an exact constant of motion
for the tippe top whether or not there is slipping and independent of the expression
for F. As we will explain later, it is this constant that to some extent controls the
motion of the spinning top. Indeed, it allows a Routhian reduction procedure (see
Sec. 2.2), resulting in relatively simple reduced equations from which we are able
to recover in full detail the stability properties of the steady states. In the specific
case that θ = 0, the Jellet is proportional to n0 = n|θ=0, the spin about the z-axis.
Since one typically sets off the tippe top at an angle θ ≈ 0, one can say that the
Jellet is proportional to the initial spin n0. Note that the spin at θ = π has an
opposite sign to the initial spin n0, meaning that, relative to a body fixed frame,
the spin is reversed when the tippe top fulfills a complete inversion.
There is a rotational symmetry for which the Jellet is the associated first integral.
The action of S1 on IR2 × SO(3) can be defined as a simultaneous rotation about
ê3 over the angle Rξ and about k̂ over the angle −ǫξ, where ξ ∈ S1 (see also [16]).
Noether’s theorem is applicable in this situation since the work of the friction force
at the point of contact vanishes under this action.
Note that the total energy of the spinning top is E = T + V is in general not
conserved. Here T is the kinetic energy with its rotational and translational part,
V = mgh(θ) is the potential energy. The orbital derivative of E is
E = vQ ·Rf ≤ 0, (2.9)
which is negative semi-definite and vanishes if and only if vQ vanishes. Observe that
E(t) decreases monotonically and hence is a suitable Lyapunov function2, see [7].
From (2.9) it follows that dissipation is due to friction.
2E(t) is analytical, therefore it is either strictly monotone or a constant. The energy E is
constant only if vQ = 0. Note that E being Lyapunov [7] implies that the limiting solutions for
t → ∞ are solutions of constant energy.
2.2 Routhian reduction
It turns out that, if we consider an approximation of the friction law, the resulting
generalized force moments assume a form that allows us to apply a Routhian re-
duction procedure, see [22] and appendix A. In turn, using the reduced equations
we are able to study in full detail the stability properties of the tippe top which
confirm the results obtained in [15], and also recover those of [16, 20].
We now ignore translational effects in the friction force, i.e. we assume that all
terms in Qθ, Qϕ, Qψ containing ẋ and ẏ are neglected. Typically this approximation
is justified by noting that for all steady states the velocity of the center of mass is
zero, and that in a neighborhood of the steady states it can be neglected. In our
situation, it allows to restrict ourselves to a system on SO(3) which is reducible
using Routh’s procedure. It is easily seen that within this approximation, if we
study the Lagrangian system on SO(3) determined by
L′ = 1
(ǫ2m sin2 θ +A)θ̇2 +A sin2 θϕ̇2 + C(ψ̇ + ϕ̇ cos θ)2
−mg(R− ǫ cos θ),
Q′ = −µR′n(R− ǫ cos θ)2θ̇dθ − µR′nǫ sin2 θ(ǫϕ̇+Rψ̇)dϕ
−µR′nR sin2 θ(Rψ̇ + ǫϕ̇)dψ,
with R′n(θ, θ̇, ϕ̇, ψ̇) = Rn(θ, ϕ, θ̇, ϕ̇, ψ̇, ẋ = 0, ẏ = 0), then we essentially study the en-
tire approximated system. Indeed, any solution (ϕ(t), θ(t), ψ(t)) to this Lagrangian
system will determine the remaining unknowns (x(t), y(t)) as solutions to the fol-
lowing system of time-dependent second order differential equations:
mẍ = Qx(ẋ, θ(t), ϕ(t), ψ(t)), mÿ = Qy(ẏ, θ(t), ϕ(t), ψ(t)). (2.10)
Our next step in the reduction procedure is to consider the Lagrangian system L′
on SO(3) with generalized force form Q′ and perform a simple coordinate transfor-
mation, determined by
(θ, ϕ, ψ) 7→ (θ, ϕ = ǫϕ+Rψ, c = Rϕ− ǫψ).
The Lagrangian L′ and the force Q′ then become
L′ = 1
(ǫ2m sin2 θ +A)θ̇2 + A sin
(ǫ2+R2)2
(ǫϕ̇+Rċ)2+
(ǫ2+R2)2
(R+ ǫ cos θ)ϕ̇+ (R cos θ − ǫ)ċ
−mg(R− ǫ cos θ)
Q′ = −µR′n
(R − ǫ cos θ)2θ̇dθ + sin2 θϕ̇dϕ
where it is understood that R′n is a function of θ, θ̇, ϕ̇ and ċ. The main reason
for writing L′ and Q′ in this form is the fact that c is a cyclic coordinate and
that Q′c = 0. Indeed, recall that Jellet integral was associated to the symmetry
determined by the vector field R∂ϕ − ǫ∂ψ, or in the above introduced coordinate
system by the vector field ∂c. In particular, we have made the symmetry generator
into a coordinate vector field, and this ensures that we can apply the Routhian
reduction procedure ([22] and appendix A), provided the coefficients of Q′ do not
depend on c. This is the case since we neglected the terms in the velocity of the
center of mass. The conserved quantity associated with the cyclic coordinate is
precisely the Jellet integral:
= RA sin
(ǫ2+R2)2
(ǫϕ̇+Rċ) +
C(R cos θ−ǫ)
(ǫ2+R2)2
(R+ ǫ cos θ)ϕ̇+ (R cos θ − ǫ)ċ
= J/(ǫ2 +R2).
The latter equality implies that
J(ǫ2 +R2)− (RAǫ sin2 θ + C(R cos θ − ǫ)(R+ ǫ cos θ))ϕ̇
R2A sin2 θ + C(R cos θ − ǫ)2
(2.11)
The Routhian reduction procedure defines a Lagrangian system
R = L′ − Jċ/(ǫ2 +R2)
with two degrees of freedom (θ, ϕ) (here every instance of ċ in R is replaced us-
ing (2.11)). The reduced equations of motion are then given by
= Q′θ = −µR′n(R− ǫ cos θ)2θ̇ (2.12)
= Q′ϕ = −µR′n sin2 θϕ̇, (2.13)
where it is understood that (2.11) is used to eliminate ċ in R′n. It takes rather
tedious but straightforward computations to show that R can be written as
R = T2 + T1 −W,
where
T2 = Tθθ(θ)θ̇
2 + Tϕϕ(θ)ϕ̇
(ǫ2m sin2 θ +A)θ̇2 +
AC sin2 θ
(R2A sin2 θ + C(R cos θ − ǫ)2)
(R2+ǫ2)
(RǫA sin2 θ+C(R+ǫ cos θ)(R cos θ−ǫ))
R2A sin2 θ+C(R cos θ−ǫ)2 ϕ̇,
W = 1
R2A sin2 θ+C(R cos θ−ǫ)2 −mgǫ cos θ.
The function W is also called the effective potential.
Remark 1 The above defined Routhian function R is not globally defined on the
sphere. In order to provide a globally defined system of differential equations for the
reduced system, we need to extract the term T1 from the Routhian and consider it
as a gyroscopic force (see e.g. [21]).
Remark 2 Observe that the effective potential W , obtained through reduction, co-
incides with the effective energy on a Jellet’s level surface as it has been used in
[15], Sec. 3.
Note that R and Q′ are independent of ϕ. This residual symmetry does not lead
to a conserved quantity (the friction does not vanish for ∂ϕ). This symmetry is
due to the approximation we carried out in the previous section; it is not present
in the original system (L,Q) or (2.2). It leads however to a zero eigenvalue of the
linearized system at equilibrium points, see Section 4.
3 Steady states
The equilibria of the reduced Routhian system (2.12) and (2.13) are determined by
θ̇ = 0, ϕ̇ = 0 and ∂W/∂θ = 0, (3.1)
i.e. they are the critical points of the effective potential (note that if θ̇ = ϕ̇ = 0 the
components of the force vanish). Equation (3.1) is satisfied if (i) sin θ = 0 or, if (ii)
f(J2, cos θ) = J
mgCR2ǫ
cos θ + ǫ
sin2 θ +
cos θ − ǫ
(3.2)
Solutions to (i) are θ = 0, π and give the vertical spinning states. Solutions to (ii)
only occur if
cos(θ) +
> 0. (3.3)
If this condition is satisfied, solutions to (ii) are the so-called intermediate states.
The existence condition only depends on A/C and ǫ/R, and will determine in our
classification the three main groups I, II and III, as it was proposed in [20]. The
values for θ for which (ii) is satisfied will depend on the Jellet, A/C and ǫ/R.
The vertical spinning states correspond to the periodic motion of the tippe top
spinning about its axle (which is in vertical position) either in the non-inverted
position (θ = 0) or inverted position (θ = π). The intermediate states correspond
to those relative equilibria in which the tippe top shows in general quasi-periodic
motion precessing about a vertical while spinning about its inclined axle rolling over
the plane without gliding (observe that the intermediate states correspond to the
tumbling solution of [7]).
The condition (3.3) for existence of intermediate states leads to a first classifi-
cation of tippe tops into three groups.
- Group I [(A/C − 1) < −ǫ/R]: the tippe tops belonging to this group do not ad-
mit intermediate states in an interval of the form [0, θc[, where θc is determined by
(A/C − 1) cos θc + ǫ/R = 0.
- Group II [−ǫ/R < (A/C−1) < ǫ/R]: intermediate states may exist for all θ ∈]0, π[.
- Group III [(A/C − 1) > ǫ/R]: tippe tops belonging to this group do not admit
intermediate state in an interval of the form ]θc, π[, where cos θc = (ǫ/R)/(1−A/C).
In the following section we refine this first classification taking into account the
stability type of the steady states and their bifurcations, explaining and giving the
details of the J2 versus θ diagrams in Fig. 1 from the introduction.
We anticipate that the subdivision in subgroups according to a change in sta-
bility type of the intermediate states is based on the simple observation that they
lie on a curve f(J2, cos θ) = 0 in the (J2, θ)-plane, and, denoting by ∂2f the partial
derivative to the second argument of f , a bifurcation point for intermediate states
is characterized as the point where
f(J2, cos θ) = 0,
∂2f(J
2, cos θ) = 0.
Note that the relation ∂2f(J
2, cos θ) = 0 is essentially the same basic relation (4.27)
in [15], however its derivation is different. As a consequence we expect the stability
results for intermediate states to confirm earlier known facts.
4 Stability analysis via the reduced equations
Determining the (linear) stability of the steady states as given above is an extremely
simple task in the reduced setting. Indeed, let (θ0, ϕ0) be an equilibrium. The
linearized equations of motion at this relative equilibrium read as
Tθθ(θ0)θ̈ =
∂ϕ̇∂θ
(θ0)ϕ̇ −
(θ0)(θ − θ0)− µmg(R− ǫ cos θ0)2θ̇
Tϕϕ(θ0)ϕ̈ = −
∂ϕ̇∂θ
(θ0)θ̇ − µmg sin2 θ0ϕ̇,
where we used that R′n equals mg (2.6) at the relative equilibria. It is not hard to
show that the characteristic polynomial of this system is
p(λ) = λ
λ3 + µmg
(R− ǫ cos θ0)2
Tθθ(θ0)
sin2 θ0
Tϕϕ(θ0)
µmg(R− ǫ cos θ0) sin θ0
∂ϕ̇∂θ
Tθθ(θ0)Tϕϕ(θ0)
Tθθ(θ0)
µmg sin2 θ0
Tθθ(θ0)Tϕϕ(θ0)
Due to the translational symmetry in ϕ, one eigenvalue is zero. In Appendix B
we show that all remaining eigenvalues have a negative real part, if and only if
∂2W/∂θ2(θ0) > 0, or if
mgǫ cos θ0 >
J2R2C
(R2A sin2 θ0+C(R cos θ0−ǫ)2)2
− (A/C − 1) sin2 θ0
cos θ0 − 4 B
sin2 θ0
(A/C) sin2 θ0+(cos θ0−ǫ/R)2
, (4.1)
with B given by B := (A−C) cos(θ0)+C ǫR .We will further manipulate this equation
to retrieve the stability results, compare also with [20]. We will retrieve six groups
depending on how the inertia ratio A/C relates to the eccentricity ǫ/R. Since in
the literature results have been expressed in terms of the spin of an initial condition
at a vertical state, we introduce n0 :=
C(R−ǫ) , which is the value of the spin n at
θ = 0 for a given Jellet J . Similarly, nπ := − JC(R+ǫ) is the spin of the solution with
Jellet J at θ = π. Note that for a fixed J these spins are related by n0 = −nπ R+ǫR−ǫ .
Vertical spinning state: θ = 0. For the vertical spinning state θ = 0, the
relation (4.1) yields
− (1 − ǫ
. (4.2)
It follows that in Group I, the vertical state θ = 0 is always stable, while for Group
II and Group III stability requires that
|n0| < n1 :=
−(1− ǫ
. (4.3)
Vertical spinning state: θ = π. For the vertical spinning state θ = π, the
relation (4.1) yields
. (4.4)
This condition is never satisfied for Group III, so θ = π is unstable; in the case of
Group I and II, when A
< (1 + ǫ
), stability requires
|nπ| > n2 =:
C[(1+ ǫ
1 + ǫ
. (4.5)
Note that for tippe tops of Group I and II n22 ≥ n2∗, with n∗ := 2
. The
equality holds when A
1 + ǫ
Intermediate states. Recall that intermediate states are determined by (3.2):
f(J2, cos θ0) = 0. Using this condition, the requirement (4.1) becomes 0 < g(cos θ0)
where we set
g(cos θ0) :=
−1) cos θ+ ǫ
sin2 θ+(cos θ− ǫ
. (4.6)
We now prove that g(cos θ0) is strictly increasing and changes sign at a bifur-
cation point for intermediate states. As we already mentioned, a bifurcation point
along the curve in the (J2, cos θ)-plane of intermediate states is determined by the
conditions
f(J2, cos θ) = 0,
∂2f(J
2, cos θ) = J
mgǫR2C
− 1) +
− 1) cos θ + ǫ
(1 − cos2 θ) + (cos θ − ǫ
)2) = 0.
An elementary substitution of the first equation into the second shows that the
function ∂2f(J
2, cos θ) along the intermediate states can also be written as
∂2f(J
2, cos θ) = g(cos θ)
mgǫR2C
Hence, bifurcation points are given by those θ such that g(cos θ0) = 0. Solving for
cos θ gives the two solutions:
1−A/C
1−A/C
1−A/C − (ǫ/R)2. (4.7)
Note that 1−A/C − (ǫ/R)2 > 0 only for tippe tops in Groups I and II. Moreover,
the solution cos θb =
1−A/C + . . . leads to a contradiction since, for tippe tops of
Group I, it is incident with the interval ]0, θc[ where no intermediate states exist
and for tippe tops in Group II satisfying 1−A/C − (ǫ/R)2 > 0, the number ǫ/R
1−A/C
is greater than 1, implying that the + solution of (4.7) can not equal a cosine. We
denote the − solution by xb, i.e.
xb :=
1−A/C −
1−A/C
1−A/C − (ǫ/R)2.
We conclude that a bifurcation point for intermediate states exists if 1 − A/C −
(ǫ/R)2 > 0 and |xb| < 1.
Before studying in further detail these two conditions, we first show that the
function g(cos θ) is strictly increasing for increasing θ. This result implies that, if
a bifurcation exists (i.e. a point θ with g(cos θ) = 0) then stability will change. On
the other hand, if no bifurcation occurs, the entire branch of intermediate states is
either stable or unstable.
Let us assume that x = cos θ. If we consider ∂2f(J
2, x) as a function on the
submanifold f(J2, x) = 0 and if we compute its derivative w.r.t x, i.e. ∂2,2f −
(∂1,2f)(∂2f/∂1f), then after some tedious computations we may conclude that the
sign of this derivative is opposite to the sign of
8((A/C − 1)x+ ǫ/R)2 + (A/C − 1)2 (A/C(1−x
2)+(x−ǫ/R)2)2
((A/C−1)x+ǫ/R)2 > 0
Hence, if there is a bifurcation at a certain x = cos(θb) in the set of intermediate
states, then we know that the intermediate states for which θ > θb are stable, while
the other branch is unstable.
It now remains to study the conditions for the bifurcation point to exist. The
first condition says that 1 − A/C − (ǫ/R)2 > 0, implying that we only have to
consider Groups I and II. We start with Group I.
Group I From (A/C − 1) < −ǫ/R, it follows that 1 − A/C − (ǫ/R)2 > 0 and
xb < 1. We have to distinguish between two subgroups: xb < −1 (Group Ia)
and xb > −1 (Group Ib). From the previous paragraph it should be clear that if
xb < −1 then the value of the function g(x) on the intermediate states −1 < x < 1
is negative, i.e. the entire branch is unstable.
Group II We define three subgroups: Group IIa is the group for which 1−A/C−
(ǫ/R)2 > 0 and |xb| < 1, Group IIc is defined by 1−A/C− (ǫ/R)2 > 0 and xb < −1
and thirdly Group IIb as the group containing the remaining tippe tops in II. Again
from the previous, we immediately conclude that the entire branch of intermediate
states is unstable in Group IIc. Group IIb can alternatively be defined as the group
containing all tippe tops for which the branch of intermediate state is entirely stable.
To show this, we first remark that the remaining tippe tops in II are characterized
by 1−A/C−(ǫ/R)2 > 0 and xb > 1 or 1−A/C−(ǫ/R)2 ≤ 0. If 1−A/C−(ǫ/R)2 > 0
and xb > 1 then the intermediate branch is entirely stable. If 1−A/C− (ǫ/R)2 = 0
then xb > 1 and the intermediate branch is stable. The remaining tippe tops we
have to consider are characterized by the condition 1−A/C− (ǫ/R)2 < 0. To show
that g(x) > 0, we consider two subcases: (i) if 1−A/C < 0 then from (4.6) it is clear
that g(x) > 0, (ii) if (ǫ/R)2 > 1−A/C > 0 then it suffices to compute g(0) (g(x) will
not change sign since there is no bifurcation point xb): from 1−A/C− (ǫ/R)2 < 0,
A/C < 1 and (ǫ/R)2 < 1 we find
g(0) = (A/C − 1) + 4 (ǫ/R)
A/C+(ǫ/R)2
> (ǫ/R)2 > 0.
The above argument also proves stability for intermediate states in Group III.
Note that tippe tops in Group II are real ‘tippe tops’ since they admit tipping
from a position near θ = 0 to the inverted state near θ = π. Tipping never occur for
tops of Group III, though they may rise up to a (stable) intermediate state. Tippe
tops of group I never flip over since the position θ = 0 is always stable.
4.1 Tippe Top Classification
The following schematic classification summarizes the previous analysis, and pre-
sented in Fig. 1. Compare also with Fig. 3 in [15].
Group I: A/C − 1 < −ǫ/R
- The non-inverted vertical position θ = 0 is stable for any value of J .
- The inverted vertical position θ = π is stable for |nπ| > n2, unstable otherwise,
with n2 given by (4.5).
- Intermediate states do not exist for all values of θ, but only for θ > θc =
arccos(( ǫ
)/(1− A
Group Ia:
1−A/C −
1−A/C
1−A/C − (ǫ/R)2 < −1 .
The entire branch of intermediate states is unstable.
Group Ib: if −1 < ǫ/R
1−A/C −
1−A/C
1−A/C − (ǫ/R)2 := cos θb .
There is a bifurcation: intermediate state are stable if θ > θb and unstable if
θ < θb.
Group II: −ǫ/R < (A/C − 1) < ǫ/R.
- If |n0| > n1 the equilibria θ = 0 become unstable, with n1 as in (4.3).
- If |nπ| > n2 the equilibria θ = π become stable.
- There are intermediate states for any θ. We distinguish the following three sub-
groups.
Group IIa: (A/C − 1) < −(ǫ/R)2 and
1−A/C −
1−A/C
1−A/C − (ǫ/R)2
< 1 .
There is a bifurcation of intermediate states.
Group IIb: either (A/C − 1) ≥ −(ǫ/R)2 or
1−A/C −
1−A/C
1−A/C − (ǫ/R)2 > 1 .
The branch of intermediate states is entirely stable.
Group IIc: (A/C − 1) < −(ǫ/R)2 and
1−A/C −
1−A/C
1−A/C − (ǫ/R)2 < −1 .
The branch of intermediate states is entirely unstable.
Group III: (A/C − 1) > ǫ/R.
- The equilibria with θ = 0 become unstable for |n0| > n1.
- The equilibria with θ = π are always unstable.
- For these tippe tops intermediate states do not exist for θ ∈]θc, π[. Bifurcations
in intermediate states do not occur and the intermediate states are stable.
Tippe tops in Group II exhibit ‘tipping’ behavior: if the initial spin satisfies |n0| >
max(n1, n2
R−ǫ )
3, then tipping is possible from θ = 0 to θ = π.
5 Remarks and Conclusions
We would like to remark that the results on the linear stability for the tippe top
presented here are equivalent to the stability properties for the model of the tippe
top without the assumption that the translational friction terms can be neglected.
In fact, the function g(cos θ) which fully determines the stability of the intermediate
states can be retrieved in the expressions for the eigenvalues of the linearized equa-
tions of motion about the relative equilibria of the full system. It is also remarkable
that the presented stability analysis does not depend on the friction coefficient µ,
which suggests that the above stability analysis is valid for a rather large class of
possible dissipative friction forces at the point of contact as was pointed out in [15].
Up until now, a linear stability analysis for the intermediate states was not available
upon our knowledge in the literature.
Most recently Ueda et al. [20] analyzed the motion of the tippe top under the
gyroscopic balance condition (gbc) and approached the stability problem by per-
turbing the system around a steady state and obtained under linear approximation
a first order ode for the perturbation of the variable θ. They derive stability crite-
ria in terms of the initial spin n given at the non-inverted position θ = 0. Possible
3About the relation between n1 and n2: in Group IIb n
> 1 holds and in Group IIc
intermediate states for tippe tops of Group I were not considered. Our approach is
based on a complete different technique, namely Routhian reduction and we obtain
a more refined and exhaustive classification of tippe tops in six groups (instead of
three). Moreover, our analysis does not exclude the possibility of launching the top
with its stem down (i.e. θ near π).
Acknowledgments
The results presented here were obtained while the first author had financial support
by the European Community’s 6th Framework Programme, Marie Curie Intraeuro-
pean Fellowship EC contract Ref. MEIF-CT-2005-515291, award Nr. MATH P00286
and the second author was Postdoctoral Fellow of the Research Foundation – Flan-
ders (FWO) at the Department of Mathematical Physics and Astronomy, Ghent
University, Belgium.
The authors wish to thank Dr. B. Malengier, Prof. F. Cantrijn, and Prof. J. Lamb
for stimulating discussions and the anonymous referee for pointing out reference [15].
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A Routhian reduction for dissipative systems.
Assume that a Lagrangian L is given, defined on the tangent space of a manifold
M on which a local coordinate system (q1, . . . , qn) is chosen. The system is not
conservative, in the sense that there is a one form on M , denoted by Q = Qidq
representing the generalized force moments of the non-conservative forces. The
equations of motion read:
= Qi for i = 1, . . . , n . (A.1)
Theorem 2 (cf. [22]) Assume that there exists a coordinate, say q1 such that
(i) ∂L
= 0, (ii) Qi(q
2, . . . , qn) is independent of q1 for all i = 1, . . . , n, (iii)
2, . . . , qn) = 0; then β = ∂L/∂q̇1 is a constant of the motion. Assume that the
latter equation is invertible and allows us to write q̇1 = f(q2, . . . , qn, q̇2, . . . , q̇n, β).
The system (A.1) is equivalent to the system
= Qi for i = 2, . . . , n , (A.2)
where the new Lagrangian R is defined by R = L− q̇1β and such that all occurrences
of q̇1 in R and Qi are replaced by f .
By equivalent systems we mean that any solution to (A.1) with fixed momentum
∂L/∂q̇1 = β is a solution to (A.2) and vice versa.
B Roots of the characteristic polynomial
We show that the roots of the polynomial
λ3 + µmg
(R− ǫ cos θ0)2
Tθθ(θ0)
sin2 θ0
Tϕϕ(θ0)
µmg(R− ǫ cos θ0) sin θ0
∂ϕ̇∂θ
Tθθ(θ0)Tϕϕ(θ0)
Tθθ(θ0)
µmg sin2 θ0
Tθθ(θ0)Tϕϕ(θ0)
all have negative real parts if and only if ∂
(θ0) > 0. For our convenience we use
the following shorthand notations
µmg(R− ǫ cos θ0)2
Tθθ(θ0)
> 0, β :=
∂ϕ̇∂θ
Tθθ(θ0)Tϕϕ(θ0)
µmg sin2 θ0
Tϕϕ(θ0)
> 0 and δ :=
Tθθ(θ0)
and the polynomial under consideration becomes
λ3 + (α+ γ)λ2 + (αγ + β2 + δ)λ+ δγ.
Note that if sin θ0 = 0 the system (2.13) is singular and a coordinate change is
necessary. The characteristic polynomial written above is not singular. Indeed, we
have that Tϕϕ(θ0) is proportional to sin
2 θ0 or that γ is well-defined in the case
sin θ0 = 0. Similarly, we have that
∂ϕ̇∂θ
(θ0) is proportional to sin θ0, and β is well-
defined. If we now write the necessary and sufficient conditions for this polynomial
to be Hurwitz [10], we obtain the conditions
(α+ γ) > 0, (α+ γ)(αγ + β2 + δ)− δγ > 0 and δγ > 0.
The first condition is trivially satisfied. We now show that the third condition δ > 0
implies the second condition. We can rewrite the second condition as
(1 + α/γ)(αγ + β2 + δ) > δ,
and this is valid if δ > 0 or if ∂
(θ0) > 0.
	1 Introduction
	2 Equations of motion
	2.1 Constants of motion: the Jellet invariant 
	2.2 Routhian reduction
	3 Steady states 
	4 Stability analysis via the reduced equations
	4.1 Tippe Top Classification
	5 Remarks and Conclusions
	A Routhian reduction for dissipative systems.
	B Roots of the characteristic polynomial
