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##  Appendix A Caveats

Here we discuss some of the key caveats that should be considered when
interpreting the results of our study.

###  A.1 Estimating the overdensity

In this study, we assume that galaxies can serve as tracers of the underlying
matter density field and measure the overdensity in terms of galaxies (e.g.
Cucciati et al. 2014). Different types of galaxies, such as quiescent or star-
forming, trace the matter density field differently. Sometimes matter density
is not traced very well by UV-selected galaxies, as in the case of Newman et
al. (2022), which finds unexpectedly low galaxy overdensity where large-scale
Ly$\alpha$ absorption is strongest indicating high matter density. Moreover,
observations might not trace the true underlying galaxy population as low
luminosity galaxies are harder to detect, especially at high redshifts. By
considering the mass of galaxies and their star formation rates, we determine
an appropriate bias factor from simulations that scales the galaxy overdensity
to the matter overdensity. For this study, we use a bias factor of $b=3.6$,
which is based on previous research (e.g. Chiang et al. 2013, Durkalec et al.
2018) and the upper and lower limit for the bias factor are 4.5 and 3.12
respectively obtained from Einasto et al. (2023) and Ata et al. (2021).

###  A.2 SED Fitting

The SFR derived from SED fitting method correlates well with the SFR derived
from independent measures such as [CII] lines at intermediate and high
redshift (e.g., Schaerer et al. 2020). We checked if this correlation holds
for galaxies that we obtained from Y. Fudamoto in _private communication_
based on serendipitously detected [CII] lines with ALMA. These three galaxies
have log($L_{\text{CII}}/L_{\odot})$ of 8.99, 9.01 and 9.10 respectively. When
we compared the [CII] derived SFR with the SFR values obtained for these three
galaxies through SED fitting using the parameter set given in Section 5.1, we
found that the SED-fit SFR values are consistently lower than the [CII]
derived SFR. If we force the fits such that the best model chooses the high
end of the E(B-V) value range, the difference between the [CII]-derived SFR
and the SED-fit value decreases. However, such fits are clearly disfavored by
comparing their reduced $\chi^{2}$. Note that recent theoretical and
observational studies suggest that [CII] is more tightly connected to the
molecular gas mass in a complicated way through Kennicut-Schmidt relation
(e.g., Zanella et al. 2018; Madden et al. 2020; Vizgan et al. 2022) than to
SFR.

###  A.3 Absence of FIR data

We tested the reliability of SFR derived from SED fitting with optical/NIR
data (used in this study). We selected 12 galaxies from ALPINE survey that
fall in the redshift range of $4.2<z<4.93$ with significant detection in the
continuum around restframe $155\mu m$. We performed the SED fitting on these
galaxies with the same parameters as listed in Table 10 along with (Casey et
al., 2012) dust template. We found that, relative to the fits that include the
FIR data, the optical/NIR-only fits show higher SFRs by $\sim 0.2$ dex. For
these galaxies, the extinction correction is generally overestimated by the
optical/NIR-only fits. Because these 12 galaxies may or may not be
representative of the true galaxy population, as these galaxies are likely the
more massive/more dusty extreme of the ALPINE population, we decided to not
make any changes to our analysis. Programs probing sub-mm galaxies in
protoclusters are needed in order to get a fuller perspective on the star
formation in protoclusters. To check how much the result of our study would
change if, indeed, what we find were a generally applicable result, we reduced
the SFR of each galaxy in our analysis, SFRs that are derived by optical/NIR-
only fitting, by 0.2 dex and performed the analysis again. The result was that
the SFRD of Taralay and the field were reduced, but well within the
uncertainties. The fractional contribution of Taralay-like systems to overall
SFRD at these redshifts remained the same.

###  A.4 Dust properties of bright and faint galaxies

Here we test our assumption that the bright and faint galaxies experience the
same level of dust extinction which let us extrapolate the SFRD results to
include the lower luminosity galaxies in Section 5.3. If this assumption is
wrong, we are either underestimating or overestimating the SFR of the fainter
galaxies. This is because the different dust extinction, through the process
of SED fitting, will cause the SFR to be different for the fainter galaxies
compared to the SFR of the brighter galaxies. To test this assumption, we
analyse the SED models for nearly 3000 galaxies at $2.5<z<3.5$ in the COSMOS
field. These galaxies make up our sample because at this redshift range we can
probe deeper than $M_{\text{FUV}}=-19.3$, the depth of our data, at
$4.2<z<4.93$. For this exercise, a galaxy is considered bright if its FUV
absolute magnitude is less than -19, the delineation point between bright and
faint.

We perform a KS test on the $E(B-V)$ distribution of both bright and faint
galaxies at $2.5<z<3.5$, and find that they prefer the same range of values
for the color excess of the nebular line light, for both young and old
populations. The range for E_BV_lines is presented in Table 10. This finding
shows that at $2.5<z<3.5$, the SFR traced by FUV photons is similarly affected
by the dust properties of both faint and bright galaxies. We assume that this
behaviour is consistent for galaxies at $4.2<z<4.93$ and make no further
changes to our FUV magnitude correction factor.

###  A.5 Comparison to Simulations

Our study diverges from the definition and characterization of protoclusters
as outlined in Chiang et al. (2017) in several key aspects. In their work, a
protocluster encompasses all the dark matter and baryonic matter that will
eventually merge into a cluster by $z=0$, with a mass exceeding
$10^{14}M_{\odot}$ within $R_{200}$. The protocluster’s volume encompasses all
the matter that will contribute to the formation of the cluster by $z=0$. In
contrast, our study defines a protocluster as a structure contained within the
$2\sigma$ density isopleth.

Furthermore, while Chiang et al. (2017) designates the most massive halo
within the protocluster as its core at any given epoch, having a size of $\sim
0.4$ cMpc at $z\sim 4.57$, our study identifies the protocluster core as the
region bounded inside a $5\sigma$ density isopleth with a size $\sim 2$ cMpc.
To put our peaks on the same footing as the core defined in Chiang et al.
(2017), our peak size will have to reduce beyond $\sigma_{\delta}>10$ where we
have spectroscopic data only for $\sim 3$ galaxies. Hence we adopt our
definition for peaks. It is crucial to acknowledge that these differing
definitions of protoclusters may contribute to the measured difference between
the observed and simulated data.

###  A.6 Using a Massive Protocluster as a Proxy

Using the density maps, the estimated mass of Taralay protocluster at $z\sim
4.57$ is $1.74\times 10^{15}$ M⊙ making this structure exceptionally massive
at these redshifts.

Using such a massive protocluster, which has highly star forming galaxies, as
a representative sample of all the protoclusters at $z\sim 4.57$ to calculate
the fraction of SFRD from protoclusters (see Figure 10) may lead to an
overestimated value for the derived contribution of protoclusters to the SFRD.
Nevertheless, there is a possibility that environmental quenching is affecting
the galaxies within this protocluster (Lemaux et al., 2018) which could result
in a decrease in the SFRD. To draw a definitive conclusion, a large collection
of protoclusters is required for comprehensively testing their contribution to
the SFRD at high redshift.

##  Appendix B Effect of parameter choices on the model spectrum

The SFR values that our results are based on come from the SED fitting
process. It is important, therefore, to discuss how the different parameters
that are used in the SED fitting process impact the spectrum of a galaxy,
affecting the estimated SFR. We discuss this impact here using two different
approaches. First, we look at the scatter on the SFR of a large sample of
galaxies which sheds light on how the SFR itself varies with different
parameters used for SED fitting. Then we look at a model SED of a single
galaxy to understand how the SED is impacted with the changes in one parameter
while keeping the rest of the parameter inputs constant.

In Figure 11 we investigate the change in SFR values by fixing metallicity and
powerlaw_slope values and by varying E_BV lines range. The sample chosen to
perform this exercise is one hundred galaxies and the same modules are chosen
to estimate their SFR with SED fitting as used in the analysis of this study,
i.e. sfhdelayed, bc03, nebular, dustatt_modified_starburst, redshifting (See
Section 5.1 for the ranges of these parameters). We find that the scatter on
the SFR resulting from changing metallicity is fairly small indicating that a
specific value of metallicity is not a strong driver of change in SFR. The
scatter on SFR from varying the powerlaw_slope and E_BV lines is larger
indicating these parameters may have an impact on the accuracy of the
estimated SFR. The largest systematic uncertainty of $\sim 0.2$ dex coming
from the powerlaw_slope value of -0.5 affecting the SFR of both the
protocluster and the coeval field galaxies equally. Assuming that factors like
IMF, E(B-V), powerlaw_slope do not dramatically change for the two
populations, this uncertainty does not change our results since we compare the
SFR of the two populations.

In the left panel of Figure 12 we compare the SFR estimated with sfhdelayed,
the SFH module chosen for perform this study, with two of the other SFH
modules available from CIGALE to see how the change in our choice of module
can impact our results. We find that the scatter on the SFR is small and a
particular choice of SFH is not likely to impact the estimated SFR
significantly. In the right panel of this figure, we change the library of SSP
to investigate the impact of using bc03 module instead of m2005 module.
Because m2005 module is not compatible with the Chabrier IMF originally used
in the SED fitting of this study, for this exercise, we use the Salpeter IMF
with both bc03 and m2005 module. We do not see a significant difference in the
estimated SFR by choosing different SSP modules.

|
---|---
Figure 11: Comparing SFR values estimated using the ranges of parameters given in Table 10 with the SFR values estimated by forcing metallicity to take on a fix value in the top left panel, changing the range of E(B-V) lines in the top right panel and forcing the powerlaw slope to a fix value in the bottom panel. For each plot, the x axis represents SFR obtained from SED fitting the galaxies using a range of a particular parameter. These ranges for metallicity, E_BV_lines and powerlaw_slope are shown on the x axis of each plot. The y axis represents SFR obtained from SED fitting galaxies using a fixed parameter. For both axes for each plot, the SFR values are in $\log(\text{SFR}/M_{\odot}\text{yr}^{-1})$. The $\sigma_{MAD}$ for a distribution in the each figure shows the scatter on the SFR value. The $\sigma_{MAD}=0$ comes from having a very few outliers with the majority of the galaxies having one to one correlation.  |
---|---
Figure 12: Comparing SFR value obtained from SED fitting using different
modules of SFH and SSP. The x axis of the first plot represents SFR obtained
from doing SED fitting with sfhdelayed module while the y axis represents SFR
obtained from doing SED fitting with sfh2exp and sfhperiodic modules. For the
second plot, x axis represents SFR obtained from doing SED fitting with bc03
module with Salpeter IMF while the y axis represents SFR obtained from doing
SED fitting with m2005 module with Salpeter IMF. For both axes for each plot,
the SFR values are in $\log(\text{SFR}/M_{\odot}\text{yr}^{-1})$. The
$\sigma_{MAD}$ for a distribution in the each figure shows the scatter on the
SFR value.

Next, we investigate the impact of using different parameter ranges on the
spectrum of a single galaxy. The SED of the example galaxy in Figure 13
changes drastically in the restframe UV (shaded region) covered by our data
for age_main of 1000 Myr and E_BV_lines of 0.5 showing that it is mainly these
two factors that can affect the spectrum of a galaxy. However, our data cannot
differentiate between the models with age_main differences of $\sim 100$ Myr
and E_BV_lines differences of $\sim 0.05$ as the variations in the models are
within the uncertainty of our data (typically of order $\sim 1e^{-5}$ mJy for
this example).

|
---|---
|
|
Figure 13:  Rest-frame SED for a single galaxy with different parameters (see
legends of each panels). The SEDs from each panel are such that they have
nearly the same SFRs while changing the parameters indicated in the legends in
the panels. Data points at $\lambda_{\text{rest}}$ = 2000 Å in each panels
indicate typical errors of the observations. The differences in the SED of the
galaxy are within the uncertainty except for the SED obtained with age_main of
1000 Myr and E_BV_lines of 0.5.

##  Appendix C SED fitting with CIGALE vs LePhare

We compare various physical parameters obtained from SED fitting with CIGALE
using the parameter set given in Section 5.1 to the best-fit parameters given
in the COSMOS2020 catalog (Weaver et al., 2022) and find no significant
difference that can be attributed to the choice of using a particular SED
fitting software. Here we show this comparison for stellar mass and star
formation rate. The sample used for comparison consists of 106 galaxies in the
redshift range of $4.2<z<4.93$ that have secure spectroscopic redshifts. This
sample has also undergone an IRAC channel 1 and/or IRAC channel 2 cut such
that the 106 galaxies have magnitudes brighter than the IRAC channel 1 and/or
IRAC channel 2 completeness limit listed in (Weaver et al., 2022).

Figure 14: Difference between stellar mass and star formation rate estimated
with CIGALE and LePhare, two SED fitting software. The $\sigma_{\text{NMAD}}$,
catastrophic outlier rate
($\eta=|\log(\text{X}_{\text{LePhare}})-\log(\text{X}_{\text{CIGALE}})|/(1+\log(\text{X}_{\text{LePhare}}$)
> $3\sigma_{\text{NMAD}}$), and bias which is defined as the median of the
difference between $\log(\text{X}_{\text{LePhare}})$ and
$\log(\text{X}_{\text{CIGALE}})$ are shown for both panels where X represents
SM/M⊙ for the left panel and SFR/M⊙ yr-1 for the right panel.


# Revisiting dissipative motion of a spinning heavy symmetric top and the rise
of the top by friction

V. Tanrıverdi
<EMAIL_ADDRESS>
Address: Bahadın Kasabası 66710 Sorgun-Yozgat TURKEY

###### Abstract

The dissipative motion and the rise of a heavy symmetrical top with a
hemispherical peg are studied. A model taking the fixed point of the top as
the center of the peg is considered when the top completely slips and the
rolling motion is ignored. This is different from existing models like
Jellet’s one. Jellett’s model and pure slipping are compared for different
tops for the rise of the top.

##  1 Introduction

The motion of a heavy symmetric top with one point fixed is one of the
interesting topics of classical mechanics. In some cases, one can consider the
fixed point as the tip of the top [2]. And, for such considerations, the tip
should be taken as a point. On the other hand, in general, the top’s tip is
not completely fixed since the tip is not a point, and the top rolls and slips
on the surface. The rolling motion occurs due to rotations of the top, and the
touchpoint can make some circle-like paths on the surface [3, 4].

One can consider two limiting cases for the motion of the top by accepting
there is not any initial translational motion. In one case, there is not any
slipping, and the top completely rolls on the plane. Rolling motion takes
place when the spin angular velocity is not high and friction is not too low.
This rolling motion can take place together with a periodic slipping [4]. And,
in general, there is not any fixed point when the rolling motion is present.
In the other limiting case, the top completely slips on the surface, and it
always touches at the same point on the surface. Contrary to rolling motion,
slipping takes place if the friction is low enough and the spin angular
velocity is high enough. For the completely slipping case, the fixed point of
the top can be taken as the center of the peg when the bottom of the peg is
hemispherical. We should note that the radius of the peg also affects rolling
and slipping conditions since it changes the velocity of the touchpoint.
Studying the motion of the top with a hemispherical peg is a previously used
method [4, 5, 6, 7].

Dissipative effects are important to explain the motion of the top in daily
life. In previous work, we considered air dissipation and friction at the
touchpoint [2], and that model does not give the rise of the top. At that
work, one can find a short summary of various works related to dissipative
effects.

In most of the situations that we encounter in daily life, friction results in
the ceasing of motion. On the contrary, for the motion of a heavy symmetric
top, friction does something unintuitive: It rises the top. According to Gray,
the first statement that the rise occurs as a result of slipping is given by
Smith [8]. Jellett explicitly gave equations for the rise of the top without
explicitly defining reaction force [6]. And, in that work, the center of mass
is considered as fixed, and by using limit, it is shown that slipping causes
an increase in the precession angular velocity which results in the rise.
Perry considered different observations related to the top and gave verbal
explanations to the rise by considering Jellett’s model [9].

Later various scientists have studied the rise of the top. Fokker observed
that the rise time is shorter for greater radii which is consistent with
Jellett’s model [3]. Hugenholtz gave an explanation to the rise of the top by
”rolling friction” without writing equations explicitly [10]. Braams
considered ”sliding friction” and its effect on the rise, and he stated that
”sliding friction” contributes to the rise of the top in the fast precession,
and he also stated that the ”rolling friction” contributes to the rise without
giving explicit equations [11]. Parkyn has considered the change of reaction
force and change of the center of mass with respect to ground and given
related equations for the motion of the top including the rise term [4]. Yogi
has defined new angular velocities to avoid singularities and considered the
change in the reaction force, and he has solved numerically resulting
equations giving the rise of the top [7]. Moffatt et al. have considered a new
reference frame which is a mixed one of stationary and body reference frames
to study the motion of symmetric rigid bodies without considering the change
of reaction force [12]. We should note that works of Jellett, Parkyn, Yogi and
Moffatt et al. consider the top’s center of mass as fixed, and the rising term
is a function of the distance between the radius of the peg and the center of
mass [4, 6, 7, 12].

Hugenholtz and Braams stated that ”rolling friction” contributes to the rise
of a heavy symmetric top, however, their statement is vague since they did not
give any explicit equations. All existing explicit models related to the rise
of a heavy symmetric top are based on the friction at the touchpoint occurring
as a result of slipping. Some authors have used ”sliding friction” to mention
this, and Parkyn used slipping to describe it. We will use slipping friction
like Parkyn since some people can understand ”sliding friction” as ordinary
sliding, which is not the cause of the rise.

In general, there is not any fixed point of a heavy symmetric top. There is a
drawing in Perry’s work showing the center of mass of the top is fixed while
rolling motion takes place, i.e. figure 32 [9]. On the other hand, one can see
from a video for a rising top, available as supplementary material in Cross’
experimental work [13], that as a result of rolling motion, the center of mass
of the top is not fixed like Perry’s drawing. From that video, one can also
see that the top also nutates and precesses. And, the position of the center
of mass changes due to these besides rolling motion. However, we should note
that there are some cases in which motion takes place similar to Perry’s
drawing, and in some of these cases, the position of the center of mass may
not change. But, this does not describe the motion properly for all cases.
Then, it is possible to say that considering the center of mass as the fixed
point is not true in general. On the other hand, the radial center of the peg
does not change during the nutation and precession, and one can accept it as
the fixed point if the rolling motion is ignored. And, in this work, we will
consider the fixed point of the top as the radial center of the peg and
rotations around it, assuming rolling motion is not present.

Another difference with previous works is that they take the reaction force as
torque affecting the motion of the top which is not in this work. This is
related to the choice of the fixed point. We already mentioned that in some of
the previous works, the reaction force is different from mass times
gravitational acceleration [4, 7]. On the other hand, Quinn and Picard
measured the mass change of a gyroscope during the rotation and find ”no
dependence on speed or sense of rotation” in their experiment [14]. We should
note that Quinn and Picard used a gyroscope with casings which is different
from the symmetric top. Nevertheless, in this work, we will assume that the
reaction of the surface is equal to mass times gravitational acceleration.

In section 2, we will derive equations defining the motion of the top with a
hemispherical peg when the rolling motion is not present and the fixed point
is the center of the hemispherical peg. We will include air dissipation and
slipping friction at the touchpoint. In section 3, we will review Jellett’s
model. In section 4, we will numerically solve obtained equations for a
dissipative experimental situation and two hypothetical tops to study the
rise, and then we will conclude in section 5.

##  2 Pure slipping model

In figure 1(a), one can see a symmetric top, body reference frame $(x,y,z)$
and stationary reference frame $(x^{\prime},y^{\prime},z^{\prime})$ together
with line of nodes $N$. Origins of the reference frames are placed to the
center of the hemispherical peg since it is taken as the fixed point by
considering pure slipping and ignoring rolling motion. In some other models, a
body reference frame whose origin is at the center of mass can be used. We
will use the name the center of peg-body reference frame to emphasize the
difference from that type of model. The hemispherical peg touches the surface
at point $T$ shown in figure 1(b). As the top rotates, the position of the
peg’s center and the point $T$ on the surface do not change. However, the
point on the peg touching the surface changes.

Figure 1: Heavy symmetric top, stationary reference frame
($x^{\prime},y^{\prime},z^{\prime}$), center of peg-body reference frame
($x,y,z$), line of nodes $N$, Euler angles ($\theta,\phi,\psi$), angular
velocities ($\dot{\theta},\dot{\phi},\dot{\psi}$) and the touchpoint T. a)
General view. b) Peg of the top.

Rotations of the symmetric top ($I_{y}=I_{x}$) can be described by Euler
equations which can be written as

$\displaystyle\tau_{x}$ $\displaystyle=$ $\displaystyle
I_{x}\dot{w}_{x}+w_{y}w_{z}(I_{z}-I_{x}),$ $\displaystyle\tau_{y}$
$\displaystyle=$ $\displaystyle I_{x}\dot{w}_{y}+w_{x}w_{z}(I_{x}-I_{z}),$ (1)
$\displaystyle\tau_{z}$ $\displaystyle=$ $\displaystyle I_{z}\dot{w}_{z},$

where $\tau_{i}$, $I_{i}$ and $w_{i}$ correspond to $i^{th}$ component of
torque, moments of inertia and angular velocity in the body coordinate system,
respectively. In terms of Euler angles, angular velocities can be written as

$\displaystyle w_{x}$ $\displaystyle=$
$\displaystyle\dot{\theta}\cos\psi+\dot{\phi}\sin\theta\sin\psi,$
$\displaystyle w_{y}$ $\displaystyle=$
$\displaystyle-\dot{\theta}\sin\psi+\dot{\phi}\sin\theta\cos\psi,$ (2)
$\displaystyle w_{z}$ $\displaystyle=$
$\displaystyle\dot{\psi}+\dot{\phi}\cos\theta,$

where $\dot{\theta}$, $\dot{\phi}$ and $\dot{\psi}$ are nutation, precession
and spin angular velocities, respectively. These can be seen in figure 1.

By considering pure slipping, one can obtain the slipping velocity of the
touchpoint of the top by using
$\vec{v}=\vec{w}\times\vec{r}=\vec{w}\times(-R_{p}\hat{z}^{\prime})$ as

$\displaystyle\vec{v}=$ $\displaystyle R_{p}$
$\displaystyle[(\dot{\theta}\cos\theta\sin\psi+\dot{\psi}\sin\theta\cos\psi)\hat{x}+(\dot{\theta}\cos\theta\cos\psi-\dot{\psi}\sin\theta\sin\psi)\hat{y}$
(3) $\displaystyle+(-\dot{\theta}\sin\theta)\hat{z}],$

where $R_{p}$ is the radius of the hemispherical peg. From this equation, it
can be seen that, as expected, the slipping velocity is independent of the
precession angular velocity $\dot{\phi}$ which takes place on the vertical
axis passing through the center of the peg and the touchpoint. The friction
should be in the reverse direction of the touch point’s velocity and can be
written as $\vec{f}=-kN\vec{v}/|\vec{v}|$, where $k$ is the positive friction
constant, and the reaction force is taken as $N=Mg$. Then, the torque due to
this friction can be obtained by using $\vec{\tau}=\vec{R}_{p}\times\vec{f}$
as

$\displaystyle\vec{\tau}=$ $\displaystyle\frac{kNR_{p}^{2}}{|\vec{v}|}$
$\displaystyle[(-\dot{\theta}\cos\psi+\dot{\psi}\sin\theta\cos\theta\sin\psi)\hat{x}$
(4)
$\displaystyle\,+(\dot{\theta}\sin\psi+\dot{\psi}\sin\theta\cot\theta\cos\psi)\hat{y}+(-\dot{\psi}\sin^{2}\theta)\hat{z}],$

where $|\vec{v}|=R_{p}\sqrt{\dot{\theta}^{2}+\dot{\psi}^{2}\sin^{2}\theta}$.

Air dissipation is another factor in the motion of the top, and
characteristics of this dissipation for the spin angular velocity and nutation
& precession angular velocities are different. Skin friction drag is mainly
responsible for air dissipation of streamlined bodies, and pressure drag is
mainly responsible for air dissipation of blunt bodies [15]. These can depend
on various factors, e.g. the density of the fluid, the cross-section of the
body, relative velocity. However, in this work, we will not go into details
and use simple models. Skin friction drag is mainly responsible from air
dissipation for the spin angular velocity, and this dissipative torque can be
modelled as
$\vec{\tau}_{\dot{\psi}}=(d_{1}\dot{\psi}+d_{2}\dot{\psi}^{2})\hat{z}$.
Pressure drag is mainly responsible for nutation & precession angular
velocities, and corresponding dissipative torque can be written as :
$\vec{\tau}_{\dot{\theta}}=c_{1}\dot{\theta}(\cos\psi\hat{x}-\sin\psi\hat{y})$
and
$\vec{\tau}_{\dot{\phi}}=c_{2}\dot{\phi}\hat{z}^{\prime}=c_{2}\dot{\phi}(\sin\theta\sin\psi\hat{x}+\sin\theta\cos\psi\hat{y}+\cos\theta\hat{z})$.
Since the origin of dissipations due to nutation & precession angular
velocities are the same and the cross-sections are the same for both
rotations, air dissipation coefficients for nutation & precession angular
velocities can be considered as the same, i.e. $c=c_{1}=c_{2}$. Signs of $c$
and $d_{1}$ should be negative, and the sign of $d_{2}$ should be negative of
the sign of $\dot{\psi}$.

In the body coordinate system, the gravitational torque can be written as

$\vec{\tau}_{g}=-Mg\tilde{l}\sin\theta(-\cos\psi\hat{x}+\sin\psi\hat{y}),$ (5)

where $\tilde{l}$ is the distance from the peg’s center to the center of mass.

If one includes all of the mentioned torques in Euler equations, one can get
the following equations

$\displaystyle\ddot{\theta}$ $\displaystyle=$
$\displaystyle-\frac{I_{z}\dot{\phi}\sin\theta}{I_{x}}(\dot{\psi}+\dot{\phi}\cos\theta)+\dot{\phi}^{2}\sin\theta\cos\theta+\frac{Mg\tilde{l}}{I_{x}}\sin\theta+\frac{c\dot{\theta}}{I_{x}}-\frac{kMgR_{p}^{2}\dot{\theta}}{I_{x}|\vec{v}|},$
$\displaystyle\ddot{\phi}$ $\displaystyle=$
$\displaystyle\frac{I_{z}\dot{\theta}}{I_{x}\sin\theta}(\dot{\psi}+\dot{\phi}\cos\theta)-\frac{2\dot{\theta}\dot{\phi}\cos\theta}{\sin\theta}+\frac{c\dot{\phi}}{I_{x}}+\frac{kMgR_{p}^{2}\dot{\psi}\cos\theta}{I_{x}|\vec{v}|},$
(6) $\displaystyle\ddot{\psi}$ $\displaystyle=$
$\displaystyle-\frac{I_{z}\dot{\theta}\cos\theta}{I_{x}\sin\theta}(\dot{\psi}+\dot{\phi}\cos\theta)+\frac{2\dot{\theta}\dot{\phi}\cos^{2}\theta}{\sin\theta}+\dot{\theta}\dot{\phi}\sin\theta+c\dot{\phi}\cos\theta\left(\frac{1}{I_{z}}-\frac{1}{I_{x}}\right)$
$\displaystyle+\frac{d_{1}\dot{\psi}+d_{2}\dot{\psi}^{2}}{I_{z}}-\frac{kMgR_{p}^{2}\dot{\psi}}{|\vec{v}|}\left(\frac{\sin^{2}\theta}{I_{z}}+\frac{\cos^{2}\theta}{I_{x}}\right),$

where moments of inertia should be calculated by considering that rotations
take place around the peg’s center. These equations describe rotations of the
heavy symmetric top when the center of the peg is fixed, air dissipation is
included according to mentioned models, the touchpoint of the peg slips on the
ground and the rolling motion is not present. The term
$kMgR_{p}^{2}\dot{\psi}\cos\theta/(I_{x}|\vec{v}|)$ is the rise term of this
model. When dissipative terms are set to zero, these equations become the same
as the ones obtained from Lagrangian [16].

One can simplify this model by ignoring air friction and friction at the
touchpoint due to motion in $\theta$ since $|\dot{\theta}|<<|\dot{\psi}|$ and
obtain

$\displaystyle\ddot{\theta}$ $\displaystyle=$
$\displaystyle-\frac{I_{z}\dot{\phi}\sin\theta}{I_{x}}(\dot{\psi}+\dot{\phi}\cos\theta)+\dot{\phi}^{2}\sin\theta\cos\theta+\frac{Mg\tilde{l}}{I_{x}}\sin\theta,$
$\displaystyle\ddot{\phi}$ $\displaystyle=$
$\displaystyle\frac{I_{z}\dot{\theta}}{I_{x}\sin\theta}(\dot{\psi}+\dot{\phi}\cos\theta)-\frac{2\dot{\theta}\dot{\phi}\cos\theta}{\sin\theta}+\frac{kMgR_{p}^{2}\dot{\psi}\cos\theta}{I_{x}|\vec{v}|},$
(7) $\displaystyle\ddot{\psi}$ $\displaystyle=$
$\displaystyle-\frac{I_{z}\dot{\theta}\cos\theta}{I_{x}\sin\theta}(\dot{\psi}+\dot{\phi}\cos\theta)+\frac{2\dot{\theta}\dot{\phi}\cos^{2}\theta}{\sin\theta}+\dot{\theta}\dot{\phi}\sin\theta$
$\displaystyle-\frac{kMgR_{p}^{2}\dot{\psi}}{|\vec{v}|}\left(\frac{\sin^{2}\theta}{I_{z}}+\frac{\cos^{2}\theta}{I_{x}}\right).$

One can see that all dissipative effects are related to motion in $\dot{\psi}$
in these equations, and they originate from the friction at the touchpoint.

##  3 Jellett’s model for the rise of the top

In this part, we will write Jellett’s model in terms of the parameters of this
work. We should note that Jellett used an upside-down center of mass-body
reference frame, and for simplicity ignored air dissipation and frictions with
respect to $\dot{\theta}$ and $\dot{\phi}$. We will write Jellett’s parameters
inside square parentheses to eliminate confusion.

Jellett gave equations for the rise of the top in equations (4) and (5) in his
work [6]. Jellett’s parameters and variables can be written in terms of
parameters and variables of this work as: $[\theta]=\theta$, $[\psi]=\phi$,
$[\phi]=-\psi$, $[R\cos\epsilon]=N$, $[\xi]=R_{p}\sin\theta$,
$[\eta]=\tilde{l}+R_{p}\cos\theta$, $[z^{\prime}]=\tilde{l}\sin\theta$,
$[z]=R_{p}+\tilde{l}\cos\theta$ and $[R\sin\epsilon]=-f$. Due to the usage of
the upside-down center of mass-body reference frame, the spin angle and
friction force get a minus sign. In Jellett’s work, the reaction force is not
specified, and we will take $N=Mg$. Then, one can obtain equations of motion
for Jellett’s model in terms of parameters and variables of this work by
considering $f=Nk$ as

$\displaystyle\ddot{\theta}$ $\displaystyle=$
$\displaystyle-\frac{\tilde{I}_{z}\dot{\phi}\sin\theta}{\tilde{I}_{x}}(\dot{\psi}+\dot{\phi}\cos\theta)+\dot{\phi}^{2}\sin\theta\cos\theta+\frac{Mg\tilde{l}}{I_{x}}\sin\theta,$
$\displaystyle\ddot{\phi}$ $\displaystyle=$
$\displaystyle\frac{\tilde{I}_{z}\dot{\theta}}{\tilde{I}_{x}\sin\theta}(\dot{\psi}+\dot{\phi}\cos\theta)-\frac{2\dot{\theta}\dot{\phi}\cos\theta}{\sin\theta}+\frac{Mgk(\tilde{l}+R_{p}\cos\theta)}{\tilde{I}_{x}\sin\theta},$
(8) $\displaystyle\ddot{\psi}$ $\displaystyle=$
$\displaystyle-\frac{\tilde{I}_{z}\dot{\theta}\cos\theta}{\tilde{I}_{x}\sin\theta}(\dot{\psi}+\dot{\phi}\cos\theta)+\frac{2\dot{\theta}\dot{\phi}\cos^{2}\theta}{\sin\theta}+\dot{\theta}\dot{\phi}\sin\theta$
$\displaystyle-\frac{Mgk\cos\theta(\tilde{l}+R_{p}\cos\theta)}{\tilde{I}_{x}\sin\theta}-\frac{MgkR_{p}\sin\theta}{\tilde{I}_{z}}.$

We have used $\tilde{I}_{i}$ to emphasize that moments of inertia should be
calculated in the center of mass-body reference frame. In Jellett’s model the
rise term is $Mgk(\tilde{l}+R_{p}\cos\theta)/(\tilde{I}_{x}\sin\theta)$ which
linearly depends on $\tilde{l}$. It can be seen that this is different from
the rise term obtained by considering pure slipping. We should note that signs
of dissipative terms should be determined by considering the sign of
$\dot{\psi}$.

##  4 Numerical solutions

In this section, we will numerically solve different cases with studied
models. Firstly, we will numerically solve a previously experimentally studied
case with the pure slipping model, which is figure 5 in Cross’ work [13]. This
case is also studied with a different model, previously [2]. To get consistent
results with the figures given in the experimental work, we will take
$\dot{\phi}$ and $\dot{\psi}$ as negative similar to previous work. Then, we
will study the rise of the top with the dissipation related parameters which
are obtained by considering the experimental case. We will also consider the
simplified pure slipping model to get a better comparison with Jellett’s model
since air dissipation and friction at the touchpoint due to motion in $\theta$
are ignored in these two models.

###  4.1 Dissipative motion

By using given values in the experimental work for figure 5, parameters of the
top becomes $M=105\,gr$, $\tilde{l}=20.9\,mm$, $R_{p}=0.1mm$,
$I_{x}=8.44\times 10^{-5}\,kg\,m^{2}$ and $I_{z}=7.23\times
10^{-5}\,kg\,m^{2}$ where moments of inertia are calculated according to the
peg’s center. The initial values are taken as $\theta_{0}=0.117\,rad$,
$\dot{\theta}_{0}=2.23\,rad\,s^{-1}$, $\dot{\phi}_{0}=-17.1\,rad\,s^{-1}$ and
$\dot{\psi}_{0}=-126\,rad\,s^{-1}$ which are the same with previous model. To
get consistent results with experimental results, dissipative constants are
taken as: $k=0.03$, $c=-6.3\times 10^{-6}\,kg\,m^{2}\,s^{-1}$,
$d_{1}=-1.1\times 10^{-6}\,kg\,m^{2}\,s^{-1}$,
$d_{2}=-sgnm(\dot{\psi})1.1\times 10^{-8}\,kg\,m^{2}$. These constants are not
the same as the previous model since models are different. The gravitational
acceleration is taken as $g=9.81\,m\,s^{-2}$.

Results of numerical solutions for $\theta$ and angular velocities can be seen
in figure 8 in the appendix. Projections of shapes for the locus can be seen
in figure 2. One can see that these are very similar to the experimental
results.

Figure 2: Projections of shapes for the locus of the figure axis. Time
intervals (in seconds) for figures as follows: (a) $[0.73,3.45]$, (b)
$[18.00,19.48]$ (c) $[23.60,24.87]$ (d) $[27.80,28.86]$. Initial values are
$\theta_{0}=0.117\,rad$, $\dot{\theta}_{0}=-2.23\,rad\,s^{-1}$,
$\dot{\phi}_{0}=-17.1\,rad\,s^{-1}$ and $\dot{\psi}_{0}=-126\,rad\,s^{-1}$.

We should note that there are some differences between the results of the
previous model and this model. Fluctuations in $\theta$ and angular velocities
are larger in the previous model, however, they are damped by the dissipative
factors at the end of 31 seconds. On the other hand, they are not damped in
this model. The main reason for this should be related to the constant
friction term in the previous model, which is considered for the friction at
the touchpoint. In this model, this friction shows itself as functions of
$\dot{\theta}/|\vec{v}|$ and $\dot{\psi}\sin\theta/|\vec{v}|$ in torque, and
$\dot{\theta}/|\vec{v}|$ is small and $\dot{\psi}\sin\theta/|\vec{v}|$ is
close to one. Then, the fluctuations in $\theta$ are damped more slowly in
this model. We should also mention that there is a slight rise in the average
value of $\theta$ which can be seen in figure 8(a) in the appendix. Such a
rise is not observed in the previous model. However, it is hard to determine
this slight rise without taking average because of the high fluctuations. On
the contrary, an increase of $\theta$ is observed in the experiment, which is
not observed in this and previous models. Nevertheless, these show themselves
with some small differences in the shapes for the locus. And, the results of
both models for the shapes for locus are similar to the experiment.

###  4.2 Rise of the top

In this part, we will first consider the rise of the top with a hypothetical
top that is slightly different from the top considered in the previous
example. We will add this hypothetical top a hemispherical peg with radius
$R_{p}=6.0\,mm$. The mass of the top will be taken $M=110\,gr$, and the
distance between the center of mass and center of the peg will be taken as
$\tilde{l}_{1}=20\,mm$. We need moments of inertia in two reference frames
which will be taken as follows $I_{x}=8.52\times 10^{-5}\,kg\,m^{2}$ and
$I_{z}=7.25\times 10^{-5}\,kg\,m^{2}$ in the center of peg-body reference
frame, and $\tilde{I}_{x}=4.08\times 10^{-5}\,kg\,m^{2}$ and
$\tilde{I}_{z}=7.25\times 10^{-5}\,kg\,m^{2}$ in the center of mass-body
reference frame.

We will numerically solve a case by the pure slipping and Jellett’s models for
this top. Then, we will consider another hypothetical top and compare the pure
slipping, the simplified pure slipping and Jellett’s models. We will cut
numerical solutions when the top reaches nearly upright position, i.e.
$\theta=0.02\,rad$, due to infinities at $\theta=0$. We will consider
configurations giving regular precession initially to reduce the fluctuations
with two exceptions which will be explained later.

####  4.2.1 Pure slipping

In this part, we will study the rise of the mentioned hypothetical top by
using equations (6). We will take initial values as follows $\theta=0.5\,rad$,
$\dot{\theta}=0$, $\dot{\phi}_{0}=-1.49\,rad\,s^{-1}$ and
$\dot{\psi}_{0}=-200\,rad\,s^{-1}$.

(a) $\theta$ (b) $\dot{\theta}$ (c) $\dot{\phi}$ (d) $\dot{\psi}$

Figure 3: Results of the numerical solution for $\theta$ (a), $\dot{\theta}$
(b), $\dot{\phi}$ (c) and $\dot{\psi}$ (d). Initial values:
$\theta_{0}=0.5\,rad$, $\dot{\theta}_{0}=0$,
$\dot{\phi}_{0}=-1.49\,rad\,s^{-1}$ and $\dot{\psi}_{0}=-200\,rad\,s^{-1}$.
The parameters of the top are given in section 4.2.

Results of the numerical solution for $\theta$, $\dot{\theta}$, $\dot{\phi}$
and $\dot{\psi}$ can be seen in figure 3. One can see from figure 3(a) that
the top rises, and it comes nearly vertical position in $21.8$ seconds. It can
be seen from figure 3(b) that there is a negligible nutation, which does not
affect the motion much. It can be seen from figure 3(d) that the magnitude of
$\dot{\psi}$ decreases which is an expected result of dissipation. Its
decrease is slightly faster in the beginning, and the main reason for this is
the air dissipation. It changes from $-200\,rad\,s^{-1}$ to $-77\,rad\,s^{-1}$
during the rise. It can be seen from figure 3(c) that the magnitude of
$\dot{\phi}$ increases which is the result of the rise term originating from
the dissipative torque at the touchpoint. Its average value changes from
$-1.49\,rad\,s^{-1}$ to $-3.9\,rad\,s^{-1}$, and there are some fluctuations
as a result of nutation.

Figure 4: Shapes for the locus (a) and its projection on to $xy$-plane (b) for
the rise of the top. Initial values are given in figure 3, and the solution is
obtained by considering the pure slipping model.

In figure 4, one can see the shapes for the locus during the rise and its
projection onto $xy$-plane. A spiral structure is observed during the rise. We
should note that this motion is different from ”spiraling motion” which can be
seen when conserved angular momenta are equal to each other [5, 17].

####  4.2.2 Jellett’s model

In this part, we will consider the rise of the top with Jellett’s model, i.e.
equations (8). Moments of inertia obtained by considering the center of mass-
body reference frame will be used, and initial values will be the same as the
previous solution except $\dot{\phi}_{0}$ which should be changed slightly in
order to get a regular precession at the beginning, i.e.
$\dot{\phi}_{0}=1.48\,rad\,s^{-1}$.

(a) $\theta$ (b) $\dot{\theta}$ (c) $\dot{\psi}$ (d) $\dot{\phi}$

Figure 5: Results of the numerical solution by considering Jellett’s model for
$\theta$ (a), $\dot{\theta}$ (b), $\dot{\psi}$ (c) and $\dot{\phi}$ (d).
Continuous (blue) lines show results of numerical solutions, and dashed
(black) lines show averages. Initial values: $\theta_{0}=0.5\,rad$,
$\dot{\theta}_{0}=0$, $\dot{\phi}_{0}=-1.48\,rad\,s^{-1}$ and
$\dot{\psi}_{0}=-200\,rad\,s^{-1}$.

The results of the numerical solution can be seen in figure 5. There are some
changes from the previous results. It can be seen that the top rises in $8.18$
seconds which is much faster than the previous model, and it is the result of
$\tilde{l}$ dependent rise term. Fluctuations of $\dot{\theta}$ increase as
time passes, which are nearly ten times more than the pure slipping model.
But, the average value of $\dot{\theta}$ does not change much, and it is
around $-1.6\,rad\,s^{-1}$. Fluctuations of $\dot{\phi}$ also increase, and
its average changes from $-1.48\,rad\,s^{-1}$ to $-1.6\,rad\,s^{-1}$ which is
less than previous model. $\dot{\psi}$ also fluctuates toward the end of the
rise and its average becomes $-194\,rad\,s^{-1}$ whose magnitudes are much
more than the one in the pure slipping as a result of ignoring air
dissipation. The growth of fluctuations occurs due to ignoring air dissipation
and fluctuations in relative small values of $\theta$.

Figure 6: Shapes for the locus (a) and its projection on to $xy$-plane (b).
Initial values are given in figure 5.

In figure 6(a), one can see the shapes for the locus for numerical results. It
can be seen that the top rises again by making a spiral shape with fewer
windings.

####  4.2.3 Comparison

In this part, we will study and compare the pure slipping model, the
simplified pure slipping model, i.e. equations (7), and Jellett’s model. We
will consider the simplified pure slipping to get a better comparison due to
the absence of air dissipation and frictional effect arising from
$\dot{\theta}$ similar to Jellett’s model. Since the most obvious difference
in the pure slipping and Jellett’s models is the dependence of the rise term
on $\tilde{l}$ in Jellett’s model, we will consider a second top whose
distance between the center of the peg and the center of mass is double of the
previously considered top. This can be done by changing the position of the
top’s disc, see figure 1. For such a change, the moment of inertia of the
symmetry axis $I_{3}$ does not change but moment of inertia of the transverse
axis $I_{1}$ does.

We have already given parameters of the previously considered top, which we
will name as top 1 from now on, and the one with doubled $\tilde{l}$ as top 2.
For the top 2, the mass, the radius of the peg and $I_{z}$ (and
$\tilde{I}_{z}$) will be the same as the top 1, and the distance between the
center of mass and the center of the peg $\tilde{l}_{t2}$ will be equal to
$40\,mm$. The moment of inertia about tranverse axis of top 2 $I_{t2,x}$ in
the center of peg-body reference frame will be taken as $22.3\times
10^{-5}\,kg\,m^{2}$, and $\tilde{I}_{t2,x}=4.64\times 10^{-5}\,kg\,m^{2}$ in
the center of mass-body reference frame.

We will use the weak top condition to get clear discrimination between the
pure slipping and Jellett’s models. The weak top condition is obtained from
regular precession and gives a relation between
$a=I_{3}(\dot{\psi}-\dot{\phi}\cos\theta)/I_{1}$ and
$\sqrt{4Mg\tilde{l}/I_{1}}$. If $|a|$ is smaller than
$\sqrt{4Mg\tilde{l}/I_{1}}$, then the top is considered as weak top [18]. We
should note that this discrimination works approximately when conserved
angular momenta are not equal to each other. For top 1:
$\sqrt{4Mg\tilde{l}_{t1}/I_{t1,x}}=31.8\,rad\,s^{-1}$ and
$\sqrt{4Mg\tilde{l}_{t1}/\tilde{I}_{t1,x}}=46.0\,rad\,s^{-1}$. For top 2:
$\sqrt{4Mg\tilde{l}_{t2}/I_{t2,x}}=27.8\,rad\,s^{-1}$ and
$\sqrt{4Mg\tilde{l}_{t2}/\tilde{I}_{t2,x}}=61.0\,rad\,s^{-1}$

(a) Pure slipping (b) Simplified pure slipping (c) Jellett’s model (d) Pure
slipping

(e) Simplified pure slipping (f) Jellett’s model

Figure 7: Results of the numerical solution for $\theta$ for top 1 (upper
three graphs) and 2 (lower three graphs) with different initial
$\dot{\psi}_{0}$ and $\dot{\phi}_{0}$ values. Continuous (black) lines show
results for $\dot{\psi}_{0}=-50\,rad\,s^{-1}$, dotted (green) lines show
results for $\dot{\psi}_{0}=-100\,rad\,s^{-1}$, dashed-dotted (blue) lines
show results for $\dot{\psi}_{0}=-150\,rad\,s^{-1}$, dashed (red) lines show
results for $\dot{\psi}_{0}=-200\,rad\,s^{-1}$, dashed-double dotted (voilet)
lines show results for $\dot{\psi}_{0}=-250\,rad\,s^{-1}$.
$\theta_{0}=0.5\,rad$ and $\dot{\theta}_{0}=0$ for all cases. $\dot{\phi}_{0}$
values are chosen very close to smaller root of
$\dot{\phi}_{1,2}=(-I_{3}\dot{\psi}-\sqrt{(I_{3}\dot{\psi})^{2}+4Mg\tilde{l}\cos\theta})/[2(I_{3}-I_{1})\cos\theta]$
except the situation $\dot{\psi}_{0}=-50\,rad\,s^{-1}$ for the pure slipping
and simplified pure slipping models, in which
$\dot{\phi}_{0}=-35\,rad\,s^{-1}$.

We will consider five different $\dot{\psi}_{0}$ values for both tops, and we
will numerically solve these 10 cases with mentioned models. For all cases,
$\theta_{0}=0.5\,rad$ and $\dot{\theta}_{0}=0$. $\dot{\psi}_{0}$ changes from
$-50\,rad\,s^{-1}$ to $-250\,rad\,s^{-1}$ with $50\,rad\,s^{-1}$ decrements,
and corresponding $\dot{\phi}_{0}$ values are taken as very close to the ones
giving regular precession with two exceptional situations. We should note that
for the pure slipping and simplified pure slipping models, $\dot{\phi}_{0}$
are taken as the same since $I_{x}$ are the same for these models, and
$\dot{\phi}_{0}$ is slightly different for Jellett’s model since
$\tilde{I}_{x}$ is different. In exceptional situations,
$\dot{\psi}=-50\,rad\,s^{-1}$ for the pure slipping and simplified pure
slipping models for top 2, the top is weak and regular precession is not
possible, and for these two situations, $\dot{\phi}_{0}$ is taken as
$-35.0\,rad\,s^{-1}$. One can see results of numerical solutions for $\theta$
for mentioned 30 situations in figure 7.

The upper three graphs of the figure 7 show results for top 1 with the order
the pure slipping, the simplified pure slipping and Jellett’s models, and the
lower three graphs show corresponding ones for top 2. It can be seen that as
the magnitude of $\dot{\psi}_{0}$ increases the rise time increases which
arises from the need for greater $\dot{\phi}$ values for the rise. The same
thing also explains the difference in rise times between the pure slipping and
simplified slipping models since air dissipation helps the decrease of
$\dot{\psi}$ in the pure slipping. The dependence of the rise term on
$\tilde{l}$ in Jellett’s model causes a faster rise of tops in all cases.

In previously mentioned exceptional two situations, top 2 does not rise and
fall. On the other hand, top 2 rises in the corresponding situation when the
solution is obtained by using Jellett’s method. For mentioned two situations,
$|a|=26.2\,rad\,s^{-1}$ which is smaller than $27.8\,rad\,s^{-1}$ in the pure
slipping and simplified pure slipping models; but in Jellett’s model,
$|a|=93.3\,rad\,s^{-1}$ which is greater than $61.0\,rad\,s^{-1}$. Then, this
case gives weak top for the pure slipping and simplified pure slipping models,
which results in fall. On the other hand, this situation is not weak in the
center of mass-body reference frame, and top 2 rises in the corresponding
situation in Jellett’s model.

Another remarkable thing is seen when top 2 is not weak and not strong enough
for the whole rise in the pure slipping model. It can be seen from figure 7(d)
that top 2 does not completely rise in the pure slipping model when
$\dot{\psi}_{0}=-100\,rad\,s^{-1}$ and $\dot{\psi}_{0}=-150\,rad\,s^{-1}$. The
top rises at the beginning, and during this process, its spin angular velocity
decreases mainly due to air dissipation, and it starts to fall around $t=4.9s$
and $t=13.9s$, respectively, and numerical results show that in these cases,
top 2 becomes weak at $t=4.7s$ and $t=13.7s$, respectively. On the other hand,
it rises entirely in Jellett’s model and simplified pure slipping. The reason
for the rise in simplified pure slipping is the absence of air dissipation.

When top 2 is strong enough for the whole rise in the pure slipping model, it
is possible to compare rise times of different models. The percentage
differences in the rise time are 15% are 11% for the last two cases in the
pure slipping model, they are 7% and 4% in the simplified pure slipping model,
and they are 43% in Jellett’s model.

##  5 Conclusion

We have studied dissipative motion and the rise of the heavy symmetrical top
with a hemispherical peg. Differently from previous works, we studied motion
by considering the radial center of the hemispherical peg as the fixed point
of the top. From the derivation, we obtained the rise term as a function of
the radius of the peg $R_{p}$, which is consistent with the experiments [3].
On the other hand, in Jellett’s model [6], and in other similar ones [4, 7,
12], the center of mass is considered as the fixed point and the rise term is
a function of $\tilde{l}$ together with $R_{p}$.

The difference between the rise terms provides the possibility of
determination of the better model with simple experiments. By using a top with
a changeable disc position, it is possible to change the position of the
center of mass which can provide necessary experimental data.

We should note that we have determined dissipative constants according to the
pure slipping model by considering an experimentally studied case. There
should be some differences in these constants if one had used any other model.
However, the differences between Jellet’s model and the pure slipping model
are bigger, and any change in the dissipative constants does not substantially
affect the following comments.

One can use strong tops and compare changes in rise times and decide the
better model. By looking at the difference between the simplified pure
slipping and pure slipping models, one can say that if one includes air
dissipation and friction due to $\dot{\theta}$ at the touchpoint in Jellett’s
model, the rise time should decrease. However, this decrease should be smaller
than the difference between the simplified pure slipping and pure slipping
models, since rise time in Jellett’s model is smaller. By considering the
results of numerical solutions, one can say that after doubling the distance
between the center of mass and the center of the peg, the rise time should
decrease. For strong tops, if the rise time changes around 15% then the pure
slipping model is better; on the other hand, if the rise time changes around
40% then Jellett’s model is better. We should note that the friction
coefficient and the radius of the peg should be convenient to get a longer
rise time which may provide better data for discrimination.

Obviously, one can use the weak top condition to decide the better model. If
one makes an experiment with a case that is weak in center of peg-reference
frame and strong in center of mass-reference frame, then one can easily decide
the better model: If the top rises, then Jellett’s model is better; and if it
falls, then the pure slipping model is better.

##  6 Appendix

Figure 8 shows results of numerical solution for dissipative motion for
$\theta$ and angular velocities.

(a) $\theta$ (b) $\dot{\theta}$

(c) $\dot{\phi}$ (d) $\dot{\psi}$

Figure 8: Results of the numerical solution for the dissipative motion for
$\theta$ (a), $\dot{\theta}$ (b), $\dot{\phi}$ (c) and $\dot{\psi}$ (d).
Continuous (blue) lines show results of numerical solution and dots (black)
show nutation average. Initial values are the same as figure 2.

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