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\DOI{10.5802/crphys.174}
\datereceived{2023-10-17}
\daterevised{2023-12-12}
\dateaccepted{2023-12-15}
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\dateposted{2024-01-26}
\begin{document}

\begin{noXML}

%\TopicEF{Research article}{Article de recherche}

\title{Left-  versus right-handed badminton slice shots: opposite spinning
of the chiral shuttlecock and Magnus effect} 

\alttitle{Slices de gaucher versus droitier au badminton : rotation
inverse du volant chiral et effet Magnus}

\author{\firstname{Eric} \lastname{Collet}\CDRorcid{0000-0003-0810-7411}}
\address{Univ Rennes, CNRS, IPR (Institut de Physique de Rennes) - UMR 6251, F-35000 Rennes, France}
\address{DYNACOM IRL2015 University of Tokyo - CNRS - UR1, Department of Chemistry, 7-3-1 Hongo, Tokyo 113-0033,  Japan}
\address{Institut universitaire de France (IUF), France}
\email{eric.collet@univ-rennes.fr} 
\urladdr{https://ipr.univ-rennes1.fr/interlocuteurs/eric-collet} 

\keywords{\kwd{Physics of badminton}
\kwd{Magnus effect}
\kwd{Symmetry}
\kwd{Chirality}
\kwd{Badminton trajectory}
\kwd{Time-{\break}resolved}}

\altkeywords{\kwd{Physique du badminton}
\kwd{Effet Magnus}
\kwd{Sym\'{e}trie}
\kwd{Chiralit\'{e}}
\kwd{Trajectoires au badminton}
\kwd{Temps r\'{e}solu}}

\thanks{This work was financed by the ``Institut Universitaire de
France'' and the Agence Nationale de la Recherche, ANR-19-CE30-0004
ELECTROPHONE}

\begin{abstract}
The chiral nature of a badminton shuttlecock is responsible for its
counter-clockwise spinning as it naturally propagates through the air.
This induces a dissymmetry between left- and right-handed players and
the resulting trajectories of the shuttlecock, which were captured in
real condition on the badminton court in slow motion at 3700~fps. The
videos clearly evidence this dissymmetry as slice shots performed by
right-handers induce a natural counter-clockwise spinning, while the
ones performed by left-handers induce a clockwise to counter-clockwise
spinning, making trajectories of shuttlecocks different. The slow
motion videos also caught a brief Magnus effect, often neglected in
badminton, lifting up the shuttlecock for both left-handers and
left-handers and affecting the effectiveness of the slice shot.
\end{abstract}

\begin{altabstract}
La nature chirale du volant de badminton est responsable de sa rotation
antihoraire lorsqu'il se propage naturellement dans l'air. Cela
induit une dissym\'{e}trie entre joueurs gauchers et droitiers et les
trajectoires du volant qui en r\'{e}sultent, captur\'{e}es en
conditions r\'{e}elles sur le terrain de badminton en slow motion \`{a}
3700~ips. Les vid\'{e}os mettent clairement en \'{e}vidence cette
dissym\'{e}trie puisque les slices effectu\'{e}s par les droitiers
induisent une rotation antihoraire naturelle, tandis que ceux
effectu\'{e}s par les gauchers induisent une rotation d'horaire \`{a}
antihoraire, rendant les trajectoires de volants diff\'{e}rentes. Les
vid\'{e}os ont \'{e}galement mis en \'{e}vidence un bref effet Magnus,
souvent n\'{e}glig\'{e} au badminton, soulevant le volant pour les
gauchers comme les droitiers et affectant l'efficacit\'{e} du slice.
\end{altabstract}

\maketitle

\end{noXML}

\section{Introduction}

The Badminton World Federation reported that badminton is one of the
most popular sports, played by over 339~million people worldwide
\cite{1}. About 23\% of them are left-handed players \cite{2}. While
in most sports, there is no geometrical difference between left-handed
and right-handed players, this is not true for badminton. This
dissymmetry comes from the chiral nature of feather shuttlecocks,
composed of an array of diverging stems, the ends of which are at the
convergent end of the skirt, joined together in an end ring.
Figure~\ref{fig1} shows the nomenclature of feather shuttlecocks. Its
global symmetry corresponds to the $\mathrm{C}_{16}$ point group, which
contains 2-, 4-, 8- and 16-fold rotation symmetry axes \cite{3}. The
feathers of a goose's left wing are typically used to make
shuttlecocks because they are more symmetrical, have a more consistent
shape and are more flexible than those from the right wing \cite{4}.
Shuttlecocks do not exhibit mirror symmetry and are chiral bodies, due
to the way the feathers are placed into the cork, which is responsible
for the natural counter-clockwise spinning of these projectiles as they
propagate through the air. The symmetry of the badminton court is
different as it corresponds to the mm2 point group. Figure~\ref{fig2}
shows mirror symmetry at the net and 2-fold symmetry at the center of
the court. When two right-handed players (or two left-handed players)
face each other (Figure~\ref{fig2}a,b), there is a global
2-fold symmetry on the court, which is also contained in the symmetry
of the shuttlecock, which spins equivalently counter-clockwise on both
sides of the court. The situation is therefore symmetry-equivalent for
both right-handed (or both left-handed) players facing each other. When
a left-handed player faces a right-handed player
(Figure~\ref{fig2}c,d) the 2-fold symmetry is lost and a
mirror symmetry appears at the net. To maintain this mirror symmetry,
the shuttlecock should spin in opposite directions on both sides of the
court. However, there is no mirror symmetry for the shuttlecock, due to
its chiral nature, and the clockwise spinning and the natural
counter-clockwise spinning are not equivalent. The situation is
therefore different by symmetry for the left-handed and right-handed
players, due to the fact that the racket manipulated by both players
can induce opposite shuttlecock spinning.



\begin{figure}
\includegraphics{fig01}
\caption{Structure and symmetry of a feathered shuttlecock. The top and
bottom views along the rotation axis (a,b) show the 16-fold
symmetry. This is due to the way the 16 bent feathers are placed into
the cork (c). The shuttlecock also exhibits a 2-fold
symmetry.\label{fig1}}
\end{figure}


\begin{figure}
\includegraphics{fig02}
\caption{Sketch of a badminton court, exhibiting 2-fold symmetry around
the court center and mirror symmetry centered on the net. A situation
where right-handed (a), or left-handed (b), players face each other
corresponds to 2-fold symmetry, and counter-clockwise spinning of the
shuttlecock on both sides of the courts. For mirror symmetry to exist,
left-handed and right-handed players would have to face each other
(c,d) and the shuttlecock to rotate equivalently clockwise and
counter-clockwise, which is not the case due to its chiral
nature.\label{fig2}}
\vspace*{1pc}
\end{figure}

Most of the studies of shuttlecock trajectories found in the literature
concern experimental measurements, including wind tunnel experimental
data of fast cameras, and/or theoretical descriptions of net shots,
serve shots, smashes and high clear shots \cite{5,6,7,8,9,10,11,12}. These
studies aimed at gaining knowledge of the equations of motion of a
shuttlecock's flying trajectory. Usually the trajectory is described
in a plane, defined by the initial position and the horizontal and
vertical components of velocities. The reproducibility and quality of
such studies are essential for quantitative analysis of the flying
properties of the different shuttlecocks. Since the rotation axis of
the shuttlecock is usually collinear with its velocity, it is often considered
that the Magnus effect \cite{13} never occurs in badminton \cite{14},
contrary to other ball sports such as baseball, volleyball and soccer
to name a few \cite{15}. Measurements of the lateral force on the
shuttlecock suggested that the spinning of the shuttlecock does induce
a Magnus effect, which may be the origin of the shuttlecock
``drifting'' noticed by players for feather shuttlecock during high
clear \cite{6,16}. In many studies, trajectories are not induced by
players because the initial trajectory of the projectile after real
impact with the racket is sometimes hard to control or to reproduce
precisely. It is also player dependent, which precludes quantitative
analysis.


In this study, we present a qualitative comparison of the shuttlecock
trajectories after slice shots performed by right-handed and
left-handed players. Badminton slicing technique is known for causing
the shuttlecock to spin and altering the angle of return. Here we
focused our attention on cross-court slice drop shots
(Figure~\ref{fig3}). Right-handers played the slice in the right-hand
corner of the court, while left-handers played the slice in the
left-hand corner of the court. A high-speed Phantom Miro 3a10 camera
was used on the badminton court, in order to capture high frame rate
(3700~fps) videos of the trajectories of the shuttlecock in real
condition. At this frame rate, the resolution of the videos was
maximized to 1024 $\times$ 768 pixels and the exposure time for each
frame was set to $268~\rmmu\mathrm{s}$. We used Yonex AS 20
shuttlecocks for the slice shots. The videos are shown in supplementary
material. Due to the fact that videos were collected on the court (3D),
it was not possible to extract precise measurements of distances (and
therefore speed) from the videos. We estimated roughly speed by scaling
lengths with shuttlecock (9~cm) or racket sizes (68~cm). Given the
expected readership, the choice is made to use popular units for speed
(km/h) and rotational speed (rps) to make this article readable by
non-physicists. In the following we discuss: shuttlecock flipping
after a smash; right-handed versus left-handed slices; magnus effect
after slice shots; shuttlecock trajectories.

\begin{figure}
\includegraphics{fig03}
\caption{A high-speed camera captured videos of the trajectories of the
shuttlecock at 3700~fps on the badminton court, as players performed
different cross-court slice shots.\label{fig3}}
\end{figure}


\section{Shuttlecock flipping after a smash}

It is well known that, due to their peculiar chiral structure,
badminton shuttlecocks spin as they fly in a high-drag. This
specificity compared to sports balls has been the subject of an
intensive research from the point of view of aerodynamics
\cite{5,6,7,8,9,10,11,12}, with unexcepted applications such as
improvements in landing accuracy of re-entry vehicles \cite{17}. The
value of the ballistic coefficient of badminton shuttlecocks is small,
which translates their ability to overcome air resistance in flight
\cite{5}. The shuttlecock is a light and extended particle, which flies
with a pure drag trajectory, so that initial velocities of shuttlecocks
of ${\simeq}$240~km/h are reduced in only ${\simeq}$0.6~s \cite{18,19},
to near the terminal velocity of ${\simeq}$25~km/h \cite{16,20}.
Women's record for fastest badminton shot is currently set to
438~km/h and belongs to P.~Tan, while the men's record at 565~km/h
belongs to S.~Rankireddy. Both are right-handed players. Kitta 
\etal~\cite{5} compared trajectories of spinning and non-spinning
feather shuttlecock and concluded that spinning shuttlecock exhibit a
marginally larger drag.

Cohen and colleagues considered the effect of the aerodynamic pressure
forces on the shuttlecock, with a global drag force applied at the
pressure center point at the level of the feather \cite{8}, which
depends on the surrounding pressure profile. When the pressure is
uniform on the feather, the aerodynamic center is the centroid of the
object. However, the center of gravity, closer to the cork, differs
from the center of pressure \cite{21}. 
Cohen and colleagues  introduced a model explaining the shuttlecock
flipping after impact with the racket, changing the direction of the
shuttlecock propagation. They have shown that the stabilizing
aerodynamic torque sets the shuttlecock nose ahead after impact. They
modeled the shuttlecock with two spheres (Figure~\ref{fig4}): one
representing the light skirt with a large cross-section (positioned in
B) and one representing the heavy small cork with smaller cross-section
(placed in C) \cite{8}. They explained that when the projectile flies
with an angle $\varphi$ with respect to the velocity $\vec {U}$, after
an impact with a racket, a torque balance around the center of gravity
$G$, due to different drag forces on the skirt and cork, makes the
shuttlecock axis collinear to velocity $\vec {U}$. For an initial
shuttlecock velocity of ${\simeq}10~\mathrm{m}{\cdot}\mathrm{s}^{-1}$
(36~km/h), they found a flipping time of ${\simeq}$20~ms, while the
stabilization of the symmetry axis of shuttlecock along the direction
of its velocity occurs typically within 150~ms and over ${\simeq}$1~m
\cite{8,22}. Video 1a \cite{addref1az} shows such flipping processes
for different shots. For example, when the initial velocity after
racket impact is ${{\simeq}}50~\mathrm{m}{\cdot}\mathrm{s}^{-1}$
(290~km/h), the flipping time is of the order of 5~ms, while the
stabilization of the symmetry axis of shuttlecock occurs within
${\simeq}$35~ms and over ${\simeq}$1~m (one racket length is about 68~cm).

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig04}
{\vspace*{-2pt}}
\caption{Left: the drag forces $F_{D}$ applied on the feather and the
cork generate a torque, stabilizing the shuttlecock axis along its
velocity (see Ref.~\cite{8}). Right: schematic representation used in
Figure~\ref{fig6}.\label{fig4}}
{\vspace*{-2pt}}
\end{figure}

Video 1b \cite{addref1bz} shows the important deformation of the
shuttlecock after impacts with the racket. When the shuttlecock
incident velocity is perpendicular to the racket, we can observe a
deformation that is akin to a blooming flower as observed in other
studies \cite{23}, or a swimming jellyfish. Indeed, part of the kinetic
energy is transferred to internal degrees of freedom of the
shuttlecock, which act as energy bumper. As shown in the
chronophotographies in Figure~\ref{fig5}a, the shuttlecock is then
exhibiting a damped breathing after impact with a pseudo-period of the
order of ${\simeq}$2.16~ms. The overall feather diameter expands from
6.7~cm to ${\simeq}$8.6~cm within ${\simeq}$1.1~ms during the first
oscillation and this breathing is damped within 4~oscillations. The
third shot in Video 1b shows that when the racket does not strike the
shuttlecock perpendicular to its axis, the deformation is more oval.
However, this is difficult to analyze because of the camera's angle
of view. For much weaker impacts, where acceleration is lower, it is
difficult to observe the deformation of the shuttlecock. This
deformation mode is similar to the breathing mode of some molecules,
revealed by femtosecond pump-probe techniques \cite{24,25,26}. The
shuttlecock breathing is due to the important deceleration during
impact, which bends the feather. In Video 1b \cite{addref1bz}, the
incident velocities of the order of
${\simeq}11~\mathrm{m}{\cdot}\mathrm{s}^{-1}$ (40~km/h) change to
${\simeq}44~\mathrm{m}{\cdot}\mathrm{s}^{-1}$ (160~km/h) within
${\simeq}$1.1~ms. This corresponds to an acceleration of the order of
$50,000~\mathrm{m}{\cdot}\mathrm{s}^{-2}$, i.e. 5000~g. During the
impact lasting 1.1~ms, the 5~g of the shuttlecock are therefore
equivalent to a weight of 25~kg. The weight of the flexible part of the
feather, which is of the order of 0.05~g, transiently corresponds
to 250~g. Figure~\ref{fig5}b shows that the feather deformation
induced by a 250~g load is similar to the one observed during
impact and in the elastic deformation regime of the feather. This
deformation is reversible after impact or 
loading in this elastic deformation regime.



\begin{figure}
\includegraphics{fig05}
\caption{(a) Chronophotographies of the shuttlecock after impacts with
the racket extracted from the Video 1b. For both shots the breathing
period is ${\simeq}$2.16~ms. (b) Reversible feather bending with a
250~g load.\label{fig5}}
\end{figure}



\section{Right-handed versus left-handed slices} 


Videos 2 \cite{addref2az,addref2bz,addref2cz} and 3
\cite{addref3az,addref3bz} show different slices performed by elite
right-handed and left-handed players. We extracted from the video
analysis the temporal evolution of the shuttlecock rotation speed
$\omega (t)$ around its symmetry axis. The cross-court slice shots were
performed as shown in Figures~\ref{fig3}, \ref{fig6}a and \ref{fig6}b,
with left-handed players at the left corner of the court, sending the
shuttlecock in diagonal at the opposite side just behind the net. We
used the mirror symmetry situation with the right-handed players
performing the slice shots from the right corner of the court towards
the opposite side behind the net. Figure~\ref{fig6}c shows the
evolution of shuttlecock rotation speed $\omega (t)$ obtained from 3
slice shots performed by right-handed players (RH1, RH2 and RH3). The
slice allows generating very high spinning, of the order of 100
revolutions per second ($\omega \simeq 100$~rps) just after contact
with the racket. The counter-clockwise rotation is chosen as positive
here, which corresponds to the natural rotation of the shuttlecock as
it flies in the air. Figure~\ref{fig6}c shows that the rotation speed
rapidly decreases, with an apparent exponential decay. After
${\simeq}$300~ms, on approaching the net, the rotation speed is of the
order of 10--20~rps. The slice shot of left-handed players (LH1-3)
generates similar fast spinning (${\simeq}$100~rps) after contact with
the racket, but the main difference is that the shuttlecock spins
clockwise ($\omega <0$)! As the rotation accelerates (decelerates in
absolute value), the shuttlecock goes out of spin after
${\simeq}$100~ms, then starts to spin counter-clockwise due to air
pressure on the chiral feather.

\begin{figure}
\vspace*{-2pt}
\includegraphics{fig06}
\vspace*{-4pt}
\caption{\fontsize{9.8}{12}\selectfont (a,b) Slice shot performed by right-handed and left-handed
players, inducing opposite shuttlecock spinning after contact with the
racket. (c) Rotation speed $\omega (t)$ after slice shots. (d)
Schematic representation of airflow (blue) with the spinning of the
shuttlecock.\label{fig6}}
\end{figure}

Around 300~ms after contact with the racket, the shuttlecock also
reaches ${\simeq}$15~rps, which is similar to the spinning observed
for right-handed players. Since experimental data show an apparent fast
exponential decay of rotation speed, followed by an almost linear
decay, we used the following empirical model to fit the curves:
{\begin{eqnarray}
\omega (t)=\omega _{i}\exp \left(-\frac{t}{\tau }\right)+\omega _{0}-at
\label{eqn1}
\end{eqnarray}}\noindent
$\omega _{i}+\omega _{0}$ corresponds to the initial rotation speed,
just after contact with the racket and $\tau $ is the spinning
deceleration time constant, which depends on shuttlecock, air density,
etc. The fast-exponential decay corresponds to the equilibration of the
shuttlecock rotation speed with its velocity. The almost linear
decrease of rotation speed shown in Figure~\ref{fig6}c ($\omega
_{0}-at$) corresponds to the natural decrease of rotation rate as
shuttlecock is slowing down. We performed independent fits for the
different slice shots (RH1-3 and LH1-3) with $\omega _{0}=23$~rps and
$a=34~\mathrm{rps}{\cdot}\mathrm{s}^{-1}$  and the results are given in
Table~\ref{tab1}. The average initial rotation just after contact with
the racket are similar for right-handed and left-handed players
(${\simeq}$120~rps). The spinning deceleration time constants are
similar for the three right-handed player shots, with an average $\tau
=32~(6)$~ms ((6) refers to the uncertainty from the fit) and globally
larger for left-handed players ($\tau =61~(10)$~ms). These different
spinning deceleration time constants are due to the aerodynamic of the
spinning shuttlecock. Figure~\ref{fig6}d shows that for a clockwise
rotation of the shuttlecock the airflow runs parallel to the feathers,
while for a counter-clockwise rotation the airflow collides with the
edges of the feathers. Friction with the air is therefore greater for
the counter-clockwise rotations induced by right-hander during slice
shots, which results in faster angular deceleration compared to
left-hander slice shots.

\begin{table}
\caption{Fitting parameters from Equation (\ref{eqn1}) for the different slice
shots for right-handed and left-handed players\label{tab1}}
\begin{tabular}{ccccccccc}
\thead
& RH1 & RH2 & RH3 & $\langle\mathrm{RH}\rangle$ & LF1 & LF2 & LF3 &
$\langle\mathrm{LF}\rangle$ \\
\endthead
$\omega _{i}$ (rps) & 103 (6) &  112 (13) &  120 (20) &  112 (13) 
& ${-}$131 (3) &  ${-}$121 (5) &  ${-}$133 (7) &  ${-}$128 (6)\0 \\
$\tau $ ($\mathrm{s}^{-1}$) & \033 (3) & \032 (6)\0 & \032 (8)\0 & \032 (6)\0 
& \mn\055 (6) & \mn\080 (7) & \mn\049 (2) & \mn\061 (10) 
\botline
\end{tabular}
\tabnote{The values in parentheses are the uncertainty.}
\end{table}

These very rapid changes in rotation speeds can be clearly seen in
Video~4a \cite{addref4az} (slowed down 500~times). To highlight the
difference between left-handed and right-handed slices, mirror symmetry
is used in Video~4b \cite{addref4bz}, which shows that the shuttlecock
slows down more on a left-handed slice than on a right-handed one. 

As slice shots are performed, players and coaches noticed different
sounds for left-handed and right-handed players. This is also due to
the orientation of the feathers: the racket of the right-handed player
rubs the shuttlecock almost parallel to the feathers, while it is
almost perpendicular for the left-handed players. It may therefore be
easier for left-handed players to spin the shuttlecock, but this
process is too fast to be studied with the present videos set-up. The
slice shot performed by both left- and right-handers generates high
initial spinning rate (${\simeq}$100~rps) of the shuttlecock, which
makes this shot very different from a smash shot. Indeed, we can see in
Video~1a \cite{addref1az} that the smash itself does not
generate rapid rotation after the impact with the racket if the racket
hits the shuttlecock perpendicular to the trajectory. Rotation is
induced by the friction of the air on the feathers, and the shuttlecock
typically reaches ${\simeq}$60~rps within 50~ms (Video~1a
\cite{addref1az}). 


The high rotational speed of the shuttlecock after slice shot is
generating a large centrifugal force. Video~5a \cite{addref5az} shows
the deformation induced by fast spinning perpendicular to velocity. The
drag coefficient for a rotating standard shuttlecock can increase due
to a large centrifugal force \cite{27}. Obayashi \etal\ investigated
the effect of a shuttlecock rotation on its skirt deflection \cite{5}.
They have shown that the skirt enlargement due to the centrifugal
forces is compensated by the effect of the aerodynamic drag when the
shuttlecock travels aligned along its rotation axis. We can see on
different videos (2a, 2b, 2c, 2a, 3a, 4b) that there is no significant
shape change when shuttlecock rotation axis is along velocity, while
shuttlecock feathers clearly bend when the spinning axis is
perpendicular to velocity, as aerodynamic drag does not compensate the
deformation induced the centrifugal force. Figure~\ref{fig7} and
Video~5b \cite{addref5bz} show the deformation of the shuttlecock,
mounted on a Dremel, with rotation speed. The deformation of the
shuttlecock is characterized by the deflection $d$ of the radius
$R(\omega)$ measured at rotation speed $\omega$, with respect to the
radius at rest $R_0$ with $d=R(\omega)-R_{0}$. This deflection is due
to the high centrifugal force, related to centrifugal acceleration
$a_c$, acting on the feather. Considering $a_{c}=R\omega^2$, with
$\omega=785~\mathrm{rad}{\cdot}\mathrm{s}^{-1}$ (125~rps) and
$R=0.0335$~m, the centrifugal acceleration at the edge of the feather
reaches $\mbox{20,643}~\mathrm{m}{\cdot}\mathrm{s}^{-2}$, which almost
corresponds to 2000~g. Even if the mass of the feather is small, the
very high g-force acting on its mass is able to bend it. If we
approximate the mass density on the feather as homogeneous along its
lengths, the deflection is then simply proportional to the g-force,
itself scaling with $\omega^{2}$. The fit in Figure~\ref{fig7},
considering $d\propto\omega^{2}$ is in quite good agreement with
experimental data, given the rough approximation considering the
feather as homogeneous. The shuttlecock deformation observed on
Video~5a \cite{addref5az} just after the slice is similar to the one
observed around 125~rps in Figure~\ref{fig7}, which confirms that its
deformation is due to its high spinning rate after slice shot.

\begin{figure}
{\vspace*{-2pt}}
\includegraphics{fig07}
{\vspace*{-2pt}}
\caption{(a) Shuttlecock deformation with rotation speed observed in
Video 5b \cite{addref5bz}. (b) The radius deflection
$d$ (dots with error bars) increases with rotation speed
$\omega$. The continuous line shows the fit of the data with
$d \propto \omega^{2}$.\label{fig7}}
{\vspace*{-2pt}}
\end{figure}

The clockwise to counter-clockwise shuttlecock rotation induced by the
slice shots of left-handed players is therefore very different from the
one of the right-handed players, as suspected from the symmetry
considerations introduced above, and clearly shown by the super slow
motion videos. For right-handed players, the fast counter-clock
spinning may accelerate the shuttlecock, which experiences after a
continuous decrease of spinning rate due to drag forces. On the
contrary, for left-handers, the initial clockwise rotation should slow
down the shuttlecock. In addition, for left-hander's slice shots the
shuttlecock transiently flies without any spin, and therefore without
rotation energy, before going into its natural counter-clockwise
spinning on approaching the net. This clearly evidences an energy
transfer from kinetic energy to rotational energy in the 100--300~ms
range, which contributes to slowing down the shuttlecock. Indeed,
during this time interval after the slice shot, the rotational energy
increases for left-handers, while it monotonously decreases for
right-handers. We will discuss how this process affects trajectories in
more detail later, because the reader may have noticed a strange
phenomenon on Videos~2
and~3~\cite{addref2az,addref2bz,addref2cz,addref3az,addref3bz}.


\begin{figure}
{\vspace*{-1pt}}
\includegraphics{fig08}
{\vspace*{-1pt}}
\caption{Magnus effect lifting up the projectile after slice shots
performed by right-handed (a, extracted from Video 2c) or left-handed
player (b, extracted from Video 3b). It occurs in the early stage of
the trajectory, as the counter-clockwise (right-handers) or clockwise
(left-handers) spinning axis $\vec{\omega}$ of the shuttlecock is
perpendicular to its velocity $\vec{U}$. The Magnus effect stops
when spinning axis gets collinear to velocity, due to the drag
torque.\label{fig8}}
\end{figure}

\section{Magnus effect after slice shots} 

Videos~2 and 3 
\cite{addref2az,addref2bz,addref2cz,addref3az,addref3bz} show that, for
some slice shots, after contact with the racket the shuttlecock is no
more oriented along its initial speed $\vec {U}$, but almost
perpendicular. For right-handers (Figure~\ref{fig8}a), the shuttlecock
points towards the player. Here again, the initial velocity towards the
net is associated with drag forces (Figure~\ref{fig4}) responsible for
a torque stabilizing the shuttlecock with the cork pointing towards its
velocity. Videos~2 \cite{addref2az,addref2bz,addref2cz} and 4b
\cite{addref4bz} and Figure~\ref{fig8}a clearly show that in addition
to flipping towards its stable orientation, the shuttlecock is lifting
up during the early stage of the fly. This phenomenon is due to the
Magnus effect \cite{13}. When the shuttlecock rotates perpendicularly
to its velocity as it flies, the surrounding air rotates. Therefore,
the air speed on one side increases and decreases on the other side.
Bernoulli's theorem indicates that an increase in fluid velocity
leads to a decrease in pressure, which results in a transverse force,
the Magnus force ($\vec{F_{M}}$), acting on the shuttlecock. The Magnus
force is given by $\vec{F_{M}} \propto \rho V \vec{\omega}\wedge
\vec{U}$, where $\rho$ is the air density and $V$ the volume of the
shuttlecock. We mentioned above that Magnus effect is usually neglected
in badminton, because the rotation axis is usually colinear with
velocity. However, the Videos 2 and 4b clearly reveal that after slice
shots the rotation axis can be almost perpendicular to the velocity
during the first 30~ms, which maximizes the Magnus force at a moment
when the rotation speed $\vec{\omega}$ is also very high (above 50~rps,
Figure~\ref{fig6}c). The angular right-hand rule is very useful to
explain the Magnus effect from a right-handed player
(Figure~\ref{fig8}a): the right index points in the direction of the
rotation ($\vec{\omega}$), the finger index points in the direction of
the velocity ($\vec{U}$) and the thumb is then pointing towards the
Magnus force ($\vec{F_{M}}$). The Magnus force is then lifting up the
shuttlecock. This effect stops within ${\simeq}$30~ms as the shuttlecock
stabilizes its spinning axis $\vec{\omega}$ collinear to its velocity
$\vec{U}$. It is therefore necessary to use super-slow-motion cameras
to observe this transient Magnus effect and the fast initial spinning
responsible for this phenomenon.


If we look at the trajectory of the shuttlecock after a left-handed
player slice shot (Figure~\ref{fig8}b), we can see that it also lifts
up. Indeed, the left-handed player spins the shuttlecock in opposite
directions, compared to the right-handed player, but the initial
velocity also changes direction (Figure~\ref{fig6}b). The cross product
is then unchanged and the Magnus force also lifts up the shuttlecock.
For the left-handed readers, in addition to the right-handed rule
mentioned above, it is suggested to use for this specific case the
angular left-hand rule (Figure~\ref{fig8}b) to understand the direction
of the Magnus effect: the left index points in the direction of the
velocity ($\vec{U}$), the finger index in the direction of the rotation
($\vec{\omega}$) and the thumb is pointing towards the Magnus force
($\vec{F_{M}}$), which is also lifting up the shuttlecock. The
timescales and trajectories of the shuttlecock lifting during the
Magnus effect are similar for right-handed and left-handed players.
This is due to the fact that during both types of slice shots the
initial rotation speeds and velocities are similar for left- and
right-handers. The magnitude of the Magnus effect observed in Videos~2
and 3 \cite{addref2az,addref2bz,addref2cz,addref3az,addref3bz} is in
the range of 20--70~cm, and we observe a weaker Magnus effect for the
slice shots of left-handers. Again, this is due to the implementation
of the feathers, which generates greater friction with the air when the
shuttlecock turns counter-clockwise on right-handed slices
(Figure~\ref{fig6}d) than when the it turns clockwise on left-handed
slices. When the player gives a large transverse component to the
initial velocity of the shuttlecock (with the titled racket or shot),
the shuttlecock flips and its rotation axis $\vec{\omega}$ gets
perpendicular to its velocity $\vec{U}$, maximizing so the cross
product behind the Magnus force, which lifts up the shuttlecock. As
this transverse component is reduced, the Magnus effect lowers in
amplitude. When the shuttlecock spinning axis is perpendicular to its
velocity, the higher the initial spinning and velocities, the stronger
the Magnus effect, which can therefore modify significantly the
trajectories for both left-handed and right-handed players.



\section{Shuttlecock trajectories}

Video~6 \cite{addref6z} and Figure~\ref{fig9} show the global
trajectories for typical slice shots performed by left-handed and
right-handed players. For slice-shots crossing the court over similar
distances, the time of flight is about 15--20\% longer for
left-handed player slice shots (0.91(4)~s versus 1.08(4)~s), as also
highlighted in Video 4b \cite{addref4bz}. This is due to the lowest
average horizontal speed as the shuttlecock is slowed down by its
clockwise initial spinning. As the shuttlecock reaches the net, the
lowest horizontal speed translates in a more vertical trajectory for
left-handers. Of course, many parameters---such as player's
size, eight of the jumps, initial horizontal/vertical speed components,
etc.---can affect the falling angle as the shuttlecock
touches the ground. In Video 6 \cite{addref6z}, where these
parameters are similar, the falling angle is higher
(${\simeq}$65\textdegree) for the left-handed player compared to the
right-handed player (${\simeq}$50\textdegree). Consequently, if the
shuttlecock passes just above the net for both types of players, it
will fall closer to the net in the case of a slice shot performed by a
left-handed player, compared to a right-handed player. The different
ratios between horizontal and vertical speeds, once the shuttlecock has
passed the net, are therefore due to the clockwise rotation of the
shuttlecock after the left-handed slice shot, which rotates in the
opposite direction to the natural rotation. Previous aerodynamic
studies of spinning shuttlecock revealed a weak variation of the drag
force with or without spinning, which may explain the slower trajectory
for left-handed players. However, in the case of the clockwise
spinning, opposite to natural rotation, the drag force and the
aerodynamics may change significantly. In addition, as mentioned above,
the kinetic energy (and therefore the speed) is reduced for the
left-hander's slice shot as part of this energy is transferred to
rotational energy, as the shuttlecock spins clockwise, then stops
spinning before spinning counter-clockwise on approaching the net. Both
phenomena qualitatively contribute to different slowing down of the
shuttlecock between left-hander's and right-hander's slice shot.

\begin{figure}
\includegraphics{fig09}
{\vspace*{-2pt}}
\caption{Typical trajectories for slice shots (yellow) performed by
left-handed (left) and right-handed (right) players. The time of flight
is longer for left-handed players and the slower trajectory translate
in a higher falling angle (red) for left-handers
(${\simeq}$65\textdegree) than right-handers
(${\simeq}$51\textdegree).\label{fig9}}
{\vspace*{-2pt}}
\end{figure}

{\vspace*{-2pt}}

\section{Conclusion}

{\vspace*{-2pt}}

Super slow motion videos provide key information on ultrafast dynamics
intrinsic to badminton physics, driven by extreme stresses applied to
the shuttlecock after racket impact. For manufacturers, it is a real
challenge to propose feather shuttlecocks able to support ultimate
spinning rates (larger than 100~rps), ultrafast flipping or breathing
deformation after impact with the racket. Indeed, there is often a
moment in the game where players have to perform strategic powerful
shots or fast slices. The videos revealed very different spinning
dynamics, caught on the fly, after slice shots performed by left- and
right-handers. A quantitative comparison is difficult because the slice
shots for left- and right-handers are different in nature due to
symmetry. On the one hand, right-handers rub the shuttlecock almost
parallel to the feathers, while left-handers rub it almost
perpendicular. On the other hand, the accelerated counter-clockwise
spinning induced by right-handed players and the clockwise to
counter-clockwise spinning induced by left-handed players intrinsically
generate different trajectories of the shuttlecock, creating a steeper
sliced shot for left-handed players. Although how much of an advantage
this gives left-handers is not clear, it is an advantage. Hopefully,
this is somewhat compensated by the possibility for the players to
perform reverse-slice shots, which are then more advantageous for
right-handers and which could be a subject for future studies. However,
the joint constraints on the arm when the forearm is pronated for a
right-handed player will modify the hitting plane and may also modify
the shuttlecock's speed and behavior. It is important to underline
that a slice shot is efficient when it skims the net. The present
results highlight the importance of balancing the transfer of
transverse speed to the shuttlecock, to limit the Magnus effect. In
fact, too much Magnus effect raises the shuttlecock more than necessary
and thus reduces the effectiveness of the slice shot. This concerns
both right-handed and left-handed players. The players are thus left to
make the right shot and as Robin Williams said, ``What's right is
what's left if you do everything else wrong.''

{\vspace*{-2pt}}
\section*{Declaration of interests}
{\vspace*{-2pt}}
The author does not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and has
declared no affiliations other than his research institution.

{\vspace*{-2pt}}
\section*{Acknowledgements}
{\vspace*{-2pt}}

EC would like to thank the players who participated in this study for
their support and patience (Faustine No\"{e}l, Loris Dufay, Carla Dubois,
Gr\'egoire Bodin, Herveline Crespel, Nicolas Latimier, Romane
Cloteaux-Foucault and Matt\'eo Justel), the coaches for their interesting
comments (Marine Dureault, J\'er\'emy Bloutin, Pierre Chagnot, Erwin
Kehlhoffner and R\'egis Bridel) as well as the Flume Ille Badminton club
and the Plus2Bad arena for their help for the video shooting. The
author would also like to thank Ana\"{\i}s Gauthier, Isabelle Cantat,
Patrick Houizot and Jean-Fran\c{c}ois Le Guennan for their technical
support for the use of the fast camera. EC would also particularly like
to thank Michael Phomsoupha, the reviewer of the article, with whom he
was able to exchange views and revise his article. EC would also like
to thank Gillian Clark and Steen Pedersen for fruitful discussions.

\CDRGrant[ANR]{ANR-19-CE30-0004}

{\vspace*{-2pt}}

\section*{Supplementary material} 

{\vspace*{-2pt}}

The videos are also available on a YouTube channel: click the links:
\vspace*{3pt}

\textbf{Videos 1a and 1b:} Shuttlecock flipping and breathing. 

\url{https://youtu.be/MxU08IsCOJs} and \url{https://youtu.be/uo_bSmp8LRI} 
\vspace*{3pt}

\textbf{Videos 2a, 2b and 2c:} slice right-handed players 

\url{https://youtu.be/9Rwyw9OSROw}, \url{https://youtu.be/UgeZOBo--5M} and\newline\hspace*{.8pc} \url{https://youtu.be/uxcP8iKRBVc} 
\vspace*{3pt}

\textbf{Videos 3a and 3b:} slice left-handed players 

\url{https://youtu.be/PivDQ0f4h5k} and \url{https://youtu.be/edMxKmq-E4M} 
\vspace*{3pt}

\textbf{Videos 4a and 4b:} time evolution of rotation speed 

\url{https://youtu.be/9KG0-MBja6s} and \url{https://youtu.be/j-fQIZEmrMc}
\vspace*{3pt}

\textbf{Video 5:} centrifugal force on the shuttlecock 

\url{https://youtu.be/jNNvOnucV0Q1001} and \url{https://youtu.be/ysxnumuBaSI}
\vspace*{3pt}

\textbf{Video 6:} shuttlecock trajectories \url{https://youtu.be/5HjanVzecrE}   

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{\vspace*{-2pt}}

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