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SIAM J. APPL. MATH .
2010 Society for Industrial and Applied Mathematics
Vol. 70, No. 7, pp. 2667–2672
JOSEPH B. KELLER
Abstract. A jogger’s ponytail swa ys from side to side as the jogger runs, although her head
does not move from side to side. The jogger’s head just moves up and down, forcing the ponytail to
do so also. We show in two ways that this vertical motion is unstable to lateral perturbations. First
we treat the ponytail as a rigid pendulum, and then we treat it as a ﬂexible string; in each case, it is
hanging from a support which is moving up and down periodically, and we solve the linear equation
for small lateral oscillation. The angular displacement of the pendulum and the amplitude of each
mode of the string satisfy Hill’s equation. This equation has solutions which grow exponentially
in time when the natural frequency of the pendulum, or that of a mode of the string, is close to
an integer multiple of half the frequency of oscillation of the support. Then the vertical motion is
unstable, and the ponytail sways.
Key words. instability, parametric resonance, Hill’s equation
AMS subject classiﬁcations. 34F15, 70J40
1. Introduction. The ponytail of a running jogger sways from side to side, but
the jogger’s head generally does not move from side to side. The head just moves
up and down, so the ponytail also moves up and down with it. But, as we shall
show, this vertical motion of the hanging ponytail is unstable to lateral perturbations.
The resulting lateral motion, the swaying, is an example of parametric excitation, a
phenomenon which is common in oscillating mechanical and electrical systems.
We shall demonstrate this instability, and analyze the resulting motion, in two
diﬀerent ways. First, in section 2, we shall represent the ponytail as a rigid pendulum
hanging from a support which is moving up and down periodically. The pendulum also
moves up and down periodically. Any small angular deviation θ(t) from the vertical
position satisﬁes Hill’s equation, a linear second order ordinary diﬀerential equation
with a periodic coeﬃcient (Stoker ). This equation has one solution, which grows
exponentially in time if the natural frequency of the pendulum is close to an integer
multiple of half the frequency of oscillation of the support (Magnus and Winkler ).
Then the purely vertical motion of the pendulum is unstable, and it sways.
Next, more realistically, in section 3 we represent the ponytail as a ﬂexible string
hanging from a vertically oscillating support. Again a purely vertical motion of the
string is possible. As was shown by Belmonte et al. , the linear equation for small
lateral perturbations of this motion has an inﬁnite number of modes of periodic vibra-
tion. Each mode amplitude satisﬁes Hill’s equation. Therefore, just like the pendulum,
a mode is unstable when its natural frequency is close to an integer multiple of half
the frequency of oscillation of the support.
A still more realistic model of a ponytail is an inextensible rod with small bending
stiﬀness, described in section 5.
2. Ponytail as a rigid ro d. Suppose that a runner moves with constant speed
U along the positive z-axis, and that her head moves up and down with the periodic
Received by the editors May 29, 2009; accepted for publication (in revised form) May 6, 2010;
published electronically July 29, 2010.
Departments of Mathematics and Mechanical Engineering, Stanford University, Stanford, CA
94305-2125 (k firstname.lastname@example.org).
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