For more on how to find the instability intervals, you can look up the annotations on ["A Pendulum Theorem" by D.J. Acheson](https://fermatslibrary.com/p/a3733a74) that has been annotated on Fermat's Library as well. Joseph Keller is a professor at Stanford where he used to go jogging around the campus. He was struck by all the women running around campus and the bouncing motion of their ponytails from side to side. In this paper Keller tries to understand how a vertical displacement of the head creates an horizontal swing of the ponytail. This paper was also awarded the [2012 Ignobel prize in Physics](https://blogs.scientificamerican.com/scicurious-brain/ignobel-prize-winner-in-physics-the-amazing-ponytail/)  You might think that if you run faster, the jogging frequency will increase, making it harder for the ponytail to swing, but that's not the case. In most of the situations, running faster will result in longer steps, while keeping the frequency constant and thus maintaining the ponytail swing. In order to derive this equation of motion, one can use the expression for the Torque  $$ \tau = I\alpha \equiv -m(g+a(t))\frac{L}{2} \sin \theta = m\frac{L^2}{2^2} \frac{d^2 \theta}{dt^2} \\ \frac{d^2 \theta}{dt^2} + (g+a(t))\frac{2}{L} \sin \theta = 0 $$ where $\tau$ is the torque, $\alpha$ is the angular acceleration and $I$ is the moment of inertia. D.J. Acheson studied the behaviour of a hanging chain of several linked pendulums, all suspended from one another, and discovered various different modes of oscillation (his paper [has been previously annotated on Fermat's Library](https://fermatslibrary.com/p/a3733a74)) Acheson proved that it was possible to take all these linked pendulums, turn them upside-down, so that they are all precariously balanced on top of one another, and then stabilise them in that position by vibrating the pivot up and down, as we can see in this awesome demonstration by Steve Mould. [](https://youtu.be/gnn21smGVrQ?t=56 "Inverted Pendulum") He started calling this gravity-defying experiment "Not quite the Indian rope trick" because of a famous magic trick performed in India during the 19th century. During this trick, a magician would hurl a rope into the air. The rope would stand erect, with no external support. His boy assistant would climb the rope and then descend. In the context of running, this means for "punk rock" hairstyle the hair will be mostly in a stable equilibrium!  In order to derive the equation of motion for a perfectly flexible string (a string that offers no resistance to bending) we start by considering an infinitesinally thin segment of string with mass density $\rho$ and mass $\rho \Delta x $. Using Newton's law for a point mass for the vertical component:  $$ F=ma \\ \rho \Delta x \frac{\partial^2 u}{\partial t^2} = T(x+\Delta x, t) \sin \theta(x+\Delta x, t)-T(x, t) \sin \theta(x, t)+\rho \Delta x g $$ where $T(x,t)$ is the magnitude of the tensile force at point $x$ and time $t$ and $g$ is the acceleration of gravity. Dividing the previous expression by $\Delta x$ and taking the limit as $\Delta x \rightarrow 0$ we get $$ \rho \frac{\partial^2 u}{\partial t^2} = \frac{\partial}{\partial x}(T(x, t) \sin \theta(x, t))+\rho g $$