Plane Pendulum and Beyond by Phase Space Geometry

Bradley Klee
By careful analysis, the small angle approximation leads to a wildly inaccurate prediction for the period of a simple plane pendulum. We make a perturbation ansatz for the phase space trajectory of a one-dimensional, anharmonic oscillator and apply conservation of energy to set undetermined functions. Iteration of the algorithm yields, to arbitrary precision, a solution to the equations of motion and the period of oscillation. Comparison with Jacobian elliptic functions leads to multidimensional applications such as the construction of approximate Seiffert spirals. Throughout we develop a quantum/classical analogy for the purpose of comparing time-independent perturbation theories.
Read more at http://arxiv.org/pdf/1605.09102v2.pdf

Duality symmetries behind solutions of the classical simple pendulum

Román Linares
The solutions that describe the motion of the classical simple pendulum have been known for very long time and are given in terms of elliptic functions, which are doubly periodic functions in the complex plane. The independent variable of the solutions is time and it can be considered either as a real variable or as a purely imaginary one, which introduces a rich symmetry structure in the space of solutions. When solutions are written in terms of the Jacobi elliptic functions the symmetry is codified in the functional form of its modulus, and is described mathematically by the six dimensional coset group Γ/Γ(2) where Γ is the modular group and Γ(2) is its congruence subgroup of second level. In this paper we discuss the physical consequences this symmetry has on the pendulum motions and it is argued they have similar properties to the ones termed as duality symmetries in other areas of physics, such as field theory and string theory. In particular a single solution of pure imaginary time for all allowed value of the total mechanical energy is given and obtained as the S-dual of a single solution of real time, where S stands for the S generator of the modular group.

I. INTRODUCTION
The simple plane pendulum constitutes an important physical system whose analytical solutions are well known.
Historically the first systematic study of the pendulum is attributed to Galileo Galilei, around 1602. Thirty years later he discovered that the period of small oscillations is approximately independent of the amplitude of the swing, property termed as isochronism, and in 1673 Huygens published the mathematical formula for this period. However, as soon as 1636, Marin Mersenne and Rene Descartes had stablished that the period in fact does depend of the amplitude. The mathematical theory to evaluate this period took longer to be established.
The Newton second law for the pendulum leads to a nonlinear differential equation of second order whose solutions are given in terms of either Jacobi elliptic functions or Weierstrass elliptic functions …
… Read more at http://arxiv.org/pdf/1601.07891v1.pdf

Aside

The Ping Pong Pendulum

Bob moves on a circular arc centered at the left pivot when swinging to the right, and on an arc centered at the right pivot when swinging to the left.

Bob moves on a circular arc centered at the left pivot when swinging to the right, and on an arc centered at the right pivot when swinging to the left.

Peter Lynch
Many damped mechanical systems oscillate with increasing frequency as the amplitude decreases. One popular example is Euler’s Disk, where the point of contact rotates with increasing rapidity as the energy is dissipated. We study a simple mechanical pendulum that exhibits this behaviour…

… Read more at http://arxiv.org/pdf/1512.03700v1.pdf

Aside

Huygens synchronization of two clocks

Henrique M. Oliveira, Luís V. Melo
The synchronization of two pendulum clocks hanging from a wall was first observed by Huygens during the XVII century. This type of synchronization is observed in other areas, and is fundamentally different from the problem of two clocks hanging from a moveable base.
We present a model explaining the phase opposition synchronization of two pendulum clocks in those conditions. The predicted behaviour is observed experimentally, validating the model.
…Read more at www.nature.com

A pendulum of horror

In 1842, the first American author of tales of horror, Edgar Allen Poe (1809-1849) wrote a short story entitled, The Pit and the Pendulum. Poe’s stories often contained a strong element of terror, in part, because he left many of the details quite vague, just as a standard technique of psychological terror is to keep the victim in ignorance as to his ultimate fate.
The Pit and the Pendulum does exactly that to both the reader and the protagonist. The main, and practically only character, whose name we never know, is brought as a prisoner before the court of the Spanish inquisition in Toledo, Spain. The trial is recalled by the prisoner during a confused dreamlike state. Subsequently, he is carried into the bowels of the earth and flung into a damp and dark dungeon.
He attempts to investigate the physical condition of his cell but exhaustion forces sleep upon him.
When our hero awakes, he is tied to a low cot with only one hand free with which to feed himself the spiced meat that is mysteriously laid beside the cot. He now notices a large pendulum high above the cot and observes the start of its oscillations.
However, the presence of rats attempting to steal his food distracts his attention from the pendulum. Meanwhile the pendulum bob, now seen to be in the shape of a large sharp metal blade, comes ever closer to his person.
The descent of the pendulum is tortuously slow giving our hero a chance to assess his situation. The strap by which he is held to the cot is a long single piece that is wound many times around his body.
His first thought is that the pendulum might eventually cut the strap and allow him to free himself before he suffers his apparently inevitable fate.
But, to his chagrin, he notes that the only place where the strap does not cover his body lies in the path of the pendulum.
Therefore he needs to devise some other method of regaining freedom of movement.
The story continues as the pendulum draws ever closer with increasingly larger amplitudes for swing.

O Botafumerio, the giant censer that hangs in the transept of the cathedral of Santigo de Compostela. The rapid motion through the air fans the hot incense coals, making copious amounts of blue smoke, and the censer itself generates a frightening swooshing sound as it passes through the bottom of its arc

Like O Botafumerio, this pendulum can also be heard to make an ominous swishing sound as it describes its increasing arc. As the pendulum nears the prisoner’s body he estimates the total range of the pendulum’s motion to be about thirty feet.
Given that the room itself is only about forty feet high, this pendular motion is of very large amplitude, again similar to O Botafumerio. At this point we refer readers to the original story in order to learn the fate of the prisoner.
However, it is of interest to ask whether Poe’s tale is realistic. Does it seem likely that the inquisition would have used something like a motorized pendulum in this context?
The answer is probably no. The time frame of the story is not clear but we may impose some limitations. The Spanish inquisition ended in the early nineteenth century and the story itself refers to capture of the city by a contain French General La Salle.
Perhaps this allusion is to the Napoleonic wars, but the time could also be much earlier. Scholars (LLorente 1823; Lea 1907) suggest that while torture was a standard and accepted means of learning truth in matters both secular and religious, it was not likely to be used gratuitously by the inquisition once sentence had been passed. And, in the story, there had been a trial and a sentence pronounced.
Furthermore, instruments of torture were fairly simple and direct. A driven pendulum that would slowly and noiselessly descend, and slowly increase its amplitude of motion, all inside and above a small dungeon, is relatively complex and hardly worth the effort even if craftsmen could be found to build such a machine.
Therefore Poe’s story is somewhat unrealistic in this regard. Some believe that the inspiration for Poe’s tale was actually a large swinging bell that Poe had observed. Nevertheless, realistic or not, the image of a sharp-edged pendulum makes it frightening.
It is the regularity and inexorability of the pendulum’s motion that contributes to the climate of terror found in this story. The pendulum’s periodicity, that is so important in other contexts, is here made to serve the cause of literary suspense…..
The pendulum: a case study in physics, Gregory L. Baker,James A. Blackburn


http://youtu.be/buRFANaKp0c