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.
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
Whereas a dry twig can be broken with a single snap, breaking a bamboo chopstick produces more than 400 crackling sounds. In a new study, researchers have found similarities between the complex acoustic emission of breaking a bamboo chopstick and the three famous power laws that describe earthquake activity. The scientists also propose that the underlying mechanism behind these laws may be simpler than currently thought.
The researchers, Sun-Ting Tsai et al., from National Tsing Hua University, have published their paper on the sounds of breaking a single bamboo chopstick in a recent issue of Physical Review Letters.
Bamboo and earthquakes Continue reading Earthquake power laws emerge in bamboo chopsticks
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
Knots have such sharp twists and turns that researchers have had trouble determining precisely how a knot’s shape affects the forces within it. But now a team has theoretically modeled a simplified knot, showing the effects on the internal forces when adding or subtracting a turn. The theoretical calculations matched the team’s experiments in which thin metal wires were tied into large open knots. The results may guide future work in characterizing more complicated knots that are pulled small and tight. Continue reading Measuring the Forces in a Knot