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
Daniel F. Styer
The Laplacian operator can be defined, not only as a differential operator, but also through its averaging properties.
Such a definition lends geometric significance to the operator:
a large Laplacian at a point reflects a “nonconformist” (i.e., different from average) character for the function there. This point of view is used to motivate the wave equation for a drumhead.
…. Read more at scitation.aip.org
A snake crawling on horizontal surfaces between two parallel walls exhibits a unique wave-like shape, which is different from the normal shape of a snake crawling without constraints. We propose that this intriguing system is analogous to a buckled beam under two lateral constraints. A new theoretical model of beam buckling, which is verified by numerical simulation, is firstly developed to account for the special boundary conditions. Under this theoretical model, the effect of geometrical parameters on the deformation shape, such as the distance between walls, length of the snake and radius of the snake, is examined. The buckling beam model is then applied to explain qualitatively the wave-like shape of the snake….
Read more at rsif.royalsocietypublishing.org