Duality symmetries behind solutions of the classical simple pendulum

Abstract

Describing the motion of the classical simple pendulum is one of the aims in every undergraduate classical mechanics course. Its analytical solutions 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 this 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. A discussion of the physical consequences that this symmetry has on the motions of the simple pendulum is presented in this contribution 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. Thus by studying deeper a very familiar mechanical system, it is possible to get an insight to more abstract physical and mathematical concepts. In particular a single solution of pure imaginary time for all allowed values 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.