Complex Trajectories of a Simple Pendulum

TL;DR

This paper investigates the complex and real trajectories of a simple pendulum, explores PT-symmetric Hamiltonians with imaginary gravity, and examines the effects of external periodic forcing on its motion.

Contribution

It introduces the analysis of complex trajectories and PT-symmetry in pendulum dynamics, extending classical models to complex and non-Hermitian Hamiltonians.

Findings

Complex trajectories are characterized in detail.

PT-symmetric Hamiltonian dynamics are analyzed.

External forcing effects on complex motion are studied.

Abstract

The motion of a classical pendulum in a gravitational field of strength g is explored. The complex trajectories as well as the real ones are determined. If g is taken to be imaginary, the Hamiltonian that describes the pendulum becomes PT-symmetric. The classical motion for this PT-symmetric Hamiltonian is examined in detail. The complex motion of this pendulum in the presence of an external periodic forcing term is also studied.