The Analysis of Rotated Vector Field for the Pendulum

TL;DR

This paper investigates the behavior of a nonlinear pendulum with damping and torque using rotated vector fields, establishing a key relation between force and periodic solutions with implications for physics phenomena.

Contribution

It introduces the application of rotated vector fields to analyze the nonlinear pendulum, deriving a critical force value in over damping conditions.

Findings

Critical force value is fixed in over damping

Relation between applied force and periodic solutions derived

Results applicable to charge-density wave studies

Abstract

The pendulum, in the presence of linear dissipation and a constant torque, is a non-integrable, nonlinear differential equation. In this paper, using the idea of rotated vector fields, derives the relation between the applied force $\beta$ and the periodic solution, and a conclusion that the critical value of $\beta$ is a fixed one in the over damping situation. These results are of practical significance in the study of charge-density waves in physics.