# Pendulum Analysis by Leaf Functions and Hyperbolic Leaf Functions

## TL;DR

This paper explores the application of leaf functions and hyperbolic leaf functions to analyze large-angle pendulums, offering potentially more accurate solutions than traditional linearization methods for nonlinear equations.

## Contribution

It introduces the use of leaf functions and hyperbolic leaf functions to model pendulum motion at large angles, extending their application beyond previously studied differential equations.

## Key findings

- Leaf functions effectively model large-angle pendulum motion.
- Hyperbolic leaf functions provide solutions for overdamped pendulums.
- New analytical approach improves understanding of nonlinear pendulum dynamics.

## Abstract

The mathematical model representing the equation of motion of a pendulum is nonlinear. Solutions that satisfy the equation cannot be represented by elementary functions, such as trigonometric functions. To solve such problems, it is common to linearize the nonlinear equations and derive approximate numerical solutions and exact solutions. Applying such linearization is limited to cases in which the angle of the pendulum is relatively small. In cases where the angle of the pendulum is large, various methods have been presented that rely on numerical solutions and the exact solutions based on Jacobian elliptic functions, to name one example. On the other hand, the author has been studying a certain type differential equation. The second derivative of a function is equal to the term multiplied by $-n$(or $n$) for terms whose function is raised to $2n-1$. Curves based on solving this differential equation are constructed as regular waves with periods. The author has termed the function satisfying this differential equation the leaf function (or the hyperbolic leaf function). In this paper, we attempt to apply this leaf function (or hyperbolic leaf function) to undamped pendulums and overdamped pendulums for large angles.