Data and Code Availability

Universal spectral structure in pendulum-like systems

Teepanis Chachiyo , Department of Physics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand.

File List

1. validation.ipynb: Python notebook producing the results and figures in the paper
2. simple_example.ipynb: simple and minimal example of ploting the exact solution
3. animation.ipynb: Python notebook generating GIF animation using the derived exact solutions
4. nonlinear_motion.gif: GIF animation showing the 3 classes of motions

The research article preprint >> https://arxiv.org/abs/2504.16816

Abstract

Pendulum-like dynamics appear in a multitude of physical and engineering applications, ranging from energy harvesting mechanisms to superconducting qubits and cold-atom tunneling platforms. Their trajectory patterns are classified into regimes, traditionally written as disjointed special functions, presented exclusively in the time domain. Therefore, their piece-wise and special mathematical nature makes them difficult to integrate into research that explores parameters across all regimes, especially in applications that require direct control over the frequency content. Here we present an exact frequency-domain formulation of the pendulum equation that applies uniformly across oscillatory, separatrix, and rotational regimes. The resulting spectral representation reveals a previously hidden unification: all regimes share the same analytic spectral structure and characteristic frequency scale. We discover that all regimes arise from a single universal spectral kernel, with parity selection distinguishing the periodic motions and the separatrix representing their discrete-to-continuum limit. Regime changes thus correspond to symmetry-driven reorganizations in frequency space rather than changes in the underlying spectral structure, with the stopping trajectory representing the continuum limit reached without system-size scaling. Beyond providing closed-form solutions, the framework reveals a transparent spectral structure underlying a broad class of classical and quantum pendulum-like systems.

Citation

In the meantime, if you use any part of this repository please cite the following preprint:

@article{Chachiyo:2026uss,
    author = "Teepanis Chachiyo",
    title = "{Universal spectral structure in pendulum-like systems}",
    eprint = "2504.16816",
    archivePrefix = "arXiv",
    primaryClass = "physics.class-ph",
    month = "4",
    year = "2026"
}

'Hello-World' example for the swinging nonlinear pendulum

# exact solution of nonlinear pendulum via spectral analyis
# https://github.com/teepanis/nonlinear-pendulum
import numpy as np
import scipy as sp
import matplotlib.pyplot as plt

# physics: amplitude, and OmegaL=sqrt(g/L)
theta0 = 179.9/180*np.pi
OmegaL = np.sqrt(9.8/1)

k = np.sin(theta0/2)
T = 4*sp.special.ellipk(k**2)/OmegaL
Omega0 = 2*np.pi/T
kappa = sp.special.ellipk(1-k**2)

t = np.linspace(0,2*T,200) + T/4
theta = np.zeros(len(t)) 

# adding odd harmonics 
for n in range(1,40,2):
    c = 4/n/np.cosh(kappa*n*Omega0/OmegaL)
    theta = theta + c*np.sin(n*Omega0*t)

plt.plot(t, theta)
plt.grid()
plt.show()

To compare with the traditional perspective we:

1. use the initial condition that the pendulum starts at rest, with an amplitude $\theta_0$.
2. For this condition, we shift the time by $T/4$.
3. For convenience, we use the form $k = \sin(\theta_0/2)$, which is equipvalent to $k = \omega_m/\omega_c$.