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

Pendulum-like dynamics is a universal motif across many areas of physics, underlying systems ranging from classical nonlinear oscillators to superconducting qubits and cold-atom tunneling platforms. 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. The spectral structure can be derived via a spectral discretization approach starting from the separatrix solution, without relying on the classical Jacobi elliptic formulation. Beyond providing closed-form solutions, the framework reveals a transparent spectral structure underlying a broad class of classical and quantum pendulum-like systems.

Pendulum Motion

3 Classes of Nonlinear Pedulum Motion

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"
}

# 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 $k = \omega_m/\omega_c$.