On the Dynamics and Stability of a Pendulum With Lateral Springs

This study characterizes the conservative dynamics of a simple pendulum interacting laterally with a pair of ideal springs—a configuration in which gravitational and elastic restoring forces manifest through nonlinear trigonometric geometry. Grounding the analysis in the Euler–Lagrange formalism, we formulate the exact equations of motion and the governing potential energy, demonstrating that the equilibrium configuration deviates from the vertical axis under asymmetric spring stiffness. Local dynamics are evaluated by linearizing the planar system about its true equilibrium, yielding the associated Jacobian matrix, the fundamental frequency of small oscillations, and a center-type classification within the conservative regime. Furthermore, nonlinear stability is established via the conserved Hamiltonian, which serves as a Lyapunov function to verify Lyapunov stability, noting the inherent absence of asymptotic stability in such Hamiltonian systems. A closed-form analytical representation of the full nonlinear model is derived in terms of Weierstrass elliptic functions. This is achieved through the first integral of energy, a tangent half-angle transformation, and a subsequent quartic-to-Weierstrass reduction. This exact solution serves as a rigorous benchmark for the validation of diverse numerical schemes, including finite differences, Chebyshev and Hermite spline collocation, and a two-step predictor–corrector integrator. Comparative results exhibit excellent agreement, with amplitude-normalized relative errors remaining below one percent across the tested regimes, while the spectral approach yields significantly higher precision. Finally, sensitivity analysis with respect to stiffness parameters reveals a qualitative transition from a gravity-dominated monostable regime to an asymmetric bistable state as elastic effects prevail. These findings establish a comprehensive analytical–numerical framework for pendulum–spring architectures, with direct implications for vibration isolation and nonlinear mechanical design.