Comparison Structure Preserving Integrator with Conventional Method in Energy Conservation From Double Pendulum

Abstract

This study investigates the long-time numerical behavior of a planar double pendulum formulated as a Hamiltonian system under different time integration schemes. The double pendulum is a benchmark chaotic system in which accurate long term energy behavior is essential for obtaining physically meaningful trajectories. The purpose of this work is to compare a conventional explicit method (classical Runge–Kutta), a Störmer–Verlet, and two structure preserving integrators (implicit midpoint and Gauss–Legendre collocation) in terms of phase space dynamics, energy conservation, and computational cost. The equations of motion are integrated numerically with identical physical parameters, time-step size, and simulation horizon for all methods. Phase plots of the angles and canonical momenta are compared with reference results from the literature to verify the correctness of the implementation. Long time energy behavior is then assessed using scalar error metrics derived from the numerical Hamiltonian, and simple runtime benchmarks are used to evaluate efficiency. The results show that the implicit midpoint and the Gauss–Legendre method provide phase portraits consistent with the reference and maintain the energy close to a single level over long times, whereas the classical Runge–Kutta scheme exhibits a mild but systematic drift. The Gauss–Legendre integrator also achieves the smallest energy errors and the highest throughput, making it the most reliable and efficient option among the methods tested. In other words, for long time simulations of Hamiltonian systems, methods that preserve the symplectic structure are more important than merely increasing the formal order of accuracy. These findings provide a quantitative benchmark for geometric integrators on chaotic systems and form a conceptual bridge toward future structure preserving, data driven models such as Hamiltonian Neural Networks.