Revealing Fractional Effects in an Asymmetric Two-Dimensional Oscillator through Variational Mechanics

Abstract

The present study is an application of a generalized Lagrangian-Hamiltonian approach to the analysis of an asymmetric two-dimensional harmonic oscillator with the use of the Caputo fractional derivative. Fractional Euler-Lagrange and Hamiltonian equations were systematically developed and then solved numerically using a predictor-corrector version of the Adams-Bashforth-Moulton method. The results show that decreasing the value of the fractional order (alpha) generates a form of memory-based damping within the system that transforms the classical closed orbits of phase space into spiral inward trajectories and causes the total energy of the system to decrease according to a power law, regardless of whether there are any dissipative forces in the system. It was demonstrated that both the degree of coupling and the degree of asymmetry can cause the system to lose energy at different rates and to have different amounts of attenuation in each direction. It was also demonstrated that the model provides a smooth transition to classical conservative behavior as alpha approaches unity, which confirms the physical validity of this representation. This demonstrates that fractional variational mechanics provides a consistent and physically meaningful way to describe the transition from conservative to dissipative behavior for coupled oscillating systems that are governed by memory and non-local effects.