Analysis of a Novel Path-Dependent Force in Classical Dynamics

Analysis of a Novel Path-Dependent Force in Classical Dynamics (Stefanou Antonios) This paper introduces and analyzes a novel form of force in classical dynamics characterized by its explicit dependence on the particle's path history. The force is composed of a driving term with a time-varying direction and a magnitude that oscillates based on the total arc length traveled, and a Lorentz-like term that induces spiraling motion. We formulate the system's integro-differential equation of motion and solve it numerically to explore its rich dynamics. The results reveal a diverse spectrum of behaviors, including stable helical motion, powerful resonant energy pumping, and distinct responses in the adiabatic (ωk≪ωc) and rapidly oscillating (ωk≫ωc) regimes. Furthermore, we identify parameter sets that lead to chaotic dynamics, confirmed via phase space analysis and a positive Lyapunov exponent. This model serves as a compelling theoretical framework for studying systems with memory effects in physics and engineering. The associated Python code for all simulations is also provided.