The parametric brachistochrone with friction, curvature, and drag

Following a variational approach, here, we present a novel dynamical model for the parametrically generalized brachistochrone for a mass point motion that includes the Coulomb friction, the path curvature, and viscous drag forces. Consideration of viscous drag resulted in an unexpectedly complex problem that, unlike friction, could not be reduced. However, this also disclosed the challenging mechanism and essence of viscous drag in determining the more comprehensive brachistochrone. We demonstrate that the friction and drag control the dynamics fundamentally differently, which is explained physically with a non-linear complex mechanical behavior of the general brachistochrone, constructed here as a second-order non-linear differential equation governing the motion of the material point. It emerges in an extensive, complex, and compact form. The physical and mathematical consistency of our general brachistochrone is demonstrated as it recovers classical brachistochrones without friction, without drag, and both. It appears that the dynamics associated with drag is much more complex than that with friction. We identified the mechanical reason for this. We show that increasing resistance, both frictional and drag, flattens the brachistochrone that controls and architects the optimal time path. Our results manifest that in the presence of friction and drag, the optimal descent curve must be carefully calculated, underscoring the necessity of incorporating essential energy dissipations in accurate modeling of real-world problems of particle motion through a viscous fluid. This study provides a more realistic, physically better explained description of the brachistochrone associated with different resistive forces, with potential applications to physical, engineering, and environmental problems.