BRACHISTOCHRONE WITH VISCOUS RESISTANCE

The brachistochrone with consideration of viscous resistance may be anissue in engineering situations. A generalized variational principle with constraints removed is deduced first, and it is also applicable to other formsof resistance and optimization of systems with multiple degrees of freedom.We then obtain the ordinary differential equations describing the functionsof the brachistochrone with viscous resistance, whose solution is reduced toroulette when the viscous resistance is neglected. The effects of theresistance on the brachistochrone are quantitatively analyzed on the basisof numerical computations by using MATLAB. When the viscous resistance isnot very large, the brachistochrone becomes more concave. With the increaseof the resistance, the brachistochrone will become flatter. A very largeresistance will finally lead to a straight-line brachistochrone.