Circular motion of a particle under friction and hydraulic dissipation

Abstract

A heavy particle travels along a rough circular path on a horizontal plane. Launched with a non-zero initial speed, due to the absence of other acting forces, but in presence of friction, it will slow down and eventually stop. The relevant transient is then solved in terms of the Jacobian elliptic functions of the time. Hydraulic air drag is the considered and the new resulting motion is ruled by the first order nonlinear ODE

$$ \dot{w}=-r\,w^{2}-C\,\sqrt{1+w^{4}}, $$

$ (r,C)>0 $, which is solved by means of the first kind Appell-Picard hypergeometric function of two variables $ F_{1}(a,b,b^{\prime };c;x,y)$.