On a class of planar geometrical curves with constant reaction forces acting to particles moving along them

In this paper, we find the dependence of the reaction force $N(y)$ of a curved trough of arbitrary shape described by a function $y(x)$. Based on the extremum condition ${dN}/{dx}$ valid for any point of the abscissa axis, we examine the equation $N(y,y',y'')=\operatorname{const}$ whose solutions determine the desired class of curves $y(x)$. We obtain an analytic solution of this equations and perform numerical simulations for various values of parameters. Examples of functions $y(x)$ for which $N=\operatorname{const}$ are presented.