ROLLING A HEAVY BALL ON A REVOLVING SURFACE

Abstract

The paper defines five theorems on the properties of newly constructed orthogonal curvilinear coordinate systems on various revolving surfaces and ten theorems on the non-linear dynamics of non-slip rolling of a heavy, homogeneous and isotropic ball on revolving surfaces. Nonlinear differential equations of rolling, without sliding, a heavy, homogeneous and isotropic ball, as well as equations of phase trajectories are derived for two special cases, when the revolving surfaces are created by the rotation of a parabola, i.e., a biquadratic parabola. It is shown that for such nonlinear rolling dynamics there is a cyclic integral, as well as one cyclic coordinate in all cases of revolving surfaces. In this paper, we present only two theorems which are part of the scientific results of the research. Based on these, five theorems on curvilinear orthogonal coordinate systems constructed over surfaces of revolution and ten theorems on the properties of the dynamics of rolling a heavy ball on surfaces of revolution were defined.