Rolling heavy ball over the sphere in real Rn3 space

Abstract

We propose that a system is in a gravitational field and that the rolling of a heavy ball, over a sphere, is activated by a force of proper weight and initial kinetic and potential energies given to the ball at the initial moment. For mathematical description of the rolling without slipping of the heavy rigid homogeneous ball over the sphere on both the inside and the outside of the sphere surface, spherical coordinates are used: angle in circular and angle in meridional directions, and angle of the ball self-rotation about radial direction. Ordinary nonlinear differential equations are derived. The angle coordinate in the circular direction is a cyclic coordinate, and an integral for the circular-cyclic coordinate is derived. Integral constant that depends on initial conditions is determined. The main nonlinear differential equation is expressed through a meridional angle coordinate, and a corresponding first integral is derived. The equation of the first integral is an equation of phase trajectory. By using this equation and the corresponding set of initial conditions, phase trajectory portraits are graphically presented. An elliptic integral is derived. By using new Hedrih’s results in theory of collision between two rolling bodies, geometry, kinematics and dynamics of successive collisions of two rolling balls over the surface of a sphere are analyzed and a methodology for investigation of vibro-impact nonlinear dynamics of vibro-impact system with rolling bodies over the sphere surface is described. A mathematical analogy between differential equations and phase trajectory portraits of dynamics of a rolling heavy ball and a material heavy mass particle moving along sphere surface is identified. New original results are visible in the first part of abstract of the manuscript. Let’s, again, point out in short list the main new original author’s results, different than published results of a number of mathematicians: (1) new nonlinear differential equations as description of the rolling, without slipping, of the heavy rigid homogeneous ball over the sphere, on both the inside and the outside, of the sphere surface, presented in spherical coordinates, (2) first integral of new nonlinear differential equation as description of the rolling, without slipping, of the heavy rigid homogeneous ball over the sphere, on both the inside and the outside, of the sphere surface, presented in spherical meridional coordinate, (3) first graphical presentation of a series of qualitative phase trajectory portraits on phase plane with phase coordinate meridional angle coordinate and meridional angular velocity, (4) proof that in the rolling, without slipping, of the heavy rigid homogeneous ball over the sphere, on both the inside and the outside, of the sphere surface, the circular angle coordinate is cyclic coordinate, and obtained cyclic integral which shows the nonlinear dynamics of a rolling heavy ball over the sphere is with constant sectorial velocity (it can be accepted as a theorem), (5) application of the new Hedrih’s results in theory of collision between two rolling bodies, in geometry, kinematics and dynamics of successive collisions of two rolling balls over the surface of a sphere, (6) a new methodology for investigation of vibro-impact nonlinear dynamics of vibro-impact system with rolling bodies over the sphere surface, (7) a mathematical analogy between nonlinear differential equations and phase trajectory portraits of dynamics of a rolling heavy ball and a material heavy mass particle moving along sphere surface is identified. Previously listed new original results of nonlinear dynamics of a ball rolling over a sphere suitable for inclusion in a university monograph on dynamics for postgraduate study are presented. Main importance of listed results is applicability in investigation of the nonlinear dynamics of the numerous engineering systems.