The Gravity First (on Reincarnation of Third Kepler’s Law)

Abstract

About four centuries ago, considering flat sections of cone x2 + y2 = z2 (along the axis of revolution on the plane Oxy), Robert Hooke wrote one fundamental differential equation \((x,y,z)^{\prime\prime} = - {{4{\pi ^2}k} \over {{{(\sqrt {{x^2} + {y^2} + {z^2}})}^3}}}\; \cdot \;(x,y,z)\), which thereafter set the foundation of the law of universal gravitation and explanation of movement of charged particle in the classical stationary Coulomb field. In this paper, differential-algebraic models arising as the result of replacement of a cone by an arbitrary quadric surface F(x, y, z) = 0 with respect to (as we call it) the Kepler parametrization of quadratic curves {F(x, y, α · x + β · y + δ)=0 | α, β, δ ∈ K}, K = ℝ, ℂ, are proposed and studied.