Abstract:
About four senturies ago considering flat sections of cone $x^2+y^2=z^2$ (along the axis of rotation on plane $Oxy$), Robert Hooke wrote one of fundamental differential equations $(x,y,z)^{\prime\prime}=-\frac{4 \pi^2k}{(\sqrt{x^2+y^2+z^2})^3}\cdot(x,y,z)$, which thereafter set the foundation of the law of universal gravitation and explanation of movement of charged particle in classical stationary Coulomb field. In the present work differential-algebraic models, arising as the result of replacement of cone with an arbitrary quadric surface $F(x,y,z)=0$ with respect to (called by us) Kepler parametrization of quadratic curves $\{F(x,y,\alpha\cdot x+\beta\cdot y+\delta)=0\:|\:\alpha,\beta,\delta\in K\},\:K=\mathbb{R},\mathbb{C}$, are proposed and studied.
Key words:flat curve, its Kepler parametrization; equations of Ptolemy, Hooke, Boltzmann; differential algebra, its rank, analytic spectrum, germ of trajectory, closure of orbit; fields of parabolic, conal, Coulomb, hyperbolic, Ampere, generalized type.
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