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

About four centuries ago, considering flat sections of cone x 2 + y 2 = z 2 (along the axis of revolution on the plane Oxy ), Robert Hooke wrote one fundamental differential equation ( x , y , z ) ″ = − 4 π 2 k ( x 2 + y 2 + z 2 ) 3 ⋅ ( x , y , z ) , which thereafter set the foundation of the law of...

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Bibliographic Details
Published in:Moscow University mathematics bulletin 2019-07, Vol.74 (4), p.147-158
Main Authors: Gerasimova, O. V., Razmyslov, Yu. P.
Format: Article
Language:English
Subjects:
Analysis
Charged particles
Differential equations
Kepler laws
Mathematics
Mathematics and Statistics
Parameterization
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Summary:About four centuries ago, considering flat sections of cone x 2 + y 2 = z 2 (along the axis of revolution on the plane Oxy ), Robert Hooke wrote one fundamental differential equation ( x , y , z ) ″ = − 4 π 2 k ( x 2 + y 2 + z 2 ) 3 ⋅ ( 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.
ISSN:0027-1322
1934-8444
DOI:10.3103/S002713221904003X