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Volume 12: Pages 39-43, 1999
Newton's Hypothetical Orbits Independently Derived
Kern E. Kenyon
4623 North Lane, Del Mar, California 92014‐4134 U.S.A.
The mathematical results of four hypothetical orbital problems from the Principia are confirmed by an independent physical method. Each orbital problem that Newton posed and solved is characterized as follows. Given the shape of the orbit and the position of the force center, find the functional form of the central attractive force that will keep a body moving around the orbit. None of Newton's hypothetical orbital problems has so far found any apparent practical application, whereas the Kepler problem, also solved by Newton in the Principia, is of great importance to physics. The Kepler problem too can be derived easily by the present method. Newton used primarily geometrical constructions and logical deductions to arrive at his force functions. In contrast to this, the present (inverse) approach is based on a force balance: as a body moves along a curved path the outward centrifugal force always balances the component of the inward attractive force that is perpendicular to the orbit. Taking the functional form for the central force derived by Newton and inserting it into the force balance, the orbital shape can be derived by solving an ordinary second‐order differential equation—the forced harmonic oscillator equation. Two of Newton's four force functions examined in this way lead to (different) fully nonlinear differential equations, which, surprisingly, can both be solved analytically and in closed form by means of the elementary functions that describe the shapes of the orbits.
Keywords: Newton's orbital solutions, independent derivation, force‐balance method
Received: February 4, 1997; Published online: December 15, 2008