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In addition, Newton had formulated, in Propositions 43–45 of Book 1 [16] and associated sections of Book 3, a sensitive test of the accuracy of the inverse square law, in which he showed that only where the law of force is calculated as the inverse square of the distance will the directions of orientation of the planets' orbital ellipses stay ...
The divergence of a vector field which is the resultant of radial inverse-square law fields with respect to one or more sources is proportional to the strength of the local sources, and hence zero outside sources. Newton's law of universal gravitation follows an inverse-square law, as do the effects of electric, light, sound, and radiation ...
Problem 4 then explores, for the case of an inverse-square law of centripetal force, how to determine the orbital ellipse for a given starting position, speed, and direction of the orbiting body. Newton points out here, that if the speed is high enough, the orbit is no longer an ellipse, but is instead a parabola or hyperbola .
The inverse square law behind the Kepler problem is the most important central force law. [1]: 92 The Kepler problem is important in celestial mechanics, since Newtonian gravity obeys an inverse square law. Examples include a satellite moving about a planet, a planet about its sun, or two binary stars about each other.
In principle, the same solutions apply to macroscopic problems involving objects interacting not only through gravity, but through any other attractive scalar force field obeying an inverse-square law, with electrostatic attraction being the obvious physical example. In practice, such problems rarely arise.
In classical physics, many important forces follow an inverse-square law, such as gravity or electrostatics. The general mathematical form of such inverse-square central forces is F = α r 2 = α u 2 {\displaystyle F={\frac {\alpha }{r^{2}}}=\alpha u^{2}} for a constant α {\displaystyle \alpha } , which is negative for an attractive force and ...
In practice, it is often parameterized to fit a specific situation, such as I = A ( d + B ) k , {\displaystyle I={\frac {A}{(d+B)^{k}}},} in which the constant A is a vertical stretching factor, B is a horizontal shift (so that the curve has a y-axis intercept at a finite value), and k is the decay power.
Informally, the case of a point charge in an arbitrary static electric field is a simple consequence of Gauss's law.For a particle to be in a stable equilibrium, small perturbations ("pushes") on the particle in any direction should not break the equilibrium; the particle should "fall back" to its previous position.