Related papers: Kepler's Differential Equations
The k-calculus was advanced by Hermann Bondi as a means of explaining special relativity using only simple algebra (Bondi H.: Relativity and Common Sense, London, Heinemann, 1964). As used by Bondi, k is Doppler shift. This paper extends…
We present an elementary deduction of the Newtonian force from Kepler's laws. We relate it to a generalization by Jacobi of the Keplerian motion, where the Euclidean form in the plane is replaced by some function with the same homogeneity.…
By numerically solving the mass distribution in a rotating disk based on Newton's laws of motion and gravitation, we demonstrate that the observed flat rotation curves for most spiral galaxies correspond to exponentially decreasing mass…
Lambert's theorem (1761) on the elapsed time along a Keplerian arc drew the attention of several prestigious mathematicians. In particular, they tried to give simple and transparent proofs of it (see our timeline \S 9). We give two new…
This paper presents a brief account of the important milestones in the historical development of the theory of differential equations. The paper begins with a discussion on the date of birth of differential equations and then touches upon…
The equations of motion of massive particles in GR are completely determined by the field equation. We utilize the particular form of Einstein's field equation and propose for the $N$-body problem of the equations that are Lorentz invariant…
We study the pages in Albert Einstein's 1916 landmark paper in the Annalen der Physik where he derived his field equations for gravity. Einstein made two heuristic and physically insightful steps. The first was to obtain the field equations…
We present a simple method to obtain the solution of a few orbital problems: the Kepler problem, the modified Kepler problem by the addition of an inverse square potential and linear force.
It was discovered by Gordon in 1977 that Keplerian ellipses in the plane are minimizers of the Lagrangian action and spectrally stable as periodic points of the associated Hamiltonian flow. The aim of this paper is to give a homotopy…
Newton's Principia, when it appeared in 1687, was received with the greatest admiration, not only by the foremost mathematicians and astronomers in Europe, but also by philosophers like Voltaire and Locke and by members of the educated…
The first three sections of this article contain a broad brush summary of the profound changes in the notion of time in fundamental physics that were brought about by three revolutions: the foundations of mechanics distilled by Newton in…
The present article deals with general mechanics in an unconventional manner. At first, Newtonian mechanics for a point particle has been described in vectorial picture, considering Cartesian, polar and tangent-normal formulations in a…
In 1755, Kant published his General Natural History and Theory of the Heavens, in which he presented his hypothesis on the formation of the solar system, known as the primitive nebula hypothesis. This original theory of the heavens was…
We present a remarkable discretization of the classical Kepler problem which preserves its trajectories and all integrals of motion. The points of any discrete orbit belong to an appropriate continuous trajectory.
While extending a famous problem asked and solved by Bertrand in 1873, Darboux found in 1877 a family of abstract surfaces of revolution, each endowed with a force function, with the striking property that all the orbits are periodic on…
Scalar fields have had a long and controversial life in gravity theories, having progressed through many deaths and resurrections. The first scientific gravity theory, Newton's, was that of a scalar potential field, so it was natural for…
In February 1700, Isaac Newton needed a precise tropical year to design a new universal calendar that would supersede the Gregorian one. However, 17th-Century astronomers were uncertain of the long-term variation in the inclination of the…
The nearly circular (mean eccentricity <e>~0.06) and coplanar (mean mutual inclination <i>~3 deg) orbits of the Solar System planets motivated Kant and Laplace to put forth the hypothesis that planets are formed in disks, which has…
I discuss the physical basis of classical mechanics, such as expressed commonly using the framework of Newton's Principia. Newton's formulation of the laws of motion is seen to have quite a few ambiguities and shortcomings. Therefore I…
General Relativity's Kerr metric is famous for its many symmetries which are responsible for the separability of the Hamilton-Jacobi equation governing the geodesic motion and of the Teukolsky equation for wave dynamics. We show that there…