Related papers: Kepler's Differential Equations
Leonhard Euler likely developed his summation formula in 1732, and soon used it to estimate the sum of the reciprocal squares to 14 digits --- a value mathematicians had been competing to determine since Leibniz's astonishing discovery that…
Euler stressed the importance of hypotheses, which he thought were the only means of arriving at a certain knowledge of the physical causes, essential to establish the laws of physics. This thought was communicated to Emilie du Chatelet in…
Here we describe a story behind the discovery of Kepler-46, which was the first exoplanetary system detected and characterized from a method known as the transit timing variations (TTVs). The TTV method relies on the gravitational…
Action at a distance in Newtonian physics is replaced by finite propagation speeds in classical physics, the physics defined by the field theories of Maxwell and Einstein. As a result, the differential equations of motion in Newtonian…
Almost all theories of physics have expressed physical laws by means of differential equations. One can ask: why differential equations? What is special about them? This article addresses these questions and is presented as an inquiry-based…
The invariance of the Lagrangian under time translations and rotations in Kepler's problem yields the conservation laws related to the energy and angular momentum. Noether's theorem reveals that these same symmetries furnish generalized…
Kepler's 2nd law, the law of the areas, is usually taught in passing, between the 1st and the 3rd laws, to be explained "later on" as a consequence of angular momentum conservation. The 1st and 3rd laws receive the bulk of attention; the…
We retrace the first steps towards understanding neutrinos, particles predicted by Pauli in 1930 to avoid a supposed violation of time-translation symmetry. Despite the tendency to reduce the whole story to his intuition and the skill of…
Hubble's name is associated closely with the idea of an expanding universe as he discovered the relation between the recession velocity and distances of galaxies. Hubble also did a lot of pioneering work on the distribution of galaxies in…
Approximately half of the planets discovered by NASA's Kepler mission are in systems where just a single planet transits its host star, and the remaining planets are observed to be in multi-planet systems. Recent analyses have reported a…
The trajectory and the orbital velocity are determined for an object moving in a gravitational system, in terms of fundamental and independent variables. In particular, considering a path on equipotential line, the elliptical orbit is…
The procedure used to "do physics" in the macroscopic world is familiar: You take an object, start it off with a particular position and velocity, subject it to known forces (say gravity or friction, or both), and follow its trajectory. You…
In one of his work, appeared in 1969, John A. Baker initiated the systematic investigation of some partial difference equations. The main purpose of this paper is to continue and to extend these investigations. Firstly, we present how such…
Ptolemy-s planetary model is an ancient geocentric astronomical model, describing the observed motion of the Sun and the planets. Ptolemy accounted for the deviations of planetary orbits from perfect circles by introducing two small and…
The Euler--Poincar\'e equations, firstly introduced by Henri Poincar\'e in 1901, arise from the application of Lagrangian mechanics to systems on Lie groups that exhibit symmetries, particularly in the contexts of classical mechanics and…
After some brief biographical notes, Wolfgang Pauli's main contributions to general relativity, quantum theory, quantum field theory, and elementary particle physics are reviewed. A detailed description is given how Pauli descovered -…
Relativistic particles in the Kepler and Coulomb potentials may have trajectories that are qualitatively different from the trajectories found in nonrelativistic mechanics. Spiral scattering trajectories were pointed out by C. G. Darwin in…
Hundreds of substellar companions to solar-type stars will be discovered with the Kepler satellite. Kepler's extreme photometric precision gives access to low-amplitude stellar variability contributed by a variety of physical processes. We…
Jacobi is one of the most famous mathematicians of his century. His name is attached to many results in various fields of mathematics and his complete works in seven volumes have been available since the end of the XIXth century and are…
Classical physics has enabled the acquisition of significant knowledge of the physical properties of nature on a standard macroscopic scale. These achievements were driven by use of the causal ontological approach (proposed originally by…