Keplers's Equation and Angular Momentum: Historical Perspective, Critical Analysis and Implications for Development of the Orbital Mechanics/Dynamics, Mathematics and Physics

Had the planet mars not existed: Kepler's equant model and its physical consequences

We examine the equant model for the motion of planets, which has been the starting point of Kepler's investigations before he modified it because of Mars observations. We show that, up to first order in eccentricity, this model implies for each orbit a velocity which satisfies Kepler's second law and Hamilton's hodograph, and a centripetal acceleration with an inverse square dependence on the distance to the sun. If this dependence is assumed to be universal, Kepler's third law follows immediately. This elementary execice in kinematics for undergraduates emphasizes the proximity of the equant …

A simple derivation of Kepler's laws without solving differential equations

Proceeding like Newton with a discrete time approach of motion and a geometrical representation of velocity and acceleration, we obtain Kepler's laws without solving differential equations. The difficult part of Newton's work, when it calls for non trivial properties of ellipses, is avoided by the introduction of polar coordinates. Then a simple reconsideration of Newton's figure naturally leads to en explicit expression of the velocity and to the equation of the trajectory. This derivation, which can be fully apprehended by beginners at university (or even before) can be considered as a first…

Computer experiments and visualization in mathematics and physics. A subjective short walk among some historical examples

In this short essay, we show how computer experiments, and especially visualization, allowed for the investigation and discovery of phenomena which would have passed unnoticed. We shall also highlight the importance of interactivity between the computer and the user. We do this by surveying several historical examples from mathematics and the physical sciences. Many pictures, and even hyperlinks to online interactive numerical experiments, are provided. Needless to say that we do not claim to be exhaustive, and that the chosen examples reflect our taste as well as our limited knowledge. Hyperl…

Challenges for the periodic systems of elements: chemical, historical and mathematical perspectives

Unveiling numerical trends among either atomic or equivalent weights that somehow preserved resemblances among elements was frequent in the 1860s. Standing out from the crowd, Meyer and Mendeleev went beyond numerical relationships, certainly motivated by a pedagogical aim. Both were after a system synthesizing the chemical knowledge of their times in an appealing way to be presented to chemistry students. Is it still the periodic system aiming at that? Is it really a map of the current chemistry landscape? Solving these questions entails addressing others such as: what is the periodic system?…

论有补格、唯一有补格与唯一有补非分配格（关于一个数学之谜的历史与认识论注记）

On complemented, uniquely complemented and uniquely complemented nondistributive lattices (a historical and epistemological note about a mathematical mystery)

本文从历史与认识论角度梳理有补格和唯一有补格的研究脉络，重点考察唯一有补非分配格这一数学之谜，并展示如何利用自由格技术构造此类结构。

Yet another approach to the inverse square law and to the circular character of the hodograph of Kepler orbits

The law of centripetal force governing the motion of celestial bodies in eccentric conic sections, has been established and thoroughly investigated by Sir Isaac Newton in his Principia Mathematica. Yet its profound implications on the understanding of such motions is still evolving. In a paper to the royal academy of science, Sir Willian Hamilton demonstrated that this law underlies the circular character of hodographs for Kepler orbits. A fact which was the object of ulterior research and exploration by Richard Feynman and many other authors [1]. In effect, a minute examination of the geometr…