Related papers: Deriving Kepler's Laws Using Quaternions
This paper gives a slight refinement of a theorem of Hamilton, which shows that the velocity of a Keplerian motion moves on a circle.
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.…
We investigate the Kepler problem using a symplectic structure consistent with the commutation rules of the noncommutative quantum mechanics. We show that a noncommutative parameter of the order of $10^{-58} \text m^2$ gives observable…
We show how the Dixon's system of first order equations of motion for the particle with inner dipole structure together with the side Mathisson constraint follows from rather general construction of the 'Hamilton system' developed by…
Newton's Theorem of Revolving Orbits derives the force that is necessary to explain a particular precession that leaves the shape of an orbit unchanged. Newton showed that for an orbiting body that is already subject to any central force,…
As scientific knowledge expands, science education may not always keep pace with the latest advancements in astrophysics. A solid scientific education is crucial for preparing students for 21st-century challenges. However, science education…
The Kepler-Heisenberg problem is that of determining the motion of a planet around a sun in the sub-Riemannian Heisenberg group. The sub-Riemannian Hamiltonian provides the kinetic energy, and the gravitational potential is given by the…
The global shape (topology) of the Universe is not derivable from General Relativity but should be determined by observations. Here we propose a method for estimation of this shape using patterns of fundamental physical parameters, for…
We study the modification of Newton's second law, upto first order in the deformation parameter $a$, in the $\kappa$-space-time. We derive the deformed Hamiltonian, expressed in terms of the commutative phase space variables, describing the…
A generalization of driven harmonic oscillator with time-dependent mass and frequency, by adding total time-derivative terms to the Lagrangian, is considered. The generalization which gives a general quadratic Hamiltonian system does not…
Over the last decades quaternions have become a crucial and very successful tool for attitude representation in robotics and aerospace. However, there is a major problem that is continuously causing trouble in practice when it comes to…
The general 4D rotation matrix is specialised to the general 3D rotation matrix by equating its leftmost top element (a00) to 1. Its associate matrix of products of the left-hand and right-hand quaternion components is specialised…
Understanding the nature of time remains a key unsolved problem in science. Newton in the Principia asserted an absolute universal time that {\it `flows equably'}. Hamilton then proposed a mathematical unification of space and time within…
The attitude space has been parameterized in various ways for practical purposes. Different representations gain preferences over others based on their intuitive understanding, ease of implementation, formulaic simplicity, and physical as…
The determination of velocities of stars from precise Doppler measurements is described here using relativistic theory of astronomical reference frames so as to determine the Keplerian and post-Keplerian parameters of binary systems. We…
This review paper is devoted to the theory of orbits. We start with the discussion of the Newtonian problem of motion then we consider the relativistic problem of motion, in particular the PN approximation and the further gravitomagnetic…
This paper describes the passage of light through a system of waveplates mathematically in terms of quaternions, an extension of the complex numbers, instead of the more usual Jones vectors and Jones matrices. Both the light beam and the…
We present a natural proof of Kepler's law of ellipses in the spirit of Euclidean geometry. Moreover we discuss two existing Euclidean geometric proofs, one by Feynman in hist Lost Lecture from 1964 and the other by Newton in the Principia…
A gravitational theory with derivative matter coupling is proposed which adopt de Sitter expansion at late times with ordinary baryonic matter. Matter components are conserved separately in the cosmological background and the Newtonian…
Using a theorem of partial differential equations, we present a general way of deriving the conserved quantities associated with a given classical point mechanical system, denoted by its Hamiltonian. Some simple examples are given to…