Related papers: Deriving Kepler's Laws Using Quaternions
The goal of this contribution is to introduce the Hamiltonian formalism of theoretical mechanics for analysing motion in generic linear and non-linear dynamical systems, including particle accelerators. This framework allows the derivation…
Within the framework of exterior algebra, the concept of time-like quaternions has been previously established. This paper advances beyond the existing structure by elucidating the procedure for constructing time-like quaternions with the…
Armed with an astrolabe and Kepler's laws one can arrive at accurate estimates of the orbits of planets.
Gravitomagnetic equations result from applying quaternionic differential operators to the energy-momentum tensor. These equations are similar to the Maxwell's EM equations. Both sets of the equations are isomorphic after changing…
We derive the perihelion precession of planetary orbits using quantum field theory extending the Standard Model to include gravity. Modeling the gravitational bound state of an electron via the Dirac equation of unified gravity [Rep. Prog.…
This paper presents an experimental study on the application of quaternions in several machine learning algorithms. Quaternion is a mathematical representation of rotation in three-dimensional space, which can be used to represent complex…
The Dirac belt trick is often employed in physics classrooms to show that a $2\pi$ rotation is not topologically equivalent to the absence of rotation whereas a $4\pi$ rotation is, mirroring a key property of quaternions and their…
The interval approach to computation of dynamics of celestial bodies in the planetary problem has been considered. It is based on the refusal from idealization of infinitely high resolving capacity of measuring tools, and forms an…
We consider secular perturbations of nearly Keplerian two-body motion under a perturbing potential that can be approximated to sufficient accuracy by expanding it to second order in the coordinates. After averaging over time to obtain the…
While Kepler was still working in Graz during 1598, some letters to his mentor Michael Maestlin demonstrate his interest in astronomical clocks and machines. The first letter, dated January 6, 1598 contains a detailed description of a…
The true and eccentric anomaly parametrizations of the Kepler motion are generalized to quasiperiodic orbits, by considering perturbations of the radial part of the kinetic energy in a form of a series of negative powers of the orbital…
Many topics in planetary studies demand an estimate of the collision probability of two objects moving on nearly Keplerian orbits. In the classic works of \"Opik (1951) and Wetherill (1967), the collision probability was derived by…
The collective dynamics of objects moving through a viscous fluid is complex and counterintuitive. A key to understanding the role of nontrivial particle shape in this complexity is the interaction of a pair of sedimenting spheroids. We…
We study the motion of a particle in a 3-dimensional lattice in the presence of a Coulomb potential, but we demonstrate semiclassicaly that the trajectories will always remain in a plane which can be taken as a rectangular lattice. The…
The paper aims to adopt the complex quaternion and octonion to formulate the field equations for electromagnetic and gravitational fields. Applying the octonionic representation enables one single definition to combine some physics contents…
It is shown that the quaternionic Hilbert space formulation of quantum mechanics allows a quantization, based on a generalized system of imprimitivity, that leads to a description of the motion of a quantum particle in the field of a…
The problem of Kepler dynamics on a conformable Poisson manifold is addressed. The Hamiltonian function is defined and the related Hamiltonian vector field governing the dynamics is derived, which leads to a modified Newton second law.…
The first integrals of the Kepler problem are used to compute preliminary orbits starting from two short observed arcs of a celestial body, which may be obtained either by optical or radar observations. We write polynomial equations for…
The functional space of biquaternions is considered on Minkovskiy space. The scalar-vector biquaternions representation is used which was offered by W. Hamilton for quaternions. With introduction of differential operator - a mutual complex…
The regularities in the structure of the planetary system, expressed by the Titius-Bode law, can be accounted also by using a more general formula, derived not by fit but from a logarithmic integrality constraint on the angular momentum…