Related papers: Deriving Kepler's Laws Using Quaternions
We show that the exact integrator for the classical Kepler motion, recently found by Kozlov ({\it J. Phys. A: Math. Theor.\} {\bf 40} (2007) 4529-4539), can be derived in a simple natural way (using well known exact discretization of the…
The paper aims to apply the complex-sedenions to explore the field equations of four fundamental interactions, which are relevant to the classical mechanics and quantum mechanics, in the curved spaces. J. C. Maxwell was the first to utilize…
It is shown that a recent result regarding the average rate of evolution of a dynamical system at equilibrium in combination with the quantization of geometric areas coming from LQG, implies the validity of Kepler's Second Law of planetary…
Can a machine or algorithm discover or learn the elliptical orbit of Mars from astronomical sightings alone? Johannes Kepler required two paradigm shifts to discover his First Law regarding the elliptical orbit of Mars. Firstly, a shift…
Isaac Newton formulated the central difference algorithm (Eur. Phys. J. Plus (2020) 135:267) when he derived his second law. The algorithm is under various names ("Verlet, leap-frog,...") the most used algorithm in simulations of complex…
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…
Neural networks have provided powerful approaches to solve various scientific problems. Many of them are even difficult for human experts who are good at accessing the physical laws from experimental data. We investigate whether neural…
In this paper we investigate the gravitational waves emission by stellar dynamical structures as complex systems in the quadrupole approximation considering bounded and unbounded orbits. Precisely, after deriving analytical expressions for…
In this paper we consider a generalized classical mechanics with fractional derivatives. The generalization is based on the time-clock randomization of momenta and coordinates taken from the conventional phase space. The fractional…
A solution to the 50 year old problem of a spinning particle in curved space has been recently derived using an extension of Clifford calculus in which each geometric element has its own coordinate. This leads us to propose that all the…
We revisit the two body problem, where one body can be deformed under the action of tides raised by the companion. Tidal deformation and consequent dissipation result in spin and orbital evolution of the system. In general, the equations of…
We use the canonical Hamiltonian formalism to generalize to spinning point particles the first law of mechanics established for binary systems of non-spinning point masses moving on circular orbits [Le Tiec, Blanchet, and Whiting, Phys.…
Assuming a suitable rotation of the Universe as a whole, we can attribute the rotation of galaxies to it. Using Lagrangian Mechanics we then can find the equations governing the rotation of a galaxy. We find in this way the azimuthal…
Six years after William Rowan Hamilton's discovery of quaternions, in 1849 James Cockle introduced the algebra of split quaternions. (He called them ``coquaternions.'') In this paper we define regular functions on split quaternions and…
The classical Hamilton equations are reinterpreted by means of complex analysis, in a non standard way. This suggests a natural extension of the Hamilton equations to the quaternionic case, extension which coincides with the one introduced…
A connection between Maxwell's equations, Newton's laws, and the special theory of relativity is established with a derivation that begins with Newton's verbal enunciation of his first two laws. Derived equations are required to be…
Towards the end of nineteenth century, Celestial Mechanics provided the most powerful tools to test Newtonian gravity in the solar system, and led also to the discovery of chaos in modern science. Nowadays, in light of general relativity,…
In this paper we present a framework which provides an analytical (i.e., infinitely differentiable) transformation between spatial coordinates and orbital elements for the solution of the gravitational two-body problem. The formalism omits…
The algebra of fourvectors is described. The fourvectors are more appropriate than the Hamilton quaternions for its use in Physics and the sciences in general. The fourvectors embrace the 3D vectors in a natural form. It is shown the…
In this paper we clarify and generalise previous work by Moser and Belbruno concerning the link between the motions in the classical Kepler problem and geodesic motion on spaces of constant curvature. Both problems can be formulated as…