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Once again, dispute has arisen over the interpretation of proposed quantum information theory experiments to probe the quantum nature of gravity by testing for gravitationally induced entanglement (GIE) between two spatially separated…
Symplectic integrators are a foundation to the study of dynamical $N$-body phenomena, at scales ranging from from planetary to cosmological. These integrators preserve the Poincar\'e invariants of Hamiltonian dynamics. The $N$-body…
Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force…
We start by formulating geometrically the Newton's law for a classical free particle in terms of Riemannian geometry, as pattern for subsequent developments. In fact, we use this scheme for further generalisation devoted to a constrained…
Gravitation, according to General Relativity, is an attribute of space-time's geometry and hence not a force in the Newtonian sense. This is a consequence of Einstein's equivalence principle, which so far passed all experimental tests with…
Special exotic class of dynamical systems~ -- the implicit maps~ -- is considered. Such maps, particularly, can appear as a result of using of implicit and semi-implicit iterative numerical methods. In the present work we propose the…
A trajectory isomorphism between the two Newtonian fixed center problem in the sphere and two associated planar two fixed center problems is constructed by performing two simultaneous gnomonic projections in $S^2$. This isomorphism converts…
Riemann's principle "force equals geometry" provided the basis for Einstein's General Relativity - the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The…
An interpretation of selected parts of Newton's Principia, with modern notation and methods. Keplers Laws are derived from an inverse square law using Newton's methods.
Invented by Kurt Hensel at the very end of 19th century on the model of power series in one indeterminate, the $p$-adic numbers have not only become an indispensable tool of contemporary arithmetic, but a research topic per se. In this…
After some more than four centuries from the formulation and publication (in Astronomia Nova) of the Kepler's Equation, which relates the eccentric (and, intermediately, the true) anomaly of the planetary trajectories to the uniformly…
The role of global topology in the dynamics of the Universe is poorly understood. Along with observational programmes for determining the topology of the Universe, some small theoretical steps have recently been made. Heuristic…
I show how Hamilton's philosophical commitments led him to a causal interpretation of classical mechanics. I argue that Hamilton's metaphysics of causation was injected into his dynamics by way of a causal interpretation of force. I then…
Hodographs for the Kepler problem are circles. This fact, known since almost two centuries ago, still provides the simplest path to derive the Kepler first law. Through Feynman `lost lecture', this derivation has now reached to a wider…
Newton's Principia, when it appeared in 1687, was received with the greatest admiration, not only by the foremost mathematicians and astronomers in Europe, but also by philosophers like Voltaire and Locke and by members of the educated…
In 1693, Gottfried Whilhelm Leibniz published in the Acta Eruditorum a geometrical proof of the fundamental theorem of the calculus. During his notorious dispute with Isaac Newton on the development of the calculus, Leibniz denied any…
A Hamiltonian framework is introduced to encompass non-rotating (but possibly charged) black holes that are ``isolated'' near future time-like infinity or for a finite time interval. The underlying space-times need not admit a stationary…
In February 1700, Isaac Newton needed a precise tropical year to design a new universal calendar that would supersede the Gregorian one. However, 17th-Century astronomers were uncertain of the long-term variation in the inclination of the…
By identifying Hamiltonian flows with geodesic flows of suitably chosen Riemannian manifolds, it is possible to explain the origin of chaos in classical Newtonian dynamics and to quantify its strength. There are several possibilities to…
We take into account the dynamics of three types of models of rotating galaxies in polar coordinates in a rotating frame. Due to non-axisymmetric potential perturbations, the angular momentum varies with time, and the kinetic energy depends…