Related papers: "Simple Dynamics" conjectures for some real Newton…
In default of a fundamental MOND theory -- a FUNDAMOND -- I advocate that, alongside searching for one, we should try to identify predictions that follow from wide classes of MOND theories, if not necessarily from all. In particular,…
Application of Newton's ideas from "Principia" gives many new results in mechanics. Here we explore the question ``What form of extra force will maintain the magnitude of a vector constant of the motion while changing its direction?''
We analyze a real one-parameter family of quasiconformal deformations of a hyperbolic rational map known as {\em spinning}. We show that under fairly general hypotheses, the limit of spinning either exists and is unique, or else converges…
In this paper we give a unified proof of the fact that the Julia set of Newton's method applied to a holomorphic function of the complex plane (a polynomial of degree large than $1$ or an entire transcendental function) is connected. The…
In this manuscript we systematically review known results of local dynamics of discrete local holomorphic dynamics near fixed points in one and several complex variables as well as the consequences in global dynamics.
This paper points out the usefulness of the concept of derivation along a map in many problems in Geometry and Physics. In particular it will be shown that this approach allows us to translate the usual concepts arising in Geometrical…
We study the complexification of the one-dimensional Newtonian particle in a monomial potential. We discuss two classes of motions on the associated Riemann surface: the rectilinear and the cyclic motions, corresponding to two different…
We survey Vojta's higher-dimensional generalizations of the $abc$ conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "$abcd$ conjecture" implies a…
This paper primarily presents numerical results for the Anderson accelerated Newton method on a set of benchmark problems. The results demonstrate superlinear convergence to solutions of both degenerate and nondegenerate problems. The…
Many papers have been published over the years that either conjecture or even (claim to) prove the universality of the form of Maxwell's equations. We present yet another derivation of Maxwell's equations and discuss the conclusions…
The central result about fast rotating-flow structures is the Taylor-Proudman theorem (TPT) which connects various aspects of the dynamics. Taylor's geometrical proof of TPT is reproduced and extended substantially, with Lie's theory for…
This paper is devoted to path-dependent kinetics equations arising, in particular, from the analysis of the coupled backward - forward systems of equations of mean field games. We present local well-posedness, global existence and some…
Several fundamental results in physics are derived from the simple starting point of two commuting orthogonal unit vectors. The combination of these unit vectors leads to spherical harmonics and Schwinger's expression of the…
Dynamics of spiral galaxies derived from a given surface mass density has been derived earlier in a classic paper. We try to transform the singular elliptic function in the integral into a compact integral with regular elliptic function.…
In this paper, we develop a systematical approach in applying an asymptotic method of moving planes to investigate qualitative properties of positive solutions for fractional parabolic equations. We first obtain a series of needed key…
We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.
We simplify the proof of some widely used theoretical theorems, extending their applicability, while correcting some erroneous results. We also generalize key results and present new results that contribute to the development of the theory.…
Various many-body models are treated, which describe $N$ points confined to move on a plane circle. Their Newtonian equations of motion ("accelerations equal forces") are integrable, i. e. they allow the explicit exhibition of $N$ constants…
Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate…
This paper presents a unified approach for inverse and direct dynamics of constrained multibody systems that can serve as a basis for analysis, simulation, and control. The main advantage of the formulation of the dynamic is that it does…