Related papers: Kovalevskaya top -- an elementary approach
In this short note, we review several one-dimensional problems such as those involving linear Schroedinger equation, variable-coefficient Helmholtz equation, Zakharov-Shabat system and Kubelka-Munk equations. We show that they all can be…
We explain the solution of the following two problems: obtaining of Kepler's laws from Newton's laws (so called two bodies problem) and obtaining the fourth Newton's law (the formula for gravitation) as a corollary of Kepler's laws. This…
We study a weighted low rank approximation that is inspired by a problem of constrained low rank approximation of matrices as initiated by the work of Golub, Hoffman, and Stewart (Linear Algebra and Its Applications, 88-89(1987), 317-327).…
S-Motzkin paths (bijective to ternary trees) and partial version of them are calculated using only elementary methods from linear algebra.
A Lax formalism for the elliptic Painlev\'e equation is presented. The construction is based on the geometry of the curves on ${\mathbb P}^1\times{\mathbb P}^1$ and described in terms of the point configurations.
Basing on a new approach to the fundamental conception of the momentum of a variable-mass point, the paper deals with the hyperreactive model of motion. The equations of motion are different in this model from the known…
The paper is a commentary of one section of the celebrated paper by Sophie Kowalewski on the motion of a rigid body with a fixed point. Its purpose is to show that the results of Kowalewski may be recovered by using the separability…
We propose three iterative methods for solving the Moser-Veselov equation, which arises in the discretization of the Euler-Arnold differential equations governing the motion of a generalized rigid body. We start by formulating the problem…
We show how the Lie group analysis method can be used in order to obtain first integrals of any system of ordinary differential equations. The method of reduction/increase of order developed by Nucci (J. Math. Phys. 37, 1772-1775 (1996)) is…
It is shown that the Jacobi problem of geodesics on ellipsoid may be reduced to more simple one, namely to the special case of the Clebsch problem. The last one may be solved directly by using Weber's approach in terms of multi-dimensional…
In this text we expose basic cases of some fundamental ideas and methods of topology. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal…
We develop a one step matrix method in order to obtain approximate solutions of first order systems and non-linear ordinary differential equations, reducible to first order systems. We find a sequence of such solutions that converge to the…
The purpose of this short note, is to rewrite Morozov's formula for correlation functions over the unitary group, in a much simpler form, involving the computation of a single determinant.
We provide a Lax-Oleinik-type representation formula for solutions of time-dependent Hamilton-Jacobi equations, posed on a network with a rather general geometry, under standard assumptions on the Hamiltonians. It depends on a given initial…
This article is devoted to one particular case of using universal accelerated proximal envelopes to obtain computationally efficient accelerated versions of methods used to solve various optimization problem setups. We propose a proximally…
Relativistically covariant form of equation of motion for real particle (body) under the action of electromagnetic radiation is derived. Equation of motion in the proper frame of the particle uses the radiation pressure cross section 3…
Contents: Introduction(3).The method of Ermakov(4).The method of Milne(7). Pinney's result(8).Lewis' results(8). The interpretation of Eliezer and Gray(14). The connection of the Ermakov invariant with N\"other's theorem(17). Possible…
This work concerns the use of the iterative algorithm (KMF algorithm) proposed by Kozlov, Mazya and Fomin to solve the Cauchy problem for Laplaces equation. This problem consists to recovering the lacking data on some part of the boundary…
A modern elementary introduction to special relativity for advanced school children or first-year university students, in Russian. I try to demonstrate that relativity does not contradict common sense; on the contrary, it follows from…
This preliminary note presents a heuristic for determining rank constrained solutions to linear matrix equations (LME). The method proposed here is based on minimizing a non-convex quadratic functional, which will hence-forth be termed as…