154 research outputs found
Uniqueness of collinear solutions for the relativistic three-body problem
Continuing work initiated in an earlier publication [Yamada, Asada, Phys.
Rev. D 82, 104019 (2010)], we investigate collinear solutions to the general
relativistic three-body problem. We prove the uniqueness of the configuration
for given system parameters (the masses and the end-to-end length). First, we
show that the equation determining the distance ratio among the three masses,
which has been obtained as a seventh-order polynomial in the previous paper,
has at most three positive roots, which apparently provide three cases of the
distance ratio. It is found, however, that, even for such cases, there exists
one physically reasonable root and only one, because the remaining two positive
roots do not satisfy the slow motion assumption in the post-Newtonian
approximation and are thus discarded. This means that, especially for the
restricted three-body problem, exactly three positions of a third body are true
even at the post-Newtonian order. They are relativistic counterparts of the
Newtonian Lagrange points L1, L2 and L3. We show also that, for the same masses
and full length, the angular velocity of the post-Newtonian collinear
configuration is smaller than that for the Newtonian case. Provided that the
masses and angular rate are fixed, the relativistic end-to-end length is
shorter than the Newtonian one.Comment: 18 pages, 1 figure; typos corrected, text improved; accepted by PR
Collinear solution to the general relativistic three-body problem
The three-body problem is reexamined in the framework of general relativity.
The Newtonian three-body problem admits Euler's collinear solution, where three
bodies move around the common center of mass with the same orbital period and
always line up. The solution is unstable. Hence it is unlikely that such a
simple configuration would exist owing to general relativistic forces dependent
not only on the masses but also on the velocity of each body. However, we show
that the collinear solution remains true with a correction to the spatial
separation between masses. Relativistic corrections to the Sun-Jupiter Lagrange
points L1, L2 and L3 are also evaluated.Comment: 12 pages, 2 figures, accepted for publication in PR
A list of all integrable 2D homogeneous polynomial potentials with a polynomial integral of order at most 4 in the momenta
We searched integrable 2D homogeneous polynomial potential with a polynomial
first integral by using the so-called direct method of searching for first
integrals. We proved that there exist no polynomial first integrals which are
genuinely cubic or quartic in the momenta if the degree of homogeneous
polynomial potentials is greater than 4.Comment: 22 pages, no figures, to appear in J. Phys. A: Math. Ge
An axiomatic approach to the non-linear theory of generalized functions and consistency of Laplace transforms
We offer an axiomatic definition of a differential algebra of generalized
functions over an algebraically closed non-Archimedean field. This algebra is
of Colombeau type in the sense that it contains a copy of the space of Schwartz
distributions. We study the uniqueness of the objects we define and the
consistency of our axioms. Next, we identify an inconsistency in the
conventional Laplace transform theory. As an application we offer a free of
contradictions alternative in the framework of our algebra of generalized
functions. The article is aimed at mathematicians, physicists and engineers who
are interested in the non-linear theory of generalized functions, but who are
not necessarily familiar with the original Colombeau theory. We assume,
however, some basic familiarity with the Schwartz theory of distributions.Comment: 23 page
Linear coupling and over-reflection phenomena of magnetohydrodynamic waves in smooth shear flows
Special features of magnetohydrodynamic waves linear dynamics in smooth shear
flows are studied. Quantitative asymptotic and numerical analysis are performed
for wide range of system parameters when basic flow has constant shear of
velocity and uniform magnetic field is parallel to the basic flow. The special
features consist of magnetohydrodynamic wave mutual transformation and
over-reflection phenomena. The transformation takes place for arbitrary shear
rates and involves all magnetohydrodynamic wave modes. While the
over-reflection occurs only for slow magnetosonic and Alfv\'en waves at high
shear rates. Studied phenomena should be decisive in the elaboration of the
self-sustaining model of magnetohydrodynamic turbulence in the shear flows
Normal families of functions and groups of pseudoconformal diffeomorphisms of quaternion and octonion variables
This paper is devoted to the specific class of pseudoconformal mappings of
quaternion and octonion variables. Normal families of functions are defined and
investigated. Four criteria of a family being normal are proven. Then groups of
pseudoconformal diffeomorphisms of quaternion and octonion manifolds are
investigated. It is proven, that they are finite dimensional Lie groups for
compact manifolds. Their examples are given. Many charactersitic features are
found in comparison with commutative geometry over or .Comment: 55 pages, 53 reference
Progress in Classical and Quantum Variational Principles
We review the development and practical uses of a generalized Maupertuis
least action principle in classical mechanics, in which the action is varied
under the constraint of fixed mean energy for the trial trajectory. The
original Maupertuis (Euler-Lagrange) principle constrains the energy at every
point along the trajectory. The generalized Maupertuis principle is equivalent
to Hamilton's principle. Reciprocal principles are also derived for both the
generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis
Principle is the classical limit of Schr\"{o}dinger's variational principle of
wave mechanics, and is also very useful to solve practical problems in both
classical and semiclassical mechanics, in complete analogy with the quantum
Rayleigh-Ritz method. Classical, semiclassical and quantum variational
calculations are carried out for a number of systems, and the results are
compared. Pedagogical as well as research problems are used as examples, which
include nonconservative as well as relativistic systems
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
Solitary waves in the Nonlinear Dirac Equation
In the present work, we consider the existence, stability, and dynamics of
solitary waves in the nonlinear Dirac equation. We start by introducing the
Soler model of self-interacting spinors, and discuss its localized waveforms in
one, two, and three spatial dimensions and the equations they satisfy. We
present the associated explicit solutions in one dimension and numerically
obtain their analogues in higher dimensions. The stability is subsequently
discussed from a theoretical perspective and then complemented with numerical
computations. Finally, the dynamics of the solutions is explored and compared
to its non-relativistic analogue, which is the nonlinear Schr{\"o}dinger
equation. A few special topics are also explored, including the discrete
variant of the nonlinear Dirac equation and its solitary wave properties, as
well as the PT-symmetric variant of the model
Small Sample Properties of the Wilcoxon Signed Rank Test with Discontinuous and Dependent Observations
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