1,039 research outputs found
Poincaré on the Foundation of Geometry in the Understanding
This paper is about PoincarĂ©âs view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, PoincarĂ©, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are âdefinitions in disguise.â I argue that this view does not accord well with PoincarĂ©âs core commitment in the philosophy of geometry: the view that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which PoincarĂ© held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to PoincarĂ©. PoincarĂ©âs view therefore contrasts sharply with Kantâs foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them
Dynamics of Black Hole Pairs I: Periodic Tables
Although the orbits of comparable mass, spinning black holes seem to defy
simple decoding, we find a means to decipher all such orbits. The dynamics is
complicated by extreme perihelion precession compounded by spin-induced
precession. We are able to quantitatively define and describe the fully three
dimensional motion of comparable mass binaries with one black hole spinning and
expose an underlying simplicity. To do so, we untangle the dynamics by
capturing the motion in the orbital plane. Our results are twofold: (1) We
derive highly simplified equations of motion in a non-orthogonal orbital basis,
and (2) we define a complete taxonomy for fully three-dimensional orbits. More
than just a naming system, the taxonomy provides unambiguous and quantitative
descriptions of the orbits, including a determination of the zoom-whirliness of
any given orbit. Through a correspondence with the rationals, we are able to
show that zoom-whirl behavior is prevalent in comparable mass binaries in the
strong-field regime. A first significant conclusion that can be drawn from this
analysis is that all generic orbits in the final stages of inspiral under
gravitational radiation losses are characterized by precessing clovers with few
leaves and that no orbit will behave like the tightly precessing ellipse of
Mercury. The gravitational waveform produced by these low-leaf clovers will
reflect the natural harmonics of the orbital basis -- harmonics that,
importantly, depend only on radius. The significance for gravitational wave
astronomy will depend on the number of windings the pair executes in the
strong-field regime and could be more conspicuous for intermediate mass pairs
than for stellar mass pairs.Comment: 19 pages, lots of figure
Classical small systems coupled to finite baths
We have studied the properties of a classical -body system coupled to a
bath containing -body harmonic oscillators, employing an model
which is different from most of the existing models with . We have
performed simulations for -oscillator systems, solving
first-order differential equations with and , in order to calculate the time-dependent energy exchange between the
system and the bath. The calculated energy in the system rapidly changes while
its envelope has a much slower time dependence. Detailed calculations of the
stationary energy distribution of the system (: an energy per
particle in the system) have shown that its properties are mainly determined by
but weakly depend on . The calculated is analyzed with the
use of the and - distributions: the latter is derived with
the superstatistical approach (SSA) and microcanonical approach (MCA) to the
nonextensive statistics, where stands for the entropic index. Based on
analyses of our simulation results, a critical comparison is made between the
SSA and MCA. Simulations have been performed also for the -body ideal-gas
system. The effect of the coupling between oscillators in the bath has been
examined by additional () models which include baths consisting of
coupled linear chains with periodic and fixed-end boundary conditions.Comment: 30 pages, 16 figures; the final version accepted in Phys. Rev.
Libration driven elliptical instability
The elliptical instability is a generic instability which takes place in any
rotating flow whose streamlines are elliptically deformed. Up to now, it has
been widely studied in the case of a constant, non-zero differential rotation
between the fluid and the elliptical distortion with applications in
turbulence, aeronautics, planetology and astrophysics. In this letter, we
extend previous analytical studies and report the first numerical and
experimental evidence that elliptical instability can also be driven by
libration, i.e. periodic oscillations of the differential rotation between the
fluid and the elliptical distortion, with a zero mean value. Our results
suggest that intermittent, space-filling turbulence due to this instability can
exist in the liquid cores and sub-surface oceans of so-called synchronized
planets and moons
Quantum effects on Lagrangian points and displaced periodic orbits in the Earth-Moon system
Recent work in the literature has shown that the one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric functions. By performing the numerical analysis of the exact formulas for the roots, we confirm and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian points the quantum corrections are of the same order of magnitude in the Earth-Moon system. Third, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The present paper investigates therefore, eventually, a restricted three-body problem involving Earth, Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist
Homoclinic Orbits around Spinning Black Holes I: Exact Solution for the Kerr Separatrix
Under the dissipative effects of gravitational radiation, black hole binaries
will transition from an inspiral to a plunge. The separatrix between bound and
plunging orbits features prominently in the transition. For equatorial Kerr
orbits, we show that the separatrix is a homoclinic orbit in one-to-one
correspondence with an energetically-bound, unstable circular orbit. After
providing a definition of homoclinic orbits, we exploit their correspondence
with circular orbits and derive exact solutions for them. This paper focuses on
homoclinic behavior in physical space, while in a companion paper we paint the
complementary phase space portrait. The exact results for the Kerr separatrix
could be useful for analytic or numerical studies of the transition from
inspiral to plunge.Comment: 21 pages, some figure
Virial theorem for rotating self-gravitating Brownian particles and two-dimensional point vortices
We derive the proper form of Virial theorem for a system of rotating
self-gravitating Brownian particles. We show that, in the two-dimensional case,
it takes a very simple form that can be used to obtain general results about
the dynamics of the system without being required to solve the
Smoluchowski-Poisson system explicitly. We also develop the analogy between
self-gravitating systems and two-dimensional point vortices and derive a
Virial-like relation for the vortex system
Integral equations PS-3 and moduli of pants
More than a hundred years ago H.Poincare and V.A.Steklov considered a problem
for the Laplace equation with spectral parameter in the boundary conditions.
Today similar problems for two adjacent domains with the spectral parameter in
the conditions on the common boundary of the domains arises in a variety of
situations: in justification and optimization of domain decomposition method,
simple 2D models of oil extraction, (thermo)conductivity of composite
materials. Singular 1D integral Poincare-Steklov equation with spectral
parameter naturally emerges after reducing this 2D problem to the common
boundary of the domains. We present a constructive representation for the
eigenvalues and eigenfunctions of this integral equation in terms of moduli of
explicitly constructed pants, one of the simplest Riemann surfaces with
boundary. Essentially the solution of integral equation is reduced to the
solution of three transcendent equations with three unknown numbers, moduli of
pants. The discreet spectrum of the equation is related to certain surgery
procedure ('grafting') invented by B.Maskit (1969), D.Hejhal (1975) and
D.Sullivan- W.Thurston (1983).Comment: 27 pages, 13 figure
The free rigid body dynamics: generalized versus classic
In this paper we analyze the normal forms of a general quadratic Hamiltonian
system defined on the dual of the Lie algebra of real -
skew - symmetric matrices, where is an arbitrary real symmetric
matrix. A consequence of the main results is that any first-order autonomous
three-dimensional differential equation possessing two independent quadratic
constants of motion which admits a positive/negative definite linear
combination, is affinely equivalent to the classical "relaxed" free rigid body
dynamics with linear controls.Comment: 12 page
Persistent Chaos in High Dimensions
An extensive statistical survey of universal approximators shows that as the
dimension of a typical dissipative dynamical system is increased, the number of
positive Lyapunov exponents increases monotonically and the number of parameter
windows with periodic behavior decreases. A subset of parameter space remains
in which topological change induced by small parameter variation is very
common. It turns out, however, that if the system's dimension is sufficiently
high, this inevitable, and expected, topological change is never catastrophic,
in the sense chaotic behavior is preserved. One concludes that deterministic
chaos is persistent in high dimensions.Comment: 4 pages, 3 figures; Changes in response to referee comment
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