173 research outputs found
Motion of a Vector Particle in a Curved Spacetime. II First Order Correction to a Geodesic in a Schwarzschild Background
The influence of spin on a photon's motion in a Schwarzschild and FRW
spacetimes is studied. The first order correction to the geodesic motion is
found. It is shown that unlike the world-lines of spinless particles, the
photons world-lines do not lie in a plane.Comment: 14 pages, LaTeX2e, second paper in the series (the first one:
gr-qc/0110067), replaced with typos and style corrected version, accepted in
MPL
Routh reduction and the class of magnetic Lagrangian systems
In this paper, some new aspects related to Routh reduction of Lagrangian
systems with symmetry are discussed. The main result of this paper is the
introduction of a new concept of transformation that is applicable to systems
obtained after Routh reduction of Lagrangian systems with symmetry, so-called
magnetic Lagrangian systems. We use these transformations in order to show
that, under suitable conditions, the reduction with respect to a (full)
semi-direct product group is equivalent to the reduction with respect to an
Abelian normal subgroup. The results in this paper are closely related to the
more general theory of Routh reduction by stages.Comment: 23 page
Restricted three-body problem in effective-field-theory models of gravity
One of the outstanding problems of classical celestial mechanics was the
restricted 3-body prob- lem, in which a planetoid of small mass is subject to
the Newtonian attraction of two celestial bodies of large mass, as it occurs,
for example, in the sun-earth-moon system. On the other hand, over the last
decades, a systematic investigation of quantum corrections to the Newtonian
potential has been carried out in the literature on quantum gravity. The
present paper studies the effect of these tiny quantum corrections on the
evaluation of equilibrium points. It is shown that, despite the extreme
smallness of the corrections, there exists no choice of sign of these
corrections for which all qualitative features of the restricted 3-body problem
in Newtonian theory remain unaffected. Moreover, first-order stability of
equilibrium points is characterized by solving a pair of algebraic equations of
fifth degree, where some coefficients depend on the Planck length. The
coordinates of stable equilibrium points are slightly changed with respect to
Newtonian theory, because the planetoid is no longer at equal distance from the
two bodies of large mass. The effect is conceptually interesting but too small
to be observed, at least for the restricted 3-body problems available in the
solar system.Comment: 20 pages, latex, 8 figure
The classical dynamics of two-electron atoms near the triple collision
The classical dynamics of two electrons in the Coulomb potential of an
attractive nucleus is chaotic in large parts of the high-dimensional phase
space. Quantum spectra of two-electron atoms, however, exhibit structures which
clearly hint at the existence of approximate symmetries in this system. In a
recent paper,(Phys. Rev. Lett. 93, 054302 (2004)), we presented a study of the
dynamics near the triple collision as a first step towards uncovering the
hidden regularity in the classical dynamics of two electron atoms. The
non-regularisable triple collision singularity is a main source of chaos in
three body Coulomb problems. Here, we will give a more detailed account of our
findings based on a study of the global structure of the stable and unstable
manifolds of the triple collision.Comment: 21 pages, 17 figure
Lagrangian Framework for Systems Composed of High-Loss and Lossless Components
Using a Lagrangian mechanics approach, we construct a framework to study the
dissipative properties of systems composed of two components one of which is
highly lossy and the other is lossless. We have shown in our previous work that
for such a composite system the modes split into two distinct classes,
high-loss and low-loss, according to their dissipative behavior. A principal
result of this paper is that for any such dissipative Lagrangian system, with
losses accounted by a Rayleigh dissipative function, a rather universal
phenomenon occurs, namely, selective overdamping: The high-loss modes are all
overdamped, i.e., non-oscillatory, as are an equal number of low-loss modes,
but the rest of the low-loss modes remain oscillatory each with an extremely
high quality factor that actually increases as the loss of the lossy component
increases. We prove this result using a new time dynamical characterization of
overdamping in terms of a virial theorem for dissipative systems and the
breaking of an equipartition of energy.Comment: 53 pages, 1 figure; Revision of our original manuscript to
incorporate suggestions from refere
Routhian reduction for quasi-invariant Lagrangians
In this paper we describe Routhian reduction as a special case of standard
symplectic reduction, also called Marsden-Weinstein reduction. We use this
correspondence to present a generalization of Routhian reduction for
quasi-invariant Lagrangians, i.e. Lagrangians that are invariant up to a total
time derivative. We show how functional Routhian reduction can be seen as a
particular instance of reduction of a quasi-invariant Lagrangian, and we
exhibit a Routhian reduction procedure for the special case of Lagrangians with
quasi-cyclic coordinates. As an application we consider the dynamics of a
charged particle in a magnetic field.Comment: 24 pages, 3 figure
Routh reduction for singular Lagrangians
This paper concerns the Routh reduction procedure for Lagrangians systems
with symmetry. It differs from the existing results on geometric Routh
reduction in the fact that no regularity conditions on either the Lagrangian
or the momentum map are required apart from the momentum being a
regular value of . The main results of this paper are: the description of
a general Routh reduction procedure that preserves the Euler-Lagrange nature of
the original system and the presentation of a presymplectic framework for Routh
reduced systems. In addition, we provide a detailed description and
interpretation of the Euler-Lagrange equations for the reduced system. The
proposed procedure includes Lagrangian systems with a non-positively definite
kinetic energy metric.Comment: 34 pages, 2 figures, accepted for publicaton in International Journal
of Geometric Methods in Modern Physics (IJGMMP
On Virtual Displacement and Virtual Work in Lagrangian Dynamics
The confusion and ambiguity encountered by students, in understanding virtual
displacement and virtual work, is discussed in this article. A definition of
virtual displacement is presented that allows one to express them explicitly
for holonomic (velocity independent), non-holonomic (velocity dependent),
scleronomous (time independent) and rheonomous (time dependent) constraints. It
is observed that for holonomic, scleronomous constraints, the virtual
displacements are the displacements allowed by the constraints. However, this
is not so for a general class of constraints. For simple physical systems, it
is shown that, the work done by the constraint forces on virtual displacements
is zero. This motivates Lagrange's extension of d'Alembert's principle to
system of particles in constrained motion. However a similar zero work
principle does not hold for the allowed displacements. It is also demonstrated
that d'Alembert's principle of zero virtual work is necessary for the
solvability of a constrained mechanical problem. We identify this special class
of constraints, physically realized and solvable, as {\it the ideal
constraints}. The concept of virtual displacement and the principle of zero
virtual work by constraint forces are central to both Lagrange's method of
undetermined multipliers, and Lagrange's equations in generalized coordinates.Comment: 12 pages, 10 figures. This article is based on an earlier article
physics/0410123. It includes new figures, equations and logical conten
Dissipative Properties of Systems Composed of High-Loss and Lossless Components
We study here dissipative properties of systems composed of two components
one of which is highly lossy and the other is lossless. A principal result of
our studies is that all the eigenmodes of such a system split into two distinct
classes characterized as high-loss and low-loss. Interestingly, this splitting
is more pronounced the higher the loss of the lossy component. In addition, the
real frequencies of the high-loss eigenmodes can become very small and even can
vanish entirely, which is the case of overdamping.Comment: Revision; Improved exposition and typos corrected; 45 pages, 4
figure
Earth-Moon Lagrangian points as a testbed for general relativity and effective field theories of gravity
We first analyse the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid. This scheme allows us to take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. A vehicle initially placed at L4 or L5 will not remain near the respective points. In particular, in the classical case the vehicle moves on a trajectory about the libration points for at least 700 days before escaping away. We show that this is true also if the modified long-distance Newtonian potential of effective gravity is employed. We also evaluate the impulse required to cancel out the perturbing force due to the Sun in order to force the spacecraft to stay precisely at L4 or L5. It turns out that this value is slightly modified with respect to the corresponding Newtonian one. In the second part of the paper, we first evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters and describe a tiny departure from the equilateral triangle. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. In other words, we deal with a theory involving quantum corrections to Einstein gravity, rather than to Newtonian gravity. By virtue of the effective-gravity correction to the long-distance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. In the latter case, for the point L1, general relativity corrects Newtonian theory by 7.61 meters, comparable, as an order of magnitude, with the lunar geodesic precession of about 3 meters per orbit. The latter is a cumulative effect accurately measured at the centimeter level through the lunar laser ranging positioning technique. Thus, it is possible to study a new laser ranging test of general relativity to measure the 7.61-meter correction to the L1 Lagrangian point, an observable never used before in the Sun-Earth-Moon system. Performing such an experiment requires controlling the propulsion to precisely reach L1, an instrumental accuracy comparable to the measurement of the lunar geodesic precession, understanding systematic effects resulting from thermal radiation and multi-body gravitational perturbations. This will then be the basis to consider a second-generation experiment to study deviations of effective field theories of gravity from general relativity in the Sun-Earth-Moon system
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