213 research outputs found
The Tulczyjew triple for classical fields
The geometrical structure known as the Tulczyjew triple has proved to be very
useful in describing mechanical systems, even those with singular Lagrangians
or subject to constraints. Starting from basic concepts of variational
calculus, we construct the Tulczyjew triple for first-order Field Theory. The
important feature of our approach is that we do not postulate {\it ad hoc} the
ingredients of the theory, but obtain them as unavoidable consequences of the
variational calculus. This picture of Field Theory is covariant and complete,
containing not only the Lagrangian formalism and Euler-Lagrange equations but
also the phase space, the phase dynamics and the Hamiltonian formalism. Since
the configuration space turns out to be an affine bundle, we have to use affine
geometry, in particular the notion of the affine duality. In our formulation,
the two maps and which constitute the Tulczyjew triple are
morphisms of double structures of affine-vector bundles. We discuss also the
Legendre transformation, i.e. the transition between the Lagrangian and the
Hamiltonian formulation of the first-order field theor
Time-dependent Mechanics and Lagrangian submanifolds of Dirac manifolds
A description of time-dependent Mechanics in terms of Lagrangian submanifolds
of Dirac manifolds (in particular, presymplectic and Poisson manifolds) is
presented. Two new Tulczyjew triples are discussed. The first one is adapted to
the restricted Hamiltonian formalism and the second one is adapted to the
extended Hamiltonian formalism
Motion of a Vector Particle in a Curved Spacetime. I. Lagrangian Approach
From the simple Lagrangian the equations of motion for the particle with spin
are derived. The spin is shown to be conserved on the particle world-line. In
the absence of a spin the equation coincides with that of a geodesic. The
equations of motion are valid for massless particles as well, since mass does
not enter the equations explicitely.Comment: 6 pages, uses mpla1.sty, published in MPLA, replaced with corrected
typo
Hamiltonian of a spinning test-particle in curved spacetime
Using a Legendre transformation, we compute the unconstrained Hamiltonian of
a spinning test-particle in a curved spacetime at linear order in the particle
spin. The equations of motion of this unconstrained Hamiltonian coincide with
the Mathisson-Papapetrou-Pirani equations. We then use the formalism of Dirac
brackets to derive the constrained Hamiltonian and the corresponding
phase-space algebra in the Newton-Wigner spin supplementary condition (SSC),
suitably generalized to curved spacetime, and find that the phase-space algebra
(q,p,S) is canonical at linear order in the particle spin. We provide explicit
expressions for this Hamiltonian in a spherically symmetric spacetime, both in
isotropic and spherical coordinates, and in the Kerr spacetime in
Boyer-Lindquist coordinates. Furthermore, we find that our Hamiltonian, when
expanded in Post-Newtonian (PN) orders, agrees with the Arnowitt-Deser-Misner
(ADM) canonical Hamiltonian computed in PN theory in the test-particle limit.
Notably, we recover the known spin-orbit couplings through 2.5PN order and the
spin-spin couplings of type S_Kerr S (and S_Kerr^2) through 3PN order, S_Kerr
being the spin of the Kerr spacetime. Our method allows one to compute the PN
Hamiltonian at any order, in the test-particle limit and at linear order in the
particle spin. As an application we compute it at 3.5PN order.Comment: Corrected typo in the ADM Hamiltonian at 3.5 PN order (eq. 6.20
Motion of test bodies in theories with nonminimal coupling
We derive the equations of motion of test bodies for a theory with nonminimal
coupling by means of a multipole method. The propagation equations for
pole-dipole particles are worked out for a gravity theory with a very general
coupling between the curvature scalar and the matter fields. Our results allow
for a systematic comparison with the equations of motion of general relativity
and other gravity theories.Comment: 5 pages, RevTex forma
The de Sitter Relativistic Top Theory
We discuss the relativistic top theory from the point of view of the de
Sitter (or anti de Sitter) group. Our treatment rests on Hanson-Regge's
spherical relativistic top lagrangian formulation. We propose an alternative
method for studying spinning objects via Kaluza-Klein theory. In particular, we
derive the relativistic top equations of motion starting with the geodesic
equation for a point particle in 4+N dimensions. We compare our approach with
the Fukuyama's formulation of spinning objects, which is also based on
Kaluza-Klein theory. We also report a generalization of our approach to a 4+N+D
dimensional theory.Comment: 25 pages, Latex,commnets and references adde
On the comparison of results regarding the post-Newtonian approximate treatment of the dynamics of extended spinning compact binaries
A brief review is given of all the Hamiltonians and effective potentials
calculated hitherto covering the post-Newtonian (pN) dynamics of a two body
system. A method is presented to compare (conservative) reduced Hamiltonians
with nonreduced potentials directly at least up to the next-to-leading-pN
order.Comment: Conference proceedings for the 7th International Conference on
Gravitation and Cosmology (ICGC2011), 4 page
Dynamics of test bodies with spin in de Sitter spacetime
We study the motion of spinning test bodies in the de Sitter spacetime of
constant positive curvature. With the help of the 10 Killing vectors, we derive
the 4-momentum and the tensor of spin explicitly in terms of the spacetime
coordinates. However, in order to find the actual trajectories, one needs to
impose the so-called supplementary condition. We discuss the dynamics of
spinning test bodies for the cases of the Frenkel and Tulczyjew conditions.Comment: 11 pages, RevTex forma
Spinning branes in Riemann-Cartan spacetime
We use the conservation law of the stress-energy and spin tensors to study
the motion of massive brane-like objects in Riemann-Cartan geometry. The
world-sheet equations and boundary conditions are obtained in a manifestly
covariant form. In the particle case, the resultant world-line equations turn
out to exhibit a novel spin-curvature coupling. In particular, the spin of a
zero-size particle does not couple to the background curvature. In the string
case, the world-sheet dynamics is studied for some special choices of spin and
torsion. As a result, the known coupling to the Kalb-Ramond antisymmetric
external field is obtained. Geometrically, the Kalb-Ramond field has been
recognized as a part of the torsion itself, rather than the torsion potential
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