226 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
A Canonical Decomposition in Collective and Relative Variables of a Klein-Gordon Field in the Rest-Frame Wigner-Covariant Instant Form
The canonical decomposition of a real Klein-Gordon field in collective and
relative variables proposed by Longhi and Materassi is reformulated on
spacelike hypersurfaces. This allows to obtain the complete canonical reduction
of the system on Wigner hyperplanes, namely in the rest-frame Wigner-covariant
instant form of dynamics. From the study of Dixon's multipoles for the
energy-momentum tensor on the Wigner hyperplanes we derive the definition of
the canonical center-of-mass variable for a Klein-Gordon field configuration:
it turns out that the Longhi-Materassi global variable should be interpreted as
a center of phase of the field configuration. A detailed study of the
kinematical "external" and "internal" properties of the field configuration on
the Wigner hyperplanes is done. The construction is then extended to charged
Klein-Gordon fields: the centers of phase of the two real components can be
combined to define a global center of phase and a collective relative variable
describing the action-reaction between the two Feshbach-Villars components of
the field with definite sign of energy and charge. The Dixon multipoles for
both the energy-momentum and the electromagnetic current are given. Also the
coupling of the Klein-Gordon field to scalar relativistic particles is studied
and it is shown that in the reduced phase space, besides the particle and field
relative variables, there is also a collective relative variable describing the
relative motion of the particle subsytem with respect to the field one.Comment: 86 pages, no figure
Tail-induced spin-orbit effect in the gravitational radiation of compact binaries
Gravitational waves contain tail effects which are due to the back-scattering
of linear waves in the curved space-time geometry around the source. In this
paper we improve the knowledge and accuracy of the two-body inspiraling
post-Newtonian (PN) dynamics and gravitational-wave signal by computing the
spin-orbit terms induced by tail effects. Notably, we derive those terms at 3PN
order in the gravitational-wave energy flux, and 2.5PN and 3PN orders in the
wave polarizations. This is then used to derive the spin-orbit tail effects in
the phasing through 3PN order. Our results can be employed to carry out more
accurate comparisons with numerical-relativity simulations and to improve the
accuracy of analytical templates aimed at describing the whole process of
inspiral, merger and ringdown.Comment: Minor corrections. To be published in Physical Review
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
Higher-order spin effects in the dynamics of compact binaries I. Equations of motion
We derive the equations of motion of spinning compact binaries including the
spin-orbit (SO) coupling terms one post-Newtonian (PN) order beyond the
leading-order effect. For black holes maximally spinning this corresponds to
2.5PN order. Our result for the equations of motion essentially confirms the
previous result by Tagoshi, Ohashi and Owen. We also compute the spin-orbit
effects up to 2.5PN order in the conserved (Noetherian) integrals of motion,
namely the energy, the total angular momentum, the linear momentum and the
center-of-mass integral. We obtain the spin precession equations at 1PN order
beyond the leading term, as well. Those results will be used in a future paper
to derive the time evolution of the binary orbital phase, providing more
accurate templates for LIGO-Virgo-LISA type interferometric detectors.Comment: transcription error in Eqs. (2.17) correcte
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
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
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
Spin-squared Hamiltonian of next-to-leading order gravitational interaction
The static, i.e., linear momentum independent, part of the next-to-leading
order (NLO) gravitational spin(1)-spin(1) interaction Hamiltonian within the
post-Newtonian (PN) approximation is calculated from a 3-dim. covariant ansatz
for the Hamilton constraint. All coefficients in this ansatz can be uniquely
fixed for black holes. The resulting Hamiltonian fits into the canonical
formalism of Arnowitt, Deser, and Misner (ADM) and is given in their
transverse-traceless (ADMTT) gauge. This completes the recent result for the
momentum dependent part of the NLO spin(1)-spin(1) ADM Hamiltonian for binary
black holes (BBH). Thus, all PN NLO effects up to quadratic order in spin for
BBH are now given in Hamiltonian form in the ADMTT gauge. The equations of
motion resulting from this Hamiltonian are an important step toward more
accurate calculations of templates for gravitational waves.Comment: REVTeX4, 10 pages, v2: minor improvements in the presentation, v3:
added omission in Eq. (4) and corrected coefficients in the result, Eq. (9);
version to appear in Phys. Rev.
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