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There are three types of dynamical equations for a post-Newtonian Lagrangian formulation of three black holes. They are approximate post-Newtonian Lagrangian equations of motion, coherent post-Newtonian Lagrangian equations of motion, and post-Newtonian Hamiltonian canonical equations. The energy integral is conserved approximately in the first model, but exactly in the second and third models. The three formulations are not exactly equivalent, while they are approximately related in the orbital dynamical behavior. Figure-eight orbits in the three problems may have different initial conditions. Imai, Chiba, and Asada adjusted the initial velocities satisfying the figure-eight orbits for given initial positions in the first model. The initial velocities are still suitable for the figure-eight orbits of the latter two models when the initial separations of the first body are enough large. However, they are not generally applicable to relatively small initial separations in the three systems. We can obtain the desired initial velocities by scanning the initial velocities of the third body in the three systems. The figure-eight orbits in these models exhibit sensitive dependence on the scanned initial velocities for smaller initial separations. Although the differences in the scanned initial velocities among the three problems are minor for the small initial separations, the scanned initial velocities for any one of the three systems do not yield the figure-eight orbits for the other two systems.

The framework of a kind of noncanonical warm inflation is introduced, and the dynamical equations of this scenario are presented. We propose the slow roll approximations and give some redefining slow roll parameters in this scenario which remain dimensionless. Performing systemic stability analysis, we calculate the slow roll conditions to guarantee that slow roll approximations hold. The slow roll conditions suggest slow roll inflation in general noncanonical warm inflationary scenario can still exist, and in addition, the slow roll approximations are more easily to be satisfied. Then, a concrete Dirac-Born-Infeld warm inflationary model is studied., 6 pages, 0figures

In this paper, we have continued the work of Herrera \emph{et al.} \cite{herrera2012dynamical} in $f(G)$ gravitational theory. For this purpose, a spherically symmetric fluid exhibiting locally anisotropic pressure along with the energy density, is taken under consideration. The perturbation scheme is imposed on modified field equations and the dynamical equations. The collapse equation is devised from these perturbed equations which stimulates to disclose the instability zone under both Newtonian and post-Newtonian constraints. It is wrapped up by concluding that dynamical instability is interpreted by the adiabatic index $\Gamma$ which relies on the anisotropic pressure, energy density and the dark source terms due to $f(G)$ gravity., Comment: 25 pages, no figure, version accepted for publication in Physical Review D

Using a Green's function approach, we compare the trajectories of classical Hamiltonian point particles in an expanding space-time to the effectively inertial trajectories in the Zel'dovich approximation. It is shown that the effective gravitational potential accelerating the particles relative to the Zel'dovich trajectories vanishes exactly initially as a consequence of the continuity equation, and acts only during a short, early period. The Green's function approach suggests an iterative scheme for improving the Zel'dovich trajectories, which can be analytically solved. We construct these trajectories explicitly and show how they interpolate between the Zel'dovich and the exact trajectories. The effective gravitational potential acting on the improved trajectories is substantially smaller at late times than the potential acting on the exact trajectories. The results may be useful for Lagrangian perturbation theory and for numerical simulations., Comment: 7 pages, 3 figures

We study the structure of scalar-tensor theories of gravity based on derivative couplings between the scalar and the matter degrees of freedom introduced through an effective metric. Such interactions are classified by their tensor structure into conformal (scalar), disformal (vector) and extended disformal (traceless tensor), as well as by the derivative order of the scalar field. Relations limited to first derivatives of the field ensure second order equations of motion in the Einstein frame and hence the absence of Ostrogradski ghost degrees of freedom. The existence of a mapping to the Jordan frame is not trivial in the general case, and can be addressed using the Jacobian of the frame transformation through its eigenvalues and eigentensors. These objects also appear in the study of different aspects of such theories, including the metric and field redefinition transformation of the path integral in the quantum mechanical description. Although sane in the Einstein frame, generic disformally coupled theories are described by higher order equations of motion in the Jordan frame. This apparent contradiction is solved by the use of a hidden constraint: the contraction of the metric equations with a Jacobian eigentensor provides a constraint relation for the higher field derivatives, which allows one to express the dynamical equations in a second order form. This signals a loophole in Horndeski's theorem and allows one to enlarge the set of scalar-tensor theories which are Ostrogradski-stable. The transformed Gauss-Bonnet terms are also discussed for the simplest conformal and disformal relations., Comment: 21 pages, 2 figures, 2 tables. Minor changes, version accepted for PRD

We apply the time-dependent variational principle of Balian and Veneroni to {Phi}{sup 4} theory. An appropriate parametrization for the variational objects allows us to write coupled dynamical equations from which we derive approximations for the two-time correlation functions involving two, three, or four field operators. {copyright} 1995 The American Physical Society.

We apply the nonlinear realizations method for constructing new Galilean conformal mechanics models. Our starting point is the Galilean conformal algebra which is a non-relativistic contraction of its relativistic counterpart. We calculate Maurer-Cartan one-forms, examine various choices of the relevant coset spaces and consider the geometric inverse Higgs-type constraints which reduce the number of the independent coset parameters and, in some cases, provide dynamical equations. New Galilean conformally invariant actions are derived in arbitrary space-time dimension D=d+1 (no central charges), as well as in the special dimension D=2+1 with one "exotic" central charge. We obtain new classical mechanics models which extend the standard (D=0+1) conformal mechanics in the presence of d non-vanishing space dimensions., Comment: v2: 1 + 20 pages, small changes in Sect.5 and two references added; the version will appear in Phys.Rev.D

We study the dynamics of a flat multidimensional anisotropic cosmological model filled with an anisotropic fluidlike medium. By an appropriate choice of variables, the dynamical equations reduce to a two-dimensional dynamical system. We present a detailed analysis of the time evolution of this system and the conditions of the existence of spacetime singularities. We investigate the conditions under which violent, exponential, and power-law inflation is possible. We show that dimensional reduction cannot proceed by anti-inflation (rapid contraction of internal space). Our model indicates that it is very difficult to achieve dimensional reduction by classical means.

Based on the dynamical equations for a nonrotating elastic deformable astronomical body in the first post-Newtonian approximation of Einstein's theory of gravity, we re-examined the boundary(junction) conditions and have proven that a term, which is missing in the customary boundary(junction) conditions, is found. This term is induced by the existence of initial equilibrium hydrostatic pressure. A physical explanation of this term is given in the Newtonian approximation as well. By using the correcting boundary conditions the relation of the free spherically symmetrical radial oscillation frequency of a nonrotating homogeneously and isotropically elastic sphere with constant density is derived.

We investigate the theories of dissipative relativistic fluids in which all of the dynamical equations can be written as total-divergence equations. Extending the analysis of Liu, Mueller, and Ruggeri, we find the general theory of this type. We discuss various features of these theories, including the causality of the full nonlinear evolution equations and the nature and stability of the equilibrium states.

An analytic approach to phenomenological models inspired by cubic string field theory is introduced and applied to some examples. We study a class of actions for a minimally coupled, homogeneous scalar field whose energy density contains infinitely many time derivatives. These nonlocal systems are systematically localized and an algorithm to find cosmological solutions of the dynamical equations is provided. Our formalism is able to define the nonlocal field in regions of the parameter space which are inaccessible by standard methods. Also, problems related to nonlocality are reinterpreted under a novel perspective and naturally overcome. We consider phenomenological models living on a Friedmann-Robertson-Walker background with power-law scale factor, both in four dimensions and on a high-energy braneworld. The quest for solutions unravels general features of nonlocal dynamics indicating several future directions of investigation., Comment: 20 pages, 3 figures. v2: references addded, typos corrected, discussion improved; v3: matches the published version

In our previous paper, based on the Carter & Quintana framework and the Damour-Soffel-Xu scheme, we deduced a complete and closed set of post-Newtonian dynamical equations for elastically deformable astronomical bodies. In this paper, we expand the general relativistic perturbation equations of elastic deformable bodies (field equations, stress-strain relation, Euler equation) in terms of Generalized Spherical Harmonics. This turns the set of complicated partial differential equations into a set of ordinary differential equations. This will be useful for numerical applications that mainly deal with the global dynamics of the Earth., 11 pages

We study the apparition of event horizons in accelerated expanding cosmologies. We give a graphical and analytical representation of the horizons using proper distances to coordinate the events. Our analysis is mainly kinematical. We show that, independently of the dynamical equations, all the event horizons tend in the future infinity to a given expression depending on the scale factor that we call asymptotic horizon. We also encounter a subclass of accelerating models without horizon. When the ingoing null geodesics do not change concavity in its cosmic evolution we recover the de Sitter and quintessence-Friedmann-Robertson-Walker models., Latex2e, 27 pages, 4 figures, submitted to Class. Quantum Grav

We calculate the off-equilibrium hydrostatic pressure for the O(N) Phi^{4} theory to the leading order in 1/N. The present paper, the first of a series, concentrates on the calculation of pressure in the non-equilibrium but translationally invariant medium. The Jaynes-Gibbs principle of maximal entropy is used to introduce the relevant density matrix which is then directly implemented into dynamical equations through generalised Kubo-Martin-Schwinger (KMS) conditions. We show that in the large N limit use of Ward identities enables the pressure to be expressed in terms of two point Green's functions. These satisfy the Kadanoff-Baym equations which are exactly solvable, and we explicitly calculate the pressure for three illustrative choices of \rho.

The one-loop contributions of conformally invariant scalar fields to the effective action are calculated for homogeneous cosmological models with small anisotropy. The dynamical equations which determine the classical geometry are displayed and the matrix elements of the stress-energy tensor between the initial and final vacuums are determined.

A scale-covariant theory of gravitation is presented which is characterized by a set of equations that are complete only after a choice of the scale function is made. Special attention is given to gauge conditions and units which allow gravitational phenomena to be described in atomic units. The generalized gravitational-field equations are derived by performing a direct scale transformation, by extending Riemannian geometry to Weyl geometry through the introduction of the notion of cotensors, and from a variation principle. Modified conservation laws are provided, a set of dynamical equations is obtained, and astrophysical consequences are considered. The theory is applied to examine certain homogeneous cosmological solutions, perihelion shifts, light deflections, secular variations of planetary orbital elements, stellar structure equations for a star in quasi-static equilibrium, and the past thermal history of earth. The possible relation of the scale-covariant theory to gauge field theories and their predictions of cosmological constants is discussed.

A program for construction of a crossing-symmetric unitary Regge theory of meson-meson scattering is proposed. The construction proceeds through solution of a nonlinear functional equation, psi = G (psi), for certain partial-wave scattering functions psi. The functional equation is analogous to a conventional dynamical equation, in that the scattering amplitude is generated from input functions which describe the primary forces between mesons and possible inelastic effects. A solution of the equation provides a scattering amplitude having Mandelstam analyticity, exact crossing symmetry, exact unitarity below the production threshold, and meromorphy of partial waves in the right-half l plane, with the consequent Regge asymptotics. Inelastic unitarity (0 < or = eta (l,s) < or = 1) is not guaranteed, but may perhaps be achieved through constraints on inputs. In any case, the partial waves are bounded throughout the physical region; such a bound was not ensured in earlier schemes based on the Mandelstam iteration. In this first paper of a series, the equations are formulated for the case of weak coupling, in which no Regge poles enter the right-half l plane. Inelastic effects are described by crossed two-particle processes and assigned input functions. Later papers will treat the case of strong coupling,more » in which Regge trajectories are generated dynamically, and the extension of the formalism to include many coupled channels.« less

We compute up to and including all the ${c}^{\ensuremath{-}2}$ terms in the dynamical equations for extended bodies interacting through electromagnetic, gravitational, or short-range fields. We show that these equations can be reduced to those of point particles with intrinsic angular momentum assuming spherical symmetry.

Whenever gyroscopic forces are present, stable static solutions to dynamical equations of motion need not minimize the energy. We show that this happens in the classical Yang-Mills theory with sources, and we identify the stable fluctuations which lower the energy. The finite form of these infinitesmal, time-dependent deformations of the known static solutions is obtained for weak external sources, and a unified description of both the static and time-varying solutions is given. Also, we demonstrate that the previously found bifurcation in the presence of strong sources is characterized by a zero-eigenvalue mode which dominates the behavior of the solutions near the bifurcation point. The stability properties of the bifurcating solutions are assessed.

In the preceding paper a geometric formulation of Kaluza-Klein theory was presented with the basic assumption that the space-time is locally and isometrically embedded in the high-dimensional space which emerged at the big bang. In the present note the Gauss-Codazzi-Ricci equations which are the integrability equations for the embedding are interpreted as the dynamical equations for a low-energy observer. The second quadratic form which results from the embedding is interpreted as a fundamental spin-two massless field. The dynamics for an observer with high-energy probes is described as usual by the Einstein-Hilbert action defined in the high-dimensional space and dimensionally reduced by integration over the internal space. The behavior of fermion masses under different gravitational field strengths is implemented by use of the mass operator defined with the second-order Casimir operator of the embedding symmetry group.

Equations for the construction of a crossing-symmetric unitary Regge theory of meson-meson scattering are described. In the case of strong coupling, Regge trajectories are to be generated dynamically as zeros of the D function in a nonlinear N/D system. This paper is concerned mainly with writing the inputs to the N/D system in such a way that a convergent theory with exact crossing symmetry is defined. The scheme demands elimination of ghosts, i.e., bound-state poles at energies below threshold where trajectories pass through zero. A method for ghost elimination is proposed which entails an s-wave subtraction constant, and allows the physical s wave to be different from the l-analytic amplitude evaluated at l = 0. A dynamical model is suggested in which the subtraction constant alone generates the meson-meson interaction. An alternative ghost-elimination scheme proposed by Gell-Mann, in which only l-analytic amplitudes are involved, can be discussed in a formalism including channels with spin.

The problem of giving a theoretical description of sequential decays is reconsidered. It is shown that the usual quantum description of such processes cannot be reconciled with the classical one. This fact suggests a more critical analysis of the actual physical situation occurring in such processes. By taking into consideration the remarkable successes which have been obtained in the description of nonsequential decays by considering the unavoidable interactions of the decaying system with the measuring apparatuses, an analogous formalism is developed for the problem under consideration. The description of the process is given in the language of the density operator. The evolution equation for such an operator has the standard form of the dynamical equations in the theory of quantum dynamical semigroups. Also in the present case, as for the nonsequential decay, the more realistic description which takes into account the actual set-up of a decay experiment turns out to yield results which agree completely with the classical ones. Some interesting problems arising within the present formalism are discussed.

The implication of structure analysis is investigated as a starting point for a nonperturbative treatment of the dynamical equations of field theory. The multiparticle ordering entails the arrangement of physical processes according to subclasses of processes embodied by the appropriate correlation functions. Closure among the correlation functions is obtained by exploiting the cluster properties of the correlation functions. The investigation culminates in a method for incorporating self-consistent feedback contributions as nested contributions which result in a method of linearization, in a channel-oriented approach, of the underlying nonlinear equations. The method of approximation exhibits a form-preserving property in that the structure of the equations is maintained in successive steps; this simulates the underlying nonlinear dynamics by allowing the system to react on itself. (AIP)

Integral equations for construction of a crossing-symmetric unitary Regge theory are extended to allow two coupled two-body channels. As in the case of a single channel, spectral functions are represented as Watson-Sommerfeld integrals over continued partial waves. A new type of partial wave is needed to represent one of the spectral functions in a region where one channel is open and the other is closed. This leads to certain difficulties in allowing realistic Regge poles.

Conditions for entropy extremum of self-bound relativistic systems of particles, in stationary axisymmetric motion, are obtained under the following constraints: (i) The system is kept isolated in a geometrical sense, implying that the total mass-energy and total angular momentum, defined by the asymptotic behavior of the metric, are kept constant, (ii) the total number of particles is kept constant, and (iii) Einstein's constraint equations are imposed on a spacelike hypersurface. It is shown that if the system is in mechanical equilibrium, the total entropy is an extremum for all trial nonequilibrium configurations that satisfy the constraints and respect the symmetry, if and only if (i) Einstein's dynamical equations are satisfied, (ii) the temperature and the chemical free energy, as seen from infinity, are constant, and (iii) the system is rigidly rotating. The proof does not depend on a particular functional expression for the mass or for the angular momentum; consequently, no related Lagrange multipliers have been used.

In parts I and II of this series, a system of partial-wave equations for construction of a crossing-symmetric unitary Regge theory of meson-meson scattering was described. Here we show that the sum of the partial waves of a solution has a representation in which crossing symmetry is apparent, all integrals converge without subtractions, double-spectral funcions have the correct support, and the contributions of Regge poles in all three channels are displayed simultaneously. We obtain the Regge asymptotic limit for s ..-->.. infinity at arbitrary fixed t by a method which avoids a difficulty in the usual heuristic argument. We also discuss the consequences at high energy of a new method of avoiding ghost poles at l = 0 on even-signature trajectories.