f(R,L m ) gravity

We generalize the f(R) type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R and of the matter Lagrangian L _m . We obtain the gravitational field equations in the metric formalism, as well as the equations of motion for test particles, which follow from the covariant divergence of the energy-momentum tensor. The equations of motion for test particles can also be derived from a variational principle in the particular case in which the Lagrangian density of the matter is an arbitrary function of the energy density of the matter only. Generally, the motion is non-geodesic, and it takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equation of motion is also considered, and a procedure for obtaining the energy-momentum tensor of the matter is presented. The gravitational field equations and the equations of motion for a particular model in which the action of the gravitational field has an exponential dependence on the standard general relativistic Hilbert–Einstein Lagrange density are also derived.     

Equations governing the energy-momentum tensor of a system of charged particles

In this paper the equations governing the energy, momentum, and angular momentum of a system of charged particles in motion are obtained as a series in inverse powers of c. The mass motion and time-symmetric part of the retarded electromagnetic field are shown to contribute to these equations a total time derivative which is an even power series in c⁻¹. The radiation-reaction terms of O(c⁻⁵) are evaluated and discussed as a prototype for higher order, odd power terms in the asymptotic expansions of the equations.     

Generalized Landau-Lifshitz models on the interval

We study the classical generalized gln Landau-Lifshitz (L-L) model with special boundary conditions that preserve integrability. We explicitly derive the first non-trivial local integral of motion, which corresponds to the boundary Hamiltonian for the sl₂ L-L model. Novel expressions of the modified Lax pairs associated to the integrals of motion are also extracted. The relevant equations of motion with the corresponding boundary conditions are determined. Dynamical integrable boundary conditions are also examined within this spirit. Then the generalized isotropic and anisotropic gln Landau-Lifshitz models are considered, and novel expressions of the boundary Hamiltonians and the relevant equations of motion and boundary conditions are derived.     

The electromagnetic two-body problem in special relativity

A system of two point charged particles is considered. Each particle moves in the electromagnetic field created by the other particle according to Maxwell's equations. A scheme of successive approximations is developed to study the field and the motion of the charges. The field (potentials and intensities) are exapanded in powers of c^?1 using a retarded time coordinate. The variables of the motion (position vectors, velocities, etc) are expanded in powers of c^?1 with coefficients depending on t only. The field is evaluated in the first three approximations. The equations of motion are derived in the same approximations and the corresponding conserved quantities are explicitly given. Thus, the usual assumption of an action-at-a-distance principle is avoided and the original nonlinear integrodifferential equations are reduced to a sequence of linear equations.     

Generalization of Nambu's mechanics

We look for a generalization of the mechanics of Hamilton and Nambu. We have found the equations of motion of a classical physical system ofS basic dynamic variables characterized byS – 1 constants of motion and by a function of the dynamical variables and the time whose value also remains constant during the evolution of the system. The numberS may be even or odd. We find that any locally invertible transformations are canonical transformations. We show that the equations of motion obtained can be put in a form similar to Nambu's equations by means of a time transformation. We study the relationship of the present formalism to Hamiltonian mechanics and consider an extension of the formalism to field theory.     

The equations of motion of macroscopic bodies in general relativity

A new approach to the problem of motion in General Relativity, based upon the systematic approximation procedure of Synge, is presented. The equations of transnational motion for a system of spherical bodies moving under their mutual gravitational attractions are derived. Approximations are based upon the weakness of the field and on the distance between any two of the bodies being considered large by comparision with their radii. The most general stress distribution consistent with maintaining the symmetry of the bodies throughout the motion is chosen. The use of controlled errors enables us to derive equations of motion applicable to a wider class of physical systems than the original equations of Einstein, Infeld and Hoffmann and Fock-Papapetrou.     

Equations of hydrodynamics in general relativity in the slow motion approximation

By means of a formal solution to the Einstein gravitational field equations a slow motion expansion in inverse powers of the speed of light is developed for the metric tensor. The formal solution, which satisfies the deDonder coordinate conditions and the Trautman outgoing radiation condition, is in the form of an integral equation which is solved iteratively. A stress-energy tensor appropriate to a perfect fluid is assumed and all orders of the metric needed to obtain the equations of motion and conserved quantities to the 21/2post-Newtonian approximation are found. The results are compared to those obtained in another gauge by S. Chandrasekhar. In addition, the relation of the fast motion approximation to the slow motion approximation is examined.     

Motion invariants for a charge in a linearly polarized wave field

Six motion integrals for a relativistic charge in the field of a transverse linearly polarized electromagnetic wave propagating with an arbitrary phase velocity u>c were obtained by solving the canonical equations of motion. On the basis of these integrals, the charge trajectory as a function of the wave phase is analyzed in a fixed coordinate system. The coordinates, time, and phase are related by elliptic functions.     

Interpretation of Einstein's theory of gravitation including the cosmological term as a de Sitter-invariant field theory on the de Sitter space

In order to remove the singular nature of Einstein's theory of gravitation including theΛ term, within the domain of field theories, it is shown that this theory can be consistently interpreted as a field theory on the fixed de Sitter space, invariant with respect to the de Sitter groupO(4,1). The corresponding field equations as well as the equations of motion are derived, their structural properties are discussed, and they are integrated for a spherical mass source of the field.     

A generally covariant formulation of classical electrodynamics for charges of finite extension

A generally covariant formulation of classical electrodynamics for charges of finite extension has been developed. The charges are required to maintain a prescribed “rigid” shape throughout the course of their motion. An action principle is formulated for the coupled system consisting of the charges plus electromagnetic and gravitational fields. The action principle yields a complicated set of coupled integro-differential equations for the motion and fields. A perturbation expansion in powers of the size of the charge distribution is obtained. In the limit that the size of the charge tends to zero, only a few kinematical features survive in the equations of motion. The resulting equations of motion have the DeWitt-Brehme [Ann. Phys.9 (1969), 220] form, but with additional curvature-coupling terms which were omitted by them owing to an algebraic error.     

Higher conservation laws for ten-dimensional supersymmetric Yang-Mills theories

It is shown that ten-dimensional supersymmetric Yang-Mills theories are integrable systems, in the (weak) sense of admitting a (superspace) Lax representation for their equations of motion. This is achieved by means of an explicit proof that the equations of motion are not only a consequence of but in fact fully equivalent to the superspace constraint F_αβ = 0. Moreover, a procedure for deriving infinite series of non-local conservation laws is outlined.     

Solitons in stimulated Raman scattering and resonant two-photon propagation

We are starting from equations of motion describing both stimulated Raman scattering and resonant two-photon absorption and emission processes. Application of the Estabrook-Wahlquist method leads to a system of differential equations whose integrability conditions are the original equations. This system is used as the starting point for applying the inverse scattering method. Implicit N-soliton formulae and explicit one-soliton formulae are derived. Possible applications under specified experimental conditions are discussed.     

W-symmetry and the rigid particle

We prove that W₃ is the gauge symmetry of the scale-invariant rigid particle, whose action is given by the integrated extrinsic curvature of its world line. This is achieved by showing that its equations of motion can be written in terms of the Boussinesq operator. The W₃ generators T and W then appear respectively as functions of the induced world line metric and the extrinsic curvature. We also show how the equations of motion for the standard relativistic particle arise from those of the rigid particle whenever it is consistent to impose the “zero-curvature gauge”, and how to rewrite them in terms of the KdV operator. The relation between particle models and integrable systems is further pursued in the case of the spinning particle, whose equations of motion are closely related to the SKdV operator. We also partially extend our analysis in the supersymmetric domain to the scale-invariant rigid particle by explicitly constructing a supercovariant version of its action.     

Quantum mechanical equations of motion of particles and spin in polarized media

Quantum mechanical equations of motion are obtained for particles and spin in media with polarized electrons in the presence of external fields. The motion of electrons and their spins is governed by the exchange interaction, while the motion of positrons and their spin is governed by the annihilation interaction. For particles with spin S ≥ 1, second-order terms in spin are taken into account. The equations obtained can be applied to describe the motion of particles and spin both in magnetic and nonmagnetic media.     

On the plane gravitational condensor with the positive gravitational constant

The equations of motion for the plane shell consisting of an ideal fluid and creating the de Sitter solution on one side and the static plane symmetric solution of Einstein equations with positive cosmological constant on the other side are derived. Solution of the equation of motion for the case of “plane gravitational condensor” is found and the necessary agreement with the solution of the problem in the caseA = 0 is demonstrated.     

Dynamic time responses of a flexible spinning disk misaligned with the axis of rotation

Using the finite element method, this study investigates the dynamic time responses of a flexible spinning disk of which axis of rotation is misaligned with the axis of symmetry. The misalignment between the axes of symmetry and rotation is one of major vibration sources in optical disk drives such as CD-ROM, CD-R, CD-RW and DVD drives. Based upon the Kirchhoff plate theory and the von Karman strain theory, three coupled equations of motion for the misaligned disk are obtained: two of the equations are for the in-plane motion while the other is for the out-of-plane motion. After transforming these equations into two weak forms for the in-plane and out-of-plane motions, the weak forms are discretized by using newly defined annular sector finite elements. Applying the generalized-α time integration method to the discretized equations, the time responses and the displacement distributions are computed and then the effects of misalignment on the responses and the distributions are analyzed. The computation results show that the misalignment has an influence on the magnitudes of the in-plane displacements. It is also found that the misalignment results in the amplitude modulation or the beat phenomenon in the time responses of the out-of-plane displacement.     

On the theory of collective motion in nuclei. III. From Hamiltonian dynamics to vortex-free fluidity

It is assumed that the Hamiltonian for collective motion in nuclei is invariant under the orthogonal group O(n, $\text{R}$). For degenerate orbits in phase space it is shown that the classical Hamiltonian equations reduce to the equations of a vortex-free fluid with a velocity field determined by independent equations of motion.     

Method of quasiclassical approximation for c-number projection in coherent states basis

A method of deriving the equations for physical average values is proposed in the form of ? orders expansion. The method is based on c-number projection of Heisenberg equations on a coherent states basis. The analysis of quantum corrections for quantum nonlinear resonance as an example is carried out. It is shown that the structure of quantum corrections results in appearance of the terms increasing in time in nonlinear systems.An example of an exactly solvable model is considered and the main times at which the expansion of ? orders is correct are obtained. The property of non-Hamiltonity of the equations of motion of the system in the phase space of projection is discussed.     

Nonequilibrium Relaxation of Bose-Einstein Condensates: Real-Time Equations of Motion and Ward Identities

We present a field-theoretical method to obtain consistently the equations of motion for small amplitude condensate perturbations in a homogeneous Bose-condensed gas directly in real time. It is based on a linear response and combines the Schwinger-Keldysh formulation of nonequilibrium quantum field theory with the Nambu-Gor'kov formalism of quasiparticle excitations in the condensed phase and the tadpole method in quantum field theory. This method leads to causal equations of motion that allow us to study the nonequilibrium evolution as an initial value problem. It also allows us to extract directly the Ward identities, which are a consequence of the underlying gauge symmetry and which in equilibrium lead to the Hugenholtz-Pines theorem. An explicit one-loop calculation of the equations of motion beyond the Hartree-Fock-Bogoliubov approximation reveals that the nonlocal, absorptive contributions to the self-energies corresponding to the Beliaev and Landau damping processes are necessary to fulfill the Ward identities in or out of equilibrium. It is argued that a consistent implementation at low temperatures must be based on the loop expansion, which is shown to fulfill the Ward identities order by order in perturbation theory.