Orbital angular momentum exchange in the interaction of twisted light with molecules

In the interaction of molecules with light endowed with orbital angular momentum, an exchange of orbital angular momentum in an electric dipole transition occurs only between the light and the center of mass motion; i.e., internal "electronic-type" motion does not participate in any exchange of orbital angular momentum in a dipole transition. A quadrupole transition is the lowest electric multipolar process in which an exchange of orbital angular momentum can occur between the light, the internal motion, and the center of mass motion. This rules out experiments seeking to observe exchange of orbital angular momentum between light beams and the internal motion in electric dipole transitions.     

Mechanism of angular momentum exchange between molecules and Laguerre-Gaussian beams

We derive the interaction Hamiltonian between a diatomic molecule and a Laguerre-Gaussian beam under the assumption of a small spread of the center of mass wave function of the molecule in comparison with the beam waist. Considering the dynamical variables of the center of mass, vibrational, rotational, and electronic motion, we show that, within the electronic dipole approximation, the orbital angular momentum of the field couples with the rotational and electronic motion. The changes in the transition probabilities and selection rules induced by the field orbital angular momentum and the applicability of the derived interaction mechanisms for polyatomic molecules are discussed.     

The analysis of rovibrational motion equations of a diatomic molecule in the linear regime

The motion equations of diatomic molecules are here extended from the absolute vibrational case to a more general and real rotational and vibrational (rovibrational) case. The rovibrational Hamiltonian is heuristically formed by substituting the respective number and angular momentum operators for the vibrational and rotational quantum numbers in the energy eigenvalues of a diatomic molecule which was first introduced by Dunham. The motion equations of observable quantities such as the position and linear momentum are then determined by implementing the well-known Heisenberg relation in quantum mechanics. We face with the second-order imaginary differential equations for describing the temporal variations of the relative position and the linear momentum of two oscillating atoms, which are coupled in the x–y horizontal plane. The possible rovibrational oscillations are distinguished by the three quantum numbers n, l and m associated with the energy and angular momentum quantities. It is finally demonstrated that the simultaneous solutions of rovibrational equations satisfy the energy conservation during all quantised oscillations of a diatomic molecule in space.     

Quadrupole test particle as a detector of gravitational waves

In this paper it is shown that in general relativity the theory of motion of quadrupole test particles (QTP's) can be used to describe the energy and angular momentum absorption by detectors of gravitational waves. By specifying the form of the quadrupole moment tensor Taub's [7] equations of motion of QTP's are simplified. In these equations the terms describing the change of the mass and of the angular momentum of a QTP due to external gravitational waves are found to occur. The limiting case of the flat space-time is also briefly discussed.     

Magnetic trapping of ultracold Rydberg atoms

We investigate the quantum dynamics of ultracold Rydberg atoms being exposed to a magnetic quadrupole field. A Hamiltonian describing the coupled dynamics of the electronic and center of mass motion is derived. Employing an adiabatic approach, the potential energy surfaces for intra-n-manifold mixing are computed. By determining the quantum states of the center of mass motion, we demonstrate that trapped states can be achieved if the total angular momentum of the atom is sufficiently large. This holds even if the extension of the electronic Rydberg state becomes equal to or even exceeds that of the ultracold center of mass motion.     

On the stationary axisymmetric Einstein-Maxwell equations

This paper is concerned with the class of exterior stationary axisymmetric solutions of the Einstein-Maxwell equations that arise from sources in which the mass is proportional to the charge and the angular momentum is proportional to the magnetic moment. With the use of Ernst's formulation the Einstein-Maxwell equations for this class are reduced to two coupled equations for two unknowns.     

Specifying Angular Momentum and Center of Mass for Vacuum Initial Data Sets

We show that it is possible to perturb arbitrary vacuum asymptotically flat spacetimes to new ones having exactly the same energy and linear momentum, but with center of mass and angular momentum equal to any preassigned values measured with respect to a fixed affine frame at infinity. This is in contrast to the axisymmetric situation where a bound on the angular momentum by the mass has been shown to hold for black hole solutions. Our construction involves changing the solution at the linear level in a shell near infinity, and perturbing to impose the vacuum constraint equations. The procedure involves the perturbation correction of an approximate solution which is given explicitly.     

Momentum,angular momentum,and equations of motion for test bodies in space-time with torsion

The definitions and transformation properties of momentum and angular momentum of test bodies possessing both macroscopic rotation and net spin are discussed. The equations of motion for momentum and angular momentum of test bodies are derived and written in a covariant form when the energy-momentum tensor is symmetric.     

Semiclassical dynamics and transport of the Dirac spin

A semiclassical theory of spin dynamics and transport is formulated using the Dirac electron model. This is done by constructing a wavepacket from the positive-energy electron band, and studying its structure and center of mass motion. The wavepacket has a minimal size equal to the Compton wavelength, and has self-rotation about the average spin angular momentum, which gives rise to the spin magnetic moment. Geometric gauge structure in the center of mass motion provides a natural explanation of the spin-orbit coupling and various Yafet terms. Applications of the spin-Hall and spin-Nernst effects are discussed.     

Approximate radiative solutions of the Einstein equations

In this paper the external field of a bounded source emitting gravitational radiation has been considered. A successive approximation method has been used to integrate the Einstein equations in Bondi's coordinates. A method of separation of angular variables has been worked out and the approximate Einstein equations have been reduced to the key equations (3.8)–(3.10). The losses of mass, momentum, and angular momentum due to gravitational multipole radiation have been found. It has been demonstrated that in the case of proper treatment a real mass occurs instead of a mass aspect in a solution of the Einstein equations. In Appendix C Bondi's news function has been given in terms of sources.     

The falling cat as a port-controlled Hamiltonian system

In this article, the falling cat is modeled as two jointed axial symmetric cylinders with arbitrary twist under the constraint of the vanishing total angular momentum. As a control system with the constraint taken into account, this model is formulated as a port-controlled Hamiltonian system defined on the cotangent bundle of the shape space for the jointed cylinders. A control is then designed as a function on the cotangent bundle, according to a standard procedure. Thus, the equations of motion are determined on the cotangent bundle together with the control. The whole motion as a vibrational motion of the falling cat is obtained after integrating the constraint equation of the vanishing total angular momentum. An example of the falling cat is given in which the model turns a somersault to approach a target state in equilibrium with an expected rotation after finishing a vibrational motion.     

The main periodicities of the motion of the center of the sun relative to the center of mass of the solar system and solar activity

The dynamic parameters of the motion of the Sun relative to the center of mass of the Solar System, viz., the distance of the center of the Sun relative to the center of mass and the angular momentum of the Sun, as well as its changes, are investigated. The frequency spectra of these parameters and of the Wolf numbers are calculated and the main spectral components are revealed. It is shown that a periodicity of 178.8 years is not predominant in the dynamic parameters of the motion of the Sun; in this frequency range a periodicity of 169 years connected with the influence of Neptune predominates. The coincidence of the periodicities in the Wolf numbers and dynamic parameters of the motion of the Sun is shown. The causes of all components of the frequency spectrum are indicated.     

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.     

Continuity and momentum equations for moist atmospheres

The moist atmosphere with occurring precipitation is considered to be a multiphase fluid composed of dry air, water vapor and hydrometeors. These compositions move with different velocities: they take a macroscopic motion with a reference velocity and a relative motion with a velocity deviated from the reference velocity. The reference velocity can be chosen as the velocities of dry air, a gas mixture and the total air mixture. The budget equations of continuity and momentum are formulated in the three reference-velocity frames. It is shown that the resulting equations are dependent on the chosen reference velocity. The diffusive flux due to compositions moving with velocities deviated from the reference velocity and the internal sources due to the phase transitions of water substances result in additional source terms in continuity and momentum equations. A continuity equation of the total mass is conserved and free of diffusive flux divergence if the reference velocity is referred to the velocity of the total air mixture. However, continuity equations in the dry-air and gasmixture frames are not conserved due to the mass diffusive flux divergence. The diffusive flux introduces additional source terms in the momentum equation. In the dry-air frame, the diffusive flux of water substances and the phase transitions of water substances contribute to the change of the total momentum. The additional sources of total momentum in the frame of a gas mixture are associated with the diffusive flux of hydrometeors, the phase transitions of hydrometeors and the gasmixture diffusive flux. In the frame of total air mixture, the contribution to the total momentum comes from the diffusive flux of all atmospheric compositions instead of the phase transitions. The continuity and momentum equations derived here are more complicated than the traditional model equations. With increasing computing power, it becomes possible to simulate atmospheric processes with these sophisticated equations. It is helpful to the improvement of precipitation forecast.     

Constraints,dissipation functions and cholesteric liquid crystals

Our recently introduced dissipation function theory of molecular liquids and solids is extended to include constrained motions. Lagrangian multipliers are used in the reversible parts of the field equations of motion. Conservation laws of mass, momentum, energy and angular momentum are derived (in both the material and spatial frames) and shown to hold under the same general conditions as in the unconstrained case. Our theory is then applied to liquid crystals. Within our formalism, constitutive relations characterizing linear and nonlinear liquid crystals are given. In the linear regime, for cholesterics, depending on whether the heat flux or the temperature gradient is used in the dissipation function we show that different reciprocal relations follow. Our results for incompressible nematics and cholesterics are compared with the Ericksen-Leslie theory and others.     

Wave Packets in a Magnetic Field

A Gaussian wave packet confined to move on a plane perpendicular to a magnetic field remains a Gaussian wave packet in its time evolution. The average position and momentum follow the Ehrenfest equations which are identical to the classical Hamilton equations. A set of nonlinear equations decoupled from the Ehrenfest equation is derived for the parameters describing the time evolution of the density distribution and phases of a wave packet. Explicit solutions are then obtained when the "internal" angular momentum of the wave packet vanishes. In this case it is shown that the motion of the wave packet is a superposition of a translational motion, a rotation and a vibration.     

Discrete gradients in discrete classical mechanics

A simple model of discrete classical mechanics is given where, starting from the continuous Hamilton equations, discrete equations of motion are established together with a proper discrete gradient definition. The conservation laws of the total discrete momentum, angular momentum, and energy are demonstrated.     

Exact solution of the classical mechanical quadratic Zeeman effect

We address the curious problem of quadratic Zeeman effect at the classical mechanical level. The problem has been very well understood for decades, but an analytical solution of the equations of motion is still to be found. This state of affairs persists because the simultaneous presence of the Coulombic and quadratic terms lowers the dynamical symmetry. Energy and orbital angular momentum are still constants of motion. We find the exact solutions by introducing the concept of an image ellipse. The quadratic effect leads to a dilation of space-time, and a one-to-one correspondence is observed for pairs of physical quantities like energy and angular momentum, and the maximum and minimum distances from the Coulomb center for the Zeeman orbit and the corresponding pairs for the image ellipse. Thus, instead of finding additional conserved quantities, we find constants of motion for an additional dynamics, namely, the image problem. The trajectory is open, in agreement with Bertrand’s theorem, but necessarily bound. A stable unbound trajectory does not exist for real values of energy and angular momentum. The radial distance, the angle covered in the plane of the orbit, and the time are uniquely determined by introducing further the concept of an image circle. While the radial distance is defined in a closed form as a transcendental function of the image-circular angle, the corresponding orbit angle and time variables are found in the form of two convergent series expansions. The latter two variables are especially contracted, thereby leading to a precession of the open cycles around the Coulomb center. It is expected that the space-time dilation effect observed here would somehow influence the solution of the quantum mechanical problem at the non-relativistic level.      

Quasi-local gravitational angular momentum and centre of mass from generalised Witten equations

Witten’s proof for the positivity of the ADM mass gives a definition of energy in terms of three-surface spinors. In this paper, we give a generalisation for the remaining six Poincaré charges at spacelike infinity, which are the angular momentum and centre of mass. The construction improves on certain three-surface spinor equations introduced by Shaw. We solve these equations asymptotically obtaining the ten Poincaré charges as integrals over the Nester–Witten two-form. We point out that the defining differential equations can be extended to three-surfaces of arbitrary signature and we study them on the entire boundary of a compact four-dimensional region of spacetime. The resulting quasi-local expressions for energy and angular momentum are integrals over a two-dimensional cross-section of the boundary. For any two consecutive such cross-sections, conservation laws are derived that determine the influx (outflow) of matter and gravitational radiation.     

Compliant structures with time-varying moment of inertia and non-zero averaged momentum and their application in angular rate microsensors

In this work we introduce a new class of fully compliant structures performing vibratory motion, yet characterized by non-zero averaged momentum, appearing due to time-dependency of the inertial parameters. The work is motivated by microelectromechancial systems (MEMS) applications, where an implementation of unidirectional, non-vibratory motion involving relative motion of parts is not desirable for reliability reasons. Instead of changing the mass, which is challenging on the microscale, the moment of inertia of the proof mass performing tilting vibrations is controlled in such a way that it is higher or lower depending on the sign of the velocity. This results in a non-zero angular momentum averaged over the period. The equations describing the dynamics of a generic structure with a time-varying inertia and in a rotating coordinate frame are derived by using a variational principle. Simple approximate expressions for the averaged momentum and steady tilting angle are obtained and validated numerically. Based on the model results for different operational scenarios, we demonstrate that these devices can be efficiently used in fully compliant actuators and vibratory angular rate sensors (microgyros) with a steady response in a sensing mode (“pseudospinning disk gyros”), as well as in a parametrically excited gyro. The structure can be viewed also as a first step toward the realization of dynamic materials (DM) which are substances with material properties that may change in space and time.