Velocity Quantization Approach of the One-Dimensional Dissipative Harmonic Oscillator

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. PACS: 03.20.+i, 03.30.+p, 03.65.−w,03.65.Ca     

Constants of Motion for Several One-Dimensional Systems and Problems Associated with Getting Their Hamiltonians

The constants of motion of the following systems are deduced: a relativistic particle with linear dissipation; a no-relativistic particle with a time explicitly depending force; a no-relativistic particle with a constant force and time depending mass; and a relativistic particle under a conservative force with position depending mass. The Hamiltonian for these systems, which is determined by getting the velocity as a function of position and generalized linear momentum, can be found explicitly at first approximation for the first system. The Hamiltonians for the other systems are kept implicitly in their expressions for their constants of motion.     

Relativistic aspects of the motion of two-particle model molecule oscillating perpendicularly to the direction of its translation

On solving exactly relativistic equations of motion for the model molecule of two particles coupled elastically, it has been shown that in the framework of relativistic mechanics this system, and in general any closed system of interacting particles, is not inertial. In particular, the translational velocity of the mass center of such a system has, as a consequence of the nonlinearity of the equations, oscillatory components reflecting its internal transverse oscillations—it is a pulsed motion. This effect can in principle be seen in the time-of-flight experiments. The force constant of elastic coupling in the system, as seen by the observer at rest, is shown to decline with increase of the total momentum of the system.     

Ambiguities on the Quantization of a One-Dimensional Dissipative System with Position Depending Dissipative Coefficient

For a one-dimensional dissipative system with position depending coefficient, two constant of motion are deduce. These constants of motion bring about two Hamiltonians to describe the dynamics of same classical system. However, their quantization describe the dynamics of two completely different quantum systems. PACS numbers: 03.20.+i; 03.30.+ p; 03.65.-w     

Restricted Constant of Motion for the One-Dimensional Harmonic Oscillator with Quadratic Dissipation and Some Consequences in Statistic and Quantum Mechanics

A restricted constant of motion, Lagrangian and Hamiltonian, for the harmonic oscillator with quadratic dissipation is deduced. The restriction comes from the consideration of the constant of motion for the velocity of the particle either for v 0 or for v < 0. A study is done about the implications that these restricted variables have on the specific heat of a thermodynamical system of oscillators with this dissipation, and on the quantization of this dissipative system.     

相对论力学中恒力作用下带电粒子的运动

将相对论力学规律的三维形式应用于带电粒子运动问题,导出恒力作用下粒子的加速度关系式及速度关系式,并据此分别得出光速c是带电粒子速度极限的结论,同时还将其与经典力学中恒力作用下的粒子运动进行了比较。     

Relativistic ponderomotive forces

The interaction of a relativistic classical electron with an inhomogeneous electromagnetic field is investigated. In second-order perturbation theory the motion is separated into fast and slow motions, and the relativistic Newtonian equation is averaged over the fast oscillations. The rate of change obtained for the slow component of the electron momentum is interpreted as a relativistic ponderomotive force. The result is generalized to the relativistic case of the wellknown expression for the Gaponov-Miller force acting on an electron at rest. The expressions obtained for the relativistic ponderomotive forces are very complicated in the general case. They simplify in the limit of a stationary field (pulses of long duration) and a small gradient. The most typical and simplest special case of an inhomogeneous field—a stationary plane-focused beam—is investigated. The main difference between relativistic ponderomotive forces and their nonrelativistic limit is they have multiple components. In addition to the usual force directed along the gradient of the field, the relativistic case is also characterized by force components that do not have the form of the gradient of a potential and are parallel to the wave vector and the direction of the field polarization. It is shown that when a relativistic electron travels in a direction close to the direction of the wave vector of a focused laser beam, these components can greatly exceed the gradient force. A force directed along the field polarization vector arises even when the gradient of the field in this direction is zero. Zh. éksp. Teor. Fiz. 116, 1198–1209 (October 1999)     

Casimir friction force and energy dissipation for moving harmonic oscillators. II

This paper is a second in a series devoted to the study of a two-oscillator system in linear relative motion (the first one published as a letter in [J.S. H?ye, I. Brevik, Europhys. Lett. 91, 60003 (2010)]). The main idea behind considering this kind of system is to use it as a simple model for Casimir friction. In the present paper we extend our previous theory so as to obtain the change in the oscillator energy to second order in the perturbation, even though we employ first order perturbation theory only. The results agree with, and confirm, our earlier results obtained via different routes. The friction force is finite at finite temperatures, whereas in the case of two oscillators moving with constant relative velocity the force becomes zero at zero temperature, due to slowly varying coupling.     

Hysteresis from dynamically pinned sliding states

We report a surprising hysteretic behavior in the dynamics of a simple one-dimensional nonlinear model inspired by the tribological problem of two sliding surfaces with a thin solid lubricant layer in between. In particular, we consider the frictional dynamics of a harmonic chain confined between two rigid incommensurate substrates which slide with a fixed relative velocity. This system was previously found, by explicit solution of the equations of motion, to possess plateaus in parameter space exhibiting a remarkable quantization of the chain center-of-mass velocity (dynamic pinning) solely determined by the interface incommensurability. Starting now from this quantized sliding state, in the underdamped regime of motion and in analogy to what ordinarily happens for static friction, the dynamics exhibits a large hysteresis under the action of an additional external driving force F_ext. A critical threshold value F_c of the adiabatically applied force F_ext is required in order to alter the robust dynamics of the plateau attractor. When the applied force is decreased and removed, the system can jump to intermediate sliding regimes (a sort of “dynamic” stick-slip motion) and eventually returns to the quantized sliding state at a much lower value of F_ext. Hysteretic behavior is also observed as a function of the external driving velocity.     

A simple model of the classicalZitterbewegung: Photon wave function

We propose a simple classical model of the zitterbewegung. In this model spin is proportional to the velocity of the particle, the component parallel top is constant and the orthogonal components are oscillating with2p frequency. The quantization of the system gives wave equations for spin,0, 1/2, 1, 3/2,…, etc. respectively. These equations are convenient for massless particles. The wave equation of the spin-1, massless free particle is equivalent to the Maxwell equations and the state functions have a probability interpretation and exhibit conserved current densities. The ground state has zero energy.     

Rigid Body Motion in Special Relativity

We study the acceleration and collisions of rigid bodies in special relativity. After a brief historical review, we give a physical definition of the term ‘rigid body’ in relativistic straight line motion. We show that the definition of ‘rigid body’ in relativity differs from the usual classical definition, so there is no difficulty in dealing with rigid bodies in relativistic motion. We then describe

1. The motion of a rigid body undergoing constant acceleration to a given velocity.
2. The acceleration of a rigid body due to an applied impulse.
3. Collisions between rigid bodies.

     

恒力作用下质点的相对论运动

研究在相对论情况下受恒力作用质点的曲线运动规律。     

Neutrino in matter and external fields

A technique for describing various processes proceeding in matter and involving neutrinos and electrons is discussed. This technique is based on “the method of exact solutions,” which implies the use of solutions to proper Dirac equations for particle wave functions in matter. Exact solutions for the neutrino and the electron in the cases of uniform nonmoving and rotating matter are discussed. On studying relativistic neutrino motion and associated neutrino-energy quantization in rotating matter, a semiclassical interpretation of particle finite motion is developed. In the general case of neutrino and electron motion in matter with varying parameters, the corresponding effective force acting on the particles is determined. The possibility of electromagnetic-wave radiation by an electron that moves in a dense neutrino flux of varying density and which is accelerated by this kind of force is predicted.     

Limit velocity of charged particles in a constant electromagnetic field under friction

It is shown that when a particle moves in a constant electromagnetic field and is subjected to frictional forces its velocity to ds to a limit independent of the initial conditions. A limit velocity exists as well for the motion of relativistic particles in constant electric fields.     

One-dimensional relativistic motion of two bodies with a constant magnitude of the interaction force

The relativistic linear motion of two bodies with a constant magnitude of the interaction force is considered. Inelastic and elastic collisions of these bodies are investigated. A mathematically exact model of the relativistic oscillator is constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 66–69, July, 1982.     

Nanomachinery: A general approach to inducing directed motion at the atomic level

The motion of bodies in a periodic potential relief with weak attenuation is considered. When subjected to various periodic external effects, the bodies may spontaneously move with a velocity uniquely defined by the frequency of a periodic action and the space period of the potential. Principles of inducing directed motion with a strictly controllable velocity that are described in this paper can be used for (1) handling individual molecules or molecular clusters on crystal surfaces, (2) creating nanomachines—objects that are free to spontaneously move both in the absence of an external force and in the presence of a force opposite to the direction of motion (and thus capable of transporting other objects), (3) designing actuators providing a strictly controllable velocity of motion, and (4) designing controllable tribological systems by appropriately profiling tribosurfaces and applying ultrasonic actions. Under periodic external perturbations, the dependence of the mean velocity of a system on the mean applied force (which macroscopically appears as the “friction law” for the system) is shown to contain plateaus of constant velocity not only when the velocity of motion is zero but also when a set of discrete equidistant velocities is present. The problem of creating totally controllable nanomachines can be posed as the problem of controlling the width and position of these plateaus.     

Linear Motion under Constant Force and Radiation Reaction

The classical relativistic equation of motion with radiation reaction is solved exactly when the motion is along the lines of force due to a constant electric field. For physically admissible solutions, there is no contribution due to the radiation reaction. The general motion without radiation reaction is not linear.     

Exact Harmonic Metric for a Uniformly Moving Schwarzschild Black Hole

The harmonic metric for Schwarzschild black hole with a uniform velocity is presented. In the limit of weak field and low velocity, this metric reduces to the post-Newtonian approximation for one moving point mass. As an application, we derive the dynamics of particle and photon in the weak-field limit for the moving Schwarzschild black hole with an arbitrary velocity. It is found that the relativistic motion of gravitational source can induce an additional centripetal force on the test particle, which may be comparable to or even larger than the conventional Newtonian gravitational force.     

The quantum field theory of laser acceleration

We determine the scattering rate and the energy loss of electrons due to a laser photon beam. From the energy loss formula we determine the force accelerating an electron by the laser photon beam and the corresponding relativistic dynamical equation describing its motion. Numerically, we calculate the velocity of electron after an acceleration time Δt = 0.1 s.     

New representation for energy-momentum and its applications to relativistic dynamics

In this paper we introduce the concept counterpart of rapidity and define energy and momentum of the relativistic particle as functions of the counterpart of rapidity. Formulae of the relativistic mechanics defined in such a way are regular near the zero-mass and speed of light state. This representation admits to attain a correct limit of the formulae of the relativistic mechanics, including the Dirac equation, at zero-mass point and explains violation of the parity at this state. On the other hand, the representation for energy-momentum can be realized as a mapping from the massless state onto the massive one which looks like a “q deformation”. Hypothesis on quantization of the energy-momentum and the velocity near the light speed is suggested. The group of transformations using the counterpart of rapidity as a parameter of transformation is constructed.