Relativistic Mechanical System Research Articles

Quantified asymptotic analysis for the relativistic quantum mechanical system with electromagnetic fields

We study the asymptotic analysis of a relativistic quantum mechanical system interacting with self-consistent electromagnetic fields, with a specific focus on the Maxwell–Klein–Gordon (MKG) model. Firstly, we show that the MKG system is globally well-posed under the Coulomb gauge condition, even in the presence of self-interacting potential. Furthermore, to derive the asymptotic analysis, we consider the non-relativistic and semi-classical regime simultaneously, introducing a single scaling parameter that serves to parameterize both the speed of light and the Planck constant. As the scaling parameter approaches zero, we obtain rigorous and quantified estimates on the asymptotic convergence of the electrostatic MKG system toward the classical Euler–Poisson system. Our analytical framework relies on the modulated energy estimate.

Construction of Eigenfunctions for the Elliptic Ruijsenaars Difference Operators

We present a perturbative construction of two kinds of eigenfunctions of the commuting family of difference operators defining the elliptic Ruijsenaars system. The first kind corresponds to elliptic deformations of the Macdonald polynomials, and the second kind generalizes asymptotically free eigenfunctions previously constructed in the trigonometric case. We obtain these eigenfunctions as infinite series which, as we show, converge in suitable domains of the variables and parameters. Our results imply that, for the domain where the elliptic Ruijsenaars operators define a relativistic quantum mechanical system, the elliptic deformations of the Macdonald polynomials provide a family of orthogonal functions with respect to the pertinent scalar product.

Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems

This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange–Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic deformations of G. Perelman entropy functionals and geometric flows evolution equations of classical mechanical systems are described. There are studied projections of such F- and W-functionals on Lorentz spacetime manifolds and three-dimensional spacelike hypersurfaces. These functionals are used for elaborating relativistic thermodynamic models for Lagrange–Hamilton geometric evolution and respective generalized Hamilton geometric flow and nonholonomic Ricci flow equations. The concept of nonholonomic W-entropy is developed as a complementary one for the classical Shannon entropy and the quantum von Neumann entropy. There are considered geometric flow generalizations of the approaches based on classical and quantum relative entropy, conditional entropy, mutual information, and related thermodynamic models. Such basic ingredients and topics of quantum geometric flow information theory are elaborated using the formalism of density matrices and measurements with quantum channels for the evolution of quantum mechanical systems.

Noncovariant Lagrangians Are Presented Which Yield Two-Component Equations of Motion for a Class of Relativistic Mechanical Systems in 1 + 1 Dimensions Including the Harmonic Oscillator

First, a Lagrangian is presented and authenticated for a Relativistic Harmonic Oscillator in 1 + 1 dimensions. It yields a two-component set of equations of motion. The time-component is the missing piece in all previous discussions of this system! The second result is that this Oscillator Langrangian generalizes to Langrangians for a class of particles in 1 + 1 dimensions subject to an arbitrary potential V which is space dependent only.

Classical and quantum geometric information flows and entanglement of relativistic mechanical systems

This article elaborates on entanglement entropy and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. A set of basic geometric and quantum mechanics and probability concepts together with methods of computation are developed in general covariant form for curved phase spaces modelled as cotangent Lorentz bundles. The constructions are based on ideas relating the Grigory Perelman's entropy for geometric flows and associated statistical thermodynamic systems to the quantum von Neumann entropy, classical and quantum relative and conditional entropy, mutual information etc. We formulate the concept of the entanglement entropy of quantum geometric information flows and study properties and inequalities for quantum, thermodynamic and geometric entropies characterising such systems.

Symmetry of Birkhoffians and conserved quantity for a relativistic mechanical system

A new symmetry of a relativistic mechanical system is put forward, and the corresponding conserved quantity is given. The new symmetry is defined in such a way that if each solution to the differential equations of motion of a relativistic mechanical system corresponding to a set of Birkhoff's dynamical functions satisfies the differential equations of motion obtained by other set of Birkhoff's dynamical functions and vice versa, then the corresponding invariance is called a symmetry of Birkhoffians. We prove that the coefficient matrix which relates to the relativistic Birkhoff's equations obtained from two sets of Birkhoff's dynamical functions, is such that the trace of all its integer powers is a conserved quantity of the system, and therefore a theorem known for nonsingular equivalent Lagrangians presented by Currie and Saletan is extended to a relativistic Birkhoffian system. Two examples are given to illustrate the application of the results.

A New Type of Conserved Quantity Deduced from Mei Symmetry for Relativistic Mechanical System in Phase Space

In this paper, a new type of conserved quantity indirectly deduced from the Mei symmetry for relativistic mechanical system in phase space is studied. The definition and the criterion of the Mei symmetry for the system are given. The condition for existence and the form of the new conserved quantity are obtained. Finally, an example is given to illustrate the application of the results.

A new type of conserved quantity of Mei symmetry for relativistic nonholonomic mechanical system in phase space

In this paper, a new type of conserved quantity induced directly from the Mei symmetry for a relativistic nonholonomic mechanical system in phase space is studied. The definition and the criterion of the Mei symmetry for the system are given. The conditions for the existence and form of the new conserved quantity are obtained. Finally, an example is given to illustrate the application of the result.

Integrating Factors and Conservation Laws for RelativisticMechanical System**The project supportedby the Natural Science Foundation of HighEducation of Jiangsu Province of China under Grant No. 04KJA130135and the ``Qing Lan'' Project Foundation of Jiangsu Province

In this paper, we present a new method to constructthe conservation laws for relativistic mechanical systems byfinding corresponding integrating factors. First, the Lagrangeequations of relativistic mechanical systems are established,and the definition of integrating factors of the systems is given;second, the necessary conditions for the existence of conservedquantities of the relativistic mechanical systems are studiedin detail, and the relation between the conservation laws andthe integrating factors of the systems is obtained and the generalizedKilling equations for the determination of the integrating factorsare given; finally, the conservation theorem and its inversefor the systems are established, and an example is given to illustratethe application of the results.

Lie Symmetrical Hojman Conserved Quantity of Relativistic Mechanical System

In this paper, we study the Lie symmetrical Hojmanconserved quantity of a relativistic mechanical system under generalinfinitesimal transformations of groups in which the time parameter isvariable. The determining equation of Lie symmetry of the system isestablished. The theorem of the Lie symmetrical Hojman conserved quantity ofthe system is presented. The above results are generalization toHojman's conclusions, in which the time parameter is not variable and the system isnon-relativistic. An example is given to illustrate the application of theresults in the last.

A new conservation law from Mei symmetry for the relativistic mechanical system

This paper studies the Mei symmetry and conservation law of the relativistic mechanical system. The definition and criterion of Mei symmetry for the relativistic mechanical system are established, which are based upon the invariance of dynamical functions under infinitesimal transformations. A new conservation law is deduced directly from the Mei symmetry of the system, and the condition under which a Mei symmetry can lead to a new conservation law is obtained and the form ofthe conservation law is presented. An example is given to illustrate the application of the results.

Geometric Quantization of Relativistic Hamiltonian Mechanics

A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.

Form invariance and Lie symmetry of relativistic mechanical system

We study the form invariance and Lie symmetry of a relativistic mechanical syste m. Firstly, we give the definition and criterion and conserved quantity of the f orm invariance and Lie symmetry in the system. Next, the relation between the fo rm invariance and Lie symmetry of the system is obtained. Finally, we give an ex ample to illustrate the application of the result.

Form invariance and Noether symmetry of a relativistic mechanical system

In this paper,we study the form invariance and Noether symmetry of a relativistic mechanical system-Firstly,we give the Noether theorem of a relativistic mechanical system,and the definition and criterion and conserved quantity of the form invariance in the system-Next,the relation between the form invariance and Noether symmetry of the system is obtained-Finally,we give an example to illustrate the application of the result-

Lie symmetries and conserved quantities of rotational relativistic systems

The Lie symmetries and conserved quantities of the rotational relativistic holonomic and nonholonomic systems were studied. By defining the infinitestinal transformations' generators and by using the invariance of the differential equations under the infinitesimal transformations, the determining equations of Lie symmetries for the rotational relativistic mechanical systems are established. The structure equations and the forms of conserved quantities are obtained. An example to illustrate the application of the results is given.

The local instability of simple relativistic systems

The invariant criterion of local instability of geodesic motion in terms of Riemannian (pseudo-Riemannian) geometry is proposed. The property of sensitive dependence on initial conditions is formulated as the local instability of nearby geodesics. In the study of sectional curvature, the bivector formalism is applied. We demonstrate the effectiveness of our method of investigating local instability in the behaviour of simple relativistic mechanical systems.

Relativistic variation principles and equation of motion for variable mass controllable mechanical system

With classical variable mass and relativistic variable mass cases being considered, the relativistic D'Alembert principles of Lagrange form, Nielsen form and Appell form for variable mass controllable mechanical system are given; the relativistic Chaplygin equation. Nielsen equation and Appell equation for variable mass controllable mechanical system in quasi-coordinates and generalized-coordinates are obtained, and the equations of motion of relativistic controllable mechanical system for holonomic system and constant mass system are discussed.

Description of chaos in simple relativistic systems.

Chaos is investigated in the context of general relativity and gravitation. We show how quantitative and global measures of chaos can be obtained from qualitative and local ones. After averaging---first, over all two-directions, and second, along the trajectory---the rate of separation of nearby trajectories (Lyapunov-like exponents) can be obtained. This gives us a tool to the invariant chaos description. The sign of the Ricci scalar serves as a criterion of the local instability in simple mechanical systems (systems with a natural Lagrange function). We also show how to reduce relativistic simple mechanical systems to the classical ones. Timelike and null geodesics in multi-black-hole cosmological spacetimes are considered. The role of relativistic systems in general relativity is emphasized. \textcopyright{} 1996 The American Physical Society.

Casimir energy for a three-piece relativistic string.

The Casimir energy for the transverse oscillations of a piecewise uniform closed string is calculated. The string consists of three pieces I, II, III of equal length, endowed with different tensions and mass densities, but adjusted in such a way that the velocity of sound always equals the velocity of light. In this sense the string forms a relativistic mechanical system. In the present paper the string is subjected to the following analysis: the dispersion function is derived and the zero-point energy is regularized using (i) a contour integration technique, being most convenient for the generalization of the theory to the case of finite temperatures, and (ii) the Hurwitz $\ensuremath{\zeta}$-function technique, being usually the most compact method when the purpose is to calculate the Casimir energy numerically at $T=0$. The energy, being always nonpositive, is shown graphically in some cases as functions of the tension ratios I/II and II/III. The generalization to finite temperature theory is also given.

Relativistic and Galilean-invariant classical mechanical systems

Relativistic and Galilean-invariant classical mechanical systems