Theorem In Classical Mechanics Research Articles

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and Its Phase-Space Formulation in Quantum Mechanics.

In classical physics, there is a well-known theorem in which it is established that the energy per degree of freedom is the same. However, in quantum mechanics, due to the non-commutativity of some pairs of observables and the possibility of having non-Markovian dynamics, the energy is not equally distributed. We propose a correspondence between what is known as the classical energy equipartition theorem and its counterpart in the phase-space formulation in quantum mechanics based on the Wigner representation. Further, we show that in the high-temperature regime, the classical result is recovered.

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and its Phase-Space Formulation in Quantum Mechanics

Correspondence between the Energy Equipartition Theorem in Classical Mechanics and its Phase-Space Formulation in Quantum Mechanics

Kolmogorov-Arnold-Moser Stability for Conserved Quantities in Finite-Dimensional Quantum Systems.

We show that for any finite-dimensional quantum systems the conserved quantities can be characterized by their robustness to small perturbations: for fragile symmetries, small perturbations can lead to large deviations over long times, while for robust symmetries, their expectation values remain close to their initial values for all times. This is in analogy with the celebrated Kolmogorov-Arnold-Moser theorem in classical mechanics. To prove this result, we introduce a resummation of a perturbation series, which generalizes the Hamiltonian of the quantum Zeno dynamics.

An approximation theorem in classical mechanics

A theorem by K. Meyer and D. Schmidt says that The reduced three-body problem in two or three dimensions with one small mass is approximately the product of the restricted problem and a harmonic oscillator [7]. This theorem was used to prove dynamical continuation results from the classical restricted circular three-body problem to the three-body problem with one small mass. We state and prove a similar theorem applicable to a larger class of mechanical systems. We present applications to spatial $(N+1)$-body systems with one small mass and gravitationally coupled systems formed by a rigid body and a small point mass.

Dual number coefficient octonion algebra, field equations and conservation laws

Starting with octonion algebra, we develop the dual number coefficient octonion (DNCO) algebra having sixteen components. DNCO forms of generalized potential, field and current equations are discussed in consistent manner. We have made an attempt to write the DNCO form of generalized Dirac–Maxwell’s equations in presence of electric and magnetic charges (dyons). Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for dyons in terms of DNCO algebra. Further, we discuss the DNCO form of linear momentum conservation law for dyons.

Split octonion reformulation of generalized linear gravitational field equations

In this paper, we describe the properties of split octonions and their connection with the 2 × 2 Zorn vector matrix containing both scalar and vector components. Starting with a brief description of gravito-dyons, we reformulate the generalized linear gravitational field equations of gravito-dyons in terms of split octonion. We express the generalized gravito-Heavisidian (GH) potentials, fields, and various wave equations of gravito-dyons in terms of split octonions variables. Accordingly, we demonstrate the work-energy theorem of classical mechanics reproducing the continuity equation for the case of gravito-dyons in terms of split octonions. Further, we discuss the split octonionic form of linear momentum conservation law for gravito-dyons in the case of linear gravitational theory. We have summarized the various split octonion equations for the case of the generalized GH-field of gravito-dyons and the generalized electromagnetic field of dyons. The unified fields of dyons and gravito-dyons have been demonstrated and corresponding field equations are discussed in unique and consistent manner in terms of split octonions.

Liouville's Theorem in Classical Mechanics and the Global Information Field

The connection between the concept of a ‘Lagrangian manifold’ and Liouville’s theorem in classical mechanics and the concept of a global information field in quantum physics is discussed.

Bound orbits and gravitational theory

It can be easily shown that bound orbits around a static source can only exist in four dimensions for any force driven by the Laplace equation. This is true not only for Maxwell's electromagnetism and Newton's gravity, but for Einstein's theory of gravitation as well. In contrast to Maxwell's electrodynamics and Newton's gravity, general relativity has a natural and remarkable generalization in higher dimensions in Lovelock gravity. However, it is not Laplace driven and hence admits bound orbits around a static black hole in all even $D=2N+2$ dimensions, where $N$ is the degree of the Lovelock polynomial action. This is as general a result as Bertrand's theorem of classical mechanics, in which the existence of closed orbits uniquely singles out the inverse square law for a long-range central force.

Dynamical Properties of a System of Heavy Lagrange Gyroscopes Formed in Closed Chain Applied in a Parallel System

One of the most important stages in the analysis of any mechanical model is to establish and find the solution of dynamical equations. Analytical Mechanics has developed a wide range of methods for obtaining equations of motion of coupled rigid body systems that represent a mechanical model. How to use the principles of the mechanics of setting up dynamic equations leads to different forms of presentation. This circumstance is often used to obtain a special form of the equation, which simplifies the study of the dynamic properties of interest to the mechanical model. In this paper we use the method Kharlamov 1 and the theorems of classical mechanics for the specific study of a parallel system, which represents a schematic model of parallel micro-manipulators, such as a finite set of bodies Si (i = 1 , ..., n). This model is to show a design of a parallel micro-platform, that it simulates the virtual movements of a flight simulator, it has three degrees of freedom, and it is a lightweight design. One of the innovations presented in this design consists of the application of a lever mechanism for the transmission of the movement.

Mechanical resistance of a compact nanoparticle in a gaseous medium

Resistance of fullerenes in air is investigated under standard conditions. The general problems of resistance of compact nanosized particles in gaseous media are solved at different temperatures and pressures. The resistance of the nanosized particle moving in a gaseous medium is calculated using the deterministic approach, direct methods of molecular dynamics, and main theorems of classical mechanics (including the theorem on the momenta of a particle and a particle ensemble). Theoretical results obtained are in complete agreement with the Lapple experimental data on the resistance of ultradispersed particles. Moreover, they describe well the resistance of particles having much larger sizes for which the molecular dynamics model disregarding collisions of molecules seems to be inapplicable.

Possible potentials responsible for stable circular relativistic orbits

Bertrand's theorem in classical mechanics of the central force fields attracts us because of its predictive power. It categorically proves that there can only be two types of forces which can produce stable, circular orbits. In this paper an attempt has been made to generalize Bertrand's theorem to the central force problem of relativistic systems. The stability criterion for potentials which can produce stable, circular orbits in the relativistic central force problem has been deduced and a general solution of it is presented. It is seen that the inverse square law passes the relativistic test but the kind of force required for simple harmonic motion does not. Special relativistic effects do not allow stable, circular orbits in the presence of a force which is proportional to the negative of the displacement of the particle from the potential centre.

An approach for generating trajectory-based dynamics which conserves the canonical distribution in the phase space formulation of quantum mechanics. I. Theories

We have reformulated and generalized our recent work [J. Liu and W. H. Miller, J. Chem. Phys. 126, 234110 (2007)] into an approach for generating a family of trajectory-based dynamics methods in the phase space formulation of quantum mechanics. The approach (equilibrium Liouville dynamics) is in the spirit of Liouville's theorem in classical mechanics. The trajectory-based dynamics is able to conserve the quantum canonical distribution for the thermal equilibrium system and approaches classical dynamics in the classical (ℏ → 0), high temperature (β → 0), and harmonic limits. Equilibrium Liouville dynamics provides the framework for the development of novel theoretical∕computational tools for studying quantum dynamical effects in large∕complex molecular systems.

Noether's theorem in classical mechanics revisited

A didatic approach to Noether's theorem in classical mechanics is derived and used to obtain the laws of conservation.

QUANTUM EQUIVALENT OF THE BERTRAND'S THEOREM

A procedure for constructing general bound state potentials is given. Analogous to the Bertrand's theorem in classical mechanics, we then identify radial eigenvalue problems possessing exact solvability and infinite number of eigenstates. Akin to the classical result, the only special cases of the general central potential, satisfying the above two conditions, are the Coulomb and harmonic oscillator potentials.

On gravitational orbits and the virial theorem

We emphasize the sharp distinctions between different one-body gravitational trajectories made by the ratio of time averagesR(t)≡−〈Ekin〉/〈Epot〉.R is calculated as a function of the eccentricity (e) and of the energy (E). Whent→∞, independently ofe andE, R→1/2 for closed orbits (this clearly illustrates the fulfillment of the virial theorem in classical mechanics); whereasR≧1, at any time, for open orbits.

Topological stability of finite-length magnetic flux tubes

It has been suggested that the activity of cosmical magnetic fields is a consequence of a general topological nonequilibrium in the neighbourhood of magnetostatic equilibria. Evidence for this suggestion can be obtained from the Kolmogorov-Arnold-Moser theorem of classical mechanics, applied to the magnetic field line flow as a Hamiltonian system. A finite-length magnetic flux tube, however, always possesses two independent sets of flux surfaces - or, equivalently, the corresponding Hamiltonian system two independent first integrals - and is topologically stable if in the volume occupied by the tube there are no singular (null) points of the magnetic field and the normal field component does not change its sign on the end faces of the tube. Therefore, the concept of nonequilibrium due to flux surface destruction is not applicable to solar atmospheric loops with each end situated in the interior of one polarity of the photospheric normal field component. Further, it seems unlikely that the tearing-mode mechanism can play a role in such loops.

Asymptotic property of solutions to the random cauchy problem for wave equations

Based on the existence and uniqueness theorem for random wave equations, we consider asymptotic behaviors of solutions to the random initial value problem. We describe conditions for the equipartition of stochastic energy (or SE for short) by making use of the random spectral theory and, according to Goldstein's semigroup method, we prove the asymptotically equipartitioned SE theorem and the so-called virial theorem of classical mechanics, and also study the probabilistic characterization of the conditions for equipartition. In addition, we show at last the equipartition of SE from a finite time onwards.

Generalizations of Noether’s Theorem in Classical Mechanics

In this paper, a review is presented of various approaches to the generalization of the version of Noether’s theorem, which is presented in most textbooks on classical mechanics. Its motivation is the controversy still persisting around the possible scope of a Noether-type theorem allowing for velocity-dependent transformations. Our analysis is centered around the one factor common to all known treatments, namely the structure of the related first integral. We first discuss the most general framework, in which a function of the above-mentioned structure constitutes a first integral of a given Lagrangian system, and show that one cannot really talk about an “interrelationship” between symmetries and first integrals there. We then compare different proposed generalizations of Noether’s theorem, by describing the nature of the restrictions which characterize them, when they are situated within the broadest framework. We prove a seemingly new equivalence-result between the two main approaches: that of invaria...

Sum rules for the quarkonium systems

In the framework of the radial Schroedinger equation we derive in a very simple way sum rules relating the potential to physical quantities such as the energy eigenvalues and the square of the lth derivative of the eigenfunctions at the origin. These sum rules contain as particular cases well-known results such as the quantum version of the Clausius theorem in classical mechanics as well as Kramers's relations for the Coulomb potential. Several illustrations are given and the possibilities of applying them to the quarkonium systems are considered.

Linear invariance and virial theorem in classical mechanics

Linear invariance and virial theorem in classical mechanics