Classical Mechanical Results Research Articles

Left-invariant grauert tubes on SU(2)

Let M be a real analytic Riemannian manifold. An adapted complex structure on TM is a complex structure on a neighborhood of the zero section such that the leaves of the Riemann foliation are complex submanifolds. This structure is called entire if it may be extended to the whole of TM. We call such manifolds Grauert tubes, or simply tubes. We consider here the case of M = G a compact connected Lie group with a left-invariant metric, and try to determine for which such metrics the associated tube is entire. It is well-known that the Grauert tube of a bi-invariant metric on a Lie group is entire. The case of the smallest group SU(2) is treated completely, thanks to the complete integrability of the geodesic flow for such a metric, a standard result in classical mechanics. Along the way we find a new obstruction to tubes being entire which is made visible by the complete integrability. (New reference and exposition shortened, 11/17/2017.)

Stationary and dynamical scattering problems and ergodic-type theorems

We prove that for the radial Schrödinger equation with Coulomb-type potential the generalized dynamical scattering operator coincides with the corresponding generalized stationary scattering operator. This fact is a quantum mechanical analogue of the ergodic results in classical mechanics.

Quantum circular billiards: Further analytical results

Analytical results for circular billiards are obtained for different quantities. In particular a two-term recursion relation for mean values of powers of radial coordinate is given and a three-term recursion relation for mean values of powers of total momentum is given. Analytical expressions for these mean values are also determined, and briefly compared with the results of classical mechanics.

Stochastic embedding of dynamical systems

Most physical systems are modeled by an ordinary or a partial differential equation, like the n-body problem in celestial mechanics. In some cases, for example, when studying the long term behavior of the solar system or for complex systems, there exist elements which can influence the dynamics of the system which are not well modeled or even known. One way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic. In this paper, we develop a theory for the stochastic embedding of ordinary differential equations. We apply this method to Lagrangian systems. In this particular case, we extend many results of classical mechanics, namely, the least action principle, the Euler-Lagrange equations, and Noether’s theorem. We also obtain a Hamiltonian formulation for our stochastic Lagrangian systems. Many applications are discussed at the end of the paper.

On internal constraints in continuum mechanics

In classical particle mechanics, it is well understood that while working with nonholonomic and nonideal constraints, one cannot expect that the constraint be workless. It is curious that in continuum mechanics, however, the implications of the result in classical mechanics have not been clearly understood. In this paper, we show that in dealing with the response of dissipative systems, one cannot require that constraints do no work or ignore the fact that the material response functions depend on the constraint reaction. An example of this is the viscosity of a fluid depending on the pressure.

Classical monopoles: Newton, NUT space, gravomagnetic lensing, and atomic spectra

Stimulated by a scholium in Newton's Principia we find some beautiful results in classical mechanics which can be interpreted in terms of the orbits in the field of a mass endowed with a gravomagnetic monopole. All the orbits lie on cones! When the cones are slit open and flattened the orbits are exactly the ellipses and hyperbolae that one would have obtained without the gravomagnetic monopole. The beauty and simplicity of these results has led us to explore the similar problems in Atomic Physics when the nuclei have an added Dirac magnetic monopole. These problems have been explored by others and we sketch the derivations and give details of the predicted spectrum of monopolar hydrogen. Finally we return to gravomagnetic monopoles in general relativity. We explain why NUT space has a non-spherical metric although NUT space itself is the spherical space-time of a mass with a gravomagnetic monopole. We demonstrate that all geodesics in NUT space lie on cones and use this result to study the gravitational lensing by bodies with gravomagnetic monopoles. We remark that just as electromagnetism would have to be extended beyond Maxwell's equations to allow for magnetic monopoles and their currents so general relativity would have to be extended to allow torsion for general distributions of gravomagnetic monopoles and their currents. Of course if monopoles were never discovered then it would be a triumph for both Maxwellian Electromagnetism and General Relativity as they stand!

Treatment of inelastic collisions of a particle with a quantum harmonic oscillator by density matrix evolution

A density matrix evolution (DME) method (Berendsen, H. J. C., and Mavri, J., 1993, J. phys. Chem., 97, 13464) to simulate the dynamics of quantum systems embedded in a classical environment is applied to study the inelastic collisions of a classical particle with a five level quantum harmonic oscillator. The results are compared with the exact quantum calculations of Secrest, D., and Johnson, B. R., 1966, J. chem. Phys., 45, 4556, and with the results of classical mechanics. The DME results are between the results of the full quantum treatment and classical mechanics but closer to the latter. Furthermore, we demonstrate the time reversibility of the DME equations of motion and discuss the importance for the time reversibility of the quantum phase described by the off-diagonal elements of the density matrix.

Localizability and covariance in analytical mechanics

Two familiar concepts in quantum mechanics, namely localizability and covariance, are investigated in the framework of classical mechanics. The analysis of such well-known problems in quantum mechanics as localizability of elementary relativistic particles gives the same result in classical mechanics as well.

The application of quantum mechanics to chemical kinetics

Much work has recently been done on the application of quantum mechanics to chemical reactions. In the majority of cases, however, the actual reaction processes have been considered as taking place according to the laws of classical mechanics, quantum-mechanical theory being only employed in calculating the interatomic forces. It has, however, been suggested by various authors that the actual transition processes involved must be treated as non-classical. Some of these authors have claimed that this method of treatment is essential for the true explanation of chemical processes, just as in the case of radioactive disintegration, where it is well established that classical considerations are unable to explain the phenomena observed. It appears, however, to be the general consensus of opinion that for chemical processes the results obtained by a strict quantum-mechanical treatment would differ negligibly from the results of classical mechanics. This opinion appears to be based only on approximate methods of treatment, and no actual figures have been published. The present paper is a contribution to a more exact knowledge of the problem. According to modern views on reaction mechanism, the reacting system passes through a maximum of potential energy in passing adiabatically from the initial to the final state. The energy difference between the initial state and the maximum is the heat of activation for the reaction (E). As a simple type of chemical reaction we may take the system shown in fig. 1. A particle of mass m passes from a to b through a region of varying potential energy V ( x ). Between a and b the potential energy reaches a maximum value E. The energy difference between a and b is Q, the heat of reaction.