Contact symmetry of time-dependent Schr\"odinger equation for a two-particle system: symmetry classification of two-body central potentials

Symmetry classification of two-body central potentials in a two-particle Schr\"{o}dinger equation in terms of contact transformations of the equation has been investigated. Explicit calculation has shown that they are of the same four different classes as for the point transformations. Thus in this problem contact transformations are not essentially different from point transformations. We have also obtained the detailed algebraic structures of the corresponding Lie algebras and the functional bases of invariants for the transformation groups in all the four classes

Geodesic motion in the neighbourhood of submanifolds embedded in warped product spaces

We study the classical geodesic motions of nonzero rest mass test particles and photons in (3+1+n)- dimensional warped product spaces. An important feature of these spaces is that they allow a natural decoupling between the motions in the (3+1)-dimensional spacetime and those in the extra n dimensions. Using this decoupling and employing phase space analysis we investigate the conditions for confinement of particles and photons to the (3+1)- spacetime submanifold. In addition to providing information regarding the motion of photons, we also show that these motions are not constrained by the value of the extrinsic curvature. We obtain the general conditions for the confinement of geodesics in the case of pseudo-Riemannian manifolds as well as establishing the conditions for the stability of such confinement. These results also generalise a recent result of the authors concerning the embeddings of hypersurfaces with codimension one.Comment: 8 pages, 1 figure. To appear in General Relativity and Gravitation as a contributed paper to Mashhoon Festschrif

Dynamics of oscillator chains

The Fermi\u2013Pasta\u2013Ulam (FPU) nonlinear oscillator chain has proved to be a seminal system for investigating problems in nonlinear dynamics. First proposed as a nonlinear system to elucidate the foundations of statistical mechanics, the initial lack of confirmation of the researchers expectations eventually led to a number of profound insights into the behavior of high-dimensional nonlinear systems. The initial numerical studies, proposed to demonstrate that energy placed in a single mode of the linearized chain would approach equipartition through nonlinear interactions, surprisingly showed recurrences. Although subsequent work showed that the origin of the recurrences is nonlinear resonance, the question of lack of equipartition remained. The attempt to understand the regularity bore fruit in a profound development in nonlinear dynamics: the birth of soliton theory. A parallel development, related to numerical observations that, at higher energies, equipartition among modes could be approached, was the understanding that the transition with increasing energy is due to resonance overlap. Further numerical investigations showed that time-scales were also important, with a transition between faster and slower evolution. This was explained in terms of mode overlap at higher energy and resonance overlap at lower energy. Numerical limitations to observing a very slow approach to equipartition and the problem of connecting high-dimensional Hamiltonian systems to lower dimensional studies of Arnold diffusion, which indicate transitions from exponentially slow diffusion along resonances to power-law diffusion across resonances, have been considered. Most of the work, both numerical and theoretical, started from low frequency (long wavelength) initial conditions. Coincident with developments to understand equipartition was another program to connect a statistical phenomenon to nonlinear dynamics, that of understanding classical heat conduction. The numerical studies were quite different, involving the excitation of a boundary oscillator with chaotic motion, rather than the excitation of the entire chain with regular motion. Although energy transitions are still important, the inability to reproduce exactly the law of classical heat conduction led to concern for the generiticity of the FPU chain and exploration of other force laws. Important concepts of unequal masses, and \u201canti-integrability,\u201d i.e. isolation of some oscillators, were considered, as well as separated optical and acoustic modes that could only communicate through very weak interactions. The importance of chains that do not allow nonlinear wave propagation in producing the Fourier heat conduction law is now recognized. A more recent development has been the exploration of energy placed on the FPU or related oscillator chains in high-frequency (short wavelength) modes and the existence of isolated structures (breathers). Breathers are found as solutions to partial differential equations, analogous to solitons at lower frequency. On oscillator chains, such as the FPU, energy initially in a single high-frequency mode is found, at higher energies, to self-organize in oscillator space to form compact structures. These structures are \u201cchaotic breathers,\u201d i.e. not completely stable, and disintegrate on longer time-scales. With the significant progress in understanding this evolution, we now have a rather complete picture of the nonlinear dynamics of the FPU and related oscillator chains, and their relation to a wide range of concepts in nonlinear dynamics. This chapter\u2019s purpose is to explicate these many concepts. After a historical perspective the basic chaos theory background is reviewed. Types of oscillators, numerical methods, and some analytical results are considered. Numerical results of studies of equipartition, both from low-frequency and high-frequency modes, are presented, together with numerical studies of heat conduction. These numerical studies are related to analytical calculations and estimates of energy transitions and time-scales to equipartition