One-dimensional relativistic dissipative system with constant force and its quantization

For a relativistic particle under a constant force and a linear velocity dissipation force, a constant of motion is found. Problems are shown for getting the Hamiltoninan of this system. Thus, the quantization of this system is carried out through the constant of motion and using the quantization of the velocity variable. The dissipative relativistic quantum bouncer is outlined within this quantization approach.Comment: 11 pages, no figure

A direct approach to the construction of standard and non-standard Lagrangians for dissipative dynamical systems with variable coefficients

We present a direct approach to the construction of Lagrangians for a large class of one-dimensional dynamical systems with a simple dependence (monomial or polynomial) on the velocity. We rederive and generalize some recent results and find Lagrangian formulations which seem to be new. Some of the considered systems (e.g., motions with the friction proportional to the velocity and to the square of the velocity) admit infinite families of different Lagrangian formulations.Comment: 17 page

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.Comment: 10 pages, 2 figure

Progress in Classical and Quantum Variational Principles

We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schr\"{o}dinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems

The Mechanics of continental lithosphere-asthenosphere coupling

The physical interaction of the convecting asthenosphere and the highly viscous lower lithosphere determines the transmission of plate driving stresses from the mantle to the plates, and controls the stress state of deep cratonic roots, which is important for cratonic diamond formation models. A growing number of studies have highlighted a high degree of heterogeneity in root zones, however, little work has been done on how a heterogenous lower lithosphere affects plate-mantle coupling. The purpose of this work is to assess the degree that physical and rheological heterogeneity in the lowermost lithosphere affects the coupling and transmission of stress from the asthenosphere to the lithosphere. Southern Africa is an area where fine scale variations in the seismic and lithological structure have been identified. Here we present numerical models of the dynamics of stylised and Southern Africa-like cratonic roots in a convecting mantle. We find lithosphere structure exerts a strong control on asthenospheric flow and this dominates the lower lithosphere stress field. Weak metasomatic zones do not systematically affect the stress regime, though they do mitigate the effects of stress extremes on root zones. Short length-scale finger-like features as seen in Southern African tomography are not stable over geological time periods for plausible rheologies, which suggests such small-scale variations in the seismic structure may have a chemical/metasomatic origin. Such chemical heterogeneities can be remarkably long-lived (order of hundreds of Myrs), despite their lack of mechanical integrity.8 page(s