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

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

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

Ruthenium in Chromite as Indicator for Magmatic Sulfide Liquid Equilibration in Mafic-Ultramafic Systems

The platinum-group element ruthenium (Ru) is an important petrogenetic tracer of Earth\u27s accretion history, core-mantle interaction, mantle evolution and the exploration for magmatic sulfide deposits. However, its geochemical behavior in mafic-ultramafic systems is still not fully understood, which limits its usefulness in the predictive modelling of geochemical systems. To further develop the use of Ru as a petrogenetic tracer, we analyzed the Ru contents of chromites from a global sample set of komatiites, komatiitic basalts, and ferropicrites by laser ablation ICP-MS and Carius tube isotope dilution ICP-MS analysis. The Ru data are combined with full major and minor element microprobe analyses. The data show that two groups of chromite can be distinguished on the basis of their Ru contents. This bimodal distribution occurs across komatiites, komatiitic basalts and ferropicrites and corresponds to the sulfide saturation state of the magma during chromite crystallization: chromites from sulfide-undersaturated magmas contain between ∼150 and 600 ppb Ru, whereas chromites that crystallized from sulfide-bearing magmas mostly contain less than ∼150 ppb Ru. The Ru contents are independent of elements that typically document a modification of chromite, suggesting that the determined Ru concentrations reflect the primary magmatic chromite composition. The Ru contents are furthermore independent of magma type (i.e. komatiites, komatiitic basalts, ferropicrites), the magma source regions (i.e. different cratons, belt and localities), the geochemical affinity (i.e. Munro-type and Karasjok-type), and age (i.e. 2.7 Ga and 2.0 Ga) and neither local fluctuations in fO2, nor emplacement styles (i.e. intrusive vs. extrusive) can account for the bimodal Ru distribution in chromite observed during this study. As a consequence, it is argued that the Ru contents of chromites from mafic and ultramafic systems are indicative of the presence or absence of a sulfide liquid during chromite crystallization. In sulfide-saturated systems, the chalcophile Ru will dominantly partition into sulfides, whereas in the absence of sulfides, Ru preferentially partitions into chromite over all other available phases. Because (i) high Ru contents in chromites are exclusively associated with sulfide-undersaturated systems, and (ii) the Ru contents of chromites can be measured via fast and cost-effective laser ablation ICP-MS, Ru variability patterns in chromites allow the identification of magmas that have equilibrated with magmatic sulfide liquids prior to or during chromite crystallization and hence have potential to host metal sulfide orebodies

Developmental differences in higher-order resting-state networks in autism spectrum disorder

Neuro Imaging Researc