From circular paths to elliptic orbits: A geometric approach to Kepler's motion

The hodograph, i.e. the path traced by a body in velocity space, was introduced by Hamilton in 1846 as an alternative for studying certain dynamical problems. The hodograph of the Kepler problem was then investigated and shown to be a circle, it was next used to investigate some other properties of the motion. We here propose a new method for tracing the hodograph and the corresponding configuration space orbit in Kepler's problem starting from the initial conditions given and trying to use no more than the methods of synthetic geometry in a sort of Newtonian approach. All of our geometric constructions require straight edge and compass only.Comment: 9 pages, 4 figure

The Forward and Backward Shift on the Hardy Space of a Tree

In this paper we initiate the study of the forward and backward shifts on the Hardy space of a tree and the little Hardy space of a tree. In particular, we investigate when these shifts are bounded, find the norm of the shifts if they are bounded, characterize the trees in which they are an isometry, compute the spectrum in some concrete examples, and completely determine when they are hypercyclic.Comment: 23 page

On the interplay between material flaws and dynamic necking

In this paper we investigate the interplay between material defects and flow localization in elastoplastic bars subjected to dynamic tension. For that task, we have developed a 10 finite difference scheme within a large deformation framework in which the material is modelled using rate-dependent J(2) plasticity. A perturbation of the initial yield stress is introduced in each node of the finite difference mesh to model localized material flaws. Numerical computations are carried out within a wide spectrum of strain rates ranging from 500 s(-1) to 2500 s(-1). On the one hand, our calculations reveal the effect of the material defects in the necking process. On the other hand, our results show that the necking inception, instead of being a random type process, is the deterministic result of the interplay between the mechanical behaviour of the material and the boundary conditions. This conclusion agrees with the experimental evidence reported by Rittel et al. [1] and Rotbaum et al. [2].The authors are indebted to the Ministerio de Economía y Competitividad de España (Project DPI2014-57989-P) for the financial support received which allowed conducting this work.Publicad

A useful form of the recurrence relation between relativistic atomic matrix elements of radial powers

Recently obtained recurrence formulae for relativistic hydrogenic radial matrix elements are cast in a simpler and perhaps more useful form. This is achieved with the help of a new relation between the $r^a$ and the $\beta r^b$ terms ($\beta$ is a $4\times 4$ Dirac matrix and $a, b$ are constants) in the atomic matrix elements.Comment: 7 pages, no figure

Assessment of Cold Ironing and LNG as Mitigation Tools of Short Sea Shipping Emissions in Port: A Spanish Case Study

[Abstract] By the end of 2025 European ports are required to provide (Directive 2014/94/EU) facilities to ensure the Liquefied Natural Gas (LNG) use and on-shore electricity supply for vessels (Cold Ironing—CI). Even though this involves considerable port investment, many uncertainties about CI and LNG performance exist because their application depends on vessel operators’ willingness. Additionally, lag times for CI connection/disconnection along with methane emissions from LNG undermine their feasibility for Short Sea Shipping (SSS). Since, among the SSS aims are the reduction in berthing times and its effectiveness for-inter-islands’ traffic where, land electricity grids are frequently dependent on the fuel burning generation by penalizing the CI performance. This paper introduces a calculation method to evaluate the pollution savings in monetary terms by CI and LNG use in SSS. The method is applied to three European routes by testing the environmental performance of two fleets: feeder and Ro-Pax vessels. The results show that feeders reach higher environmental improvements by using port mitigation than Ro-Pax vessels. Additionally, the need for ensuring the sustainability of on-shore grids before the CI implementation was evinced, especially in insularity frameworks, where the environmental benefits from LNG use proved to be more effectiveThis publication has been supported by the H2020 project FLOTANT, which is coordinated by the Oceanic Platform of the Canary Islands (PLOCAN). The project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 81528

Recurrence relation for relativistic atomic matrix elements

Recurrence formulae for arbitrary hydrogenic radial matrix elements are obtained in the Dirac form of relativistic quantum mechanics. Our approach is inspired on the relativistic extension of the second hypervirial method that has been succesfully employed to deduce an analogous relationship in non relativistic quantum mechanics. We obtain first the relativistic extension of the second hypervirial and then the relativistic recurrence relation. Furthermore, we use such relation to deduce relativistic versions of the Pasternack-Sternheimer rule and of the virial theorem.Comment: 10 pages, no figure