Search results for "Mathematica"

Analytische Geometrie nach Prof. Dr. F. Schur

1892

Physical model, theoretical aspects and applications of the flight of a ball in the atmosphere. Part III: Theory in the case of vertical angular freq…

1995

If a ball is viewed as a rigid body, its flight in the atmosphere can be described by six ordinary differential equations, which has been derived in the first part of this paper. In this following third part, some further theoretical aspects in the case of vertical angular frequency will be pointed out using an unknown transformation of the original independent variable, i.e. the time, as indicated in Part II. Last, but not least, the general case of angular frequency is to be treated. A rough qualitative discussion of the solutions is given as well as—if the equations are viewed as a three-dimensional dynamical system—the unique stable equilibrium, which depends on the spin. This equilibri…

The planar two-body problem for spheroids and disks

2021

We outline a new method suggested by Conway (2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral, and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not invol…

X-ray measurements of charge transfer reactions involving cold, very highly charged ions

1999

The magnetic trapping mode of the Livermore high-energy Electron Beam Ion Trap is exploited to study charge transfer reactions between cold (few eV/amu) highly charged ions and gases. By selectively puffing neutral gases and monitoring the x-ray emission, state-selective measurements of the charge transfer reaction channels are possible. The observed K-shell x-ray spectra show prominent emission from high-n levels decaying to the n = 1 ground level, which is enabled by electron capture into states with low orbital angular momentum. A comparison with modeling calculations, therefore, allows a determination of the range of principal and angular momentum quantum numbers involved in the reactio…

Angular Pseudomomentum Theory for the Generalized Nonlinear Schr\"{o}dinger Equation in Discrete Rotational Symmetry Media

2009

We develop a complete mathematical theory for the symmetrical solutions of the generalized nonlinear Schr\"odinger equation based on the new concept of angular pseudomomentum. We consider the symmetric solitons of a generalized nonlinear Schr\"odinger equation with a nonlinearity depending on the modulus of the field. We provide a rigorous proof of a set of mathematical results justifying that these solitons can be classified according to the irreducible representations of a discrete group. Then we extend this theory to non-stationary solutions and study the relationship between angular momentum and pseudomomentum. We illustrate these theoretical results with numerical examples. Finally, we…

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

2013

Abstract This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations …

A gradient elasticity theory for second-grade materials and higher order inertia

2012

Abstract Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the …

Experimental and Theoretical Studies of Nonclassical d 0 Cyclopentadienyl Polyhydride Complexes of Molybdenum and Tungsten

1998

Low-temperature protonation of compounds Cp{sup *}MH{sub 5}(PMe{sub 3}) (M = Mo, 1; W, 2) by HBF{sub 4}{center_dot}Et{sub 2}O in CD{sub 2}Cl{sub 2} or CDFCl{sub 2} affords the thermally unstable hexahydride derivatives [Cp{sup *}MH{sub 6}(PMe{sub 3})]{sup +} (M = Mo, 3; W, 4). The corresponding protonation of 1- and 2-d{sup 5} affords 3- and 4-d{sup 5}, respectively. The {Delta}{delta} on going from H{sub 6} to HD{sub 5} is small for both compounds, but positive for 3 and negative for 4, and no isotopic perturbation of resonance (IPR) is observed. The T{sub 1min} at 400 MHz for [Cp{sup *}MH{sub 6}(PMe{sub 3})]{sup +} apparently doubles on going from Mo to W (52 ms for 3 and approximately 10…

Exact non-Markovian dynamics of Gaussian quantum channels: Finite-time and asymptotic regimes

2018

We investigate the Markovian and non-Markovian dynamics of Gaussian quantum channels, exploiting a recently introduced necessary and sufficient criterion and the ensuing measure of non-Markovianity based on the violation of the divisibility property of the dynamical map. We compare the paradigmatic instances of Quantum Brownian motion (QBM) and Pure Damping (PD) channels, and for the former we find that the exact dynamical evolution is always non-Markovian in the finite-time as well as in the asymptotic regimes, for any nonvanishing value of the non-Markovianity parameter. If one resorts to the rotating wave approximated (RWA) form of the QBM, that neglects the anomalous diffusion contribut…

Diffusion Acceleration in Randomly Switching Sawtooth Potential

2005

We investigate an overdamped Brownian motion in symmetric sawtooth periodic potential switched by Markovian dichotomous noise between two configurations. The two configurations differ each other by a translation of half of period. The calculation of the effective diffusion coefficient is reduced to the mean first‐passage time problem, and we obtain the exact expression valid for arbitrary mean rate of switchings and arbitrary intensity of white Gaussian noise. We find the area at parameters plane where acceleration of diffusion in comparison with the free diffusion case takes place.