Motion of a particle in three-dimensional fields

2020 ◽  
pp. 91-138
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Angular Momentum ◽  
Zeeman Effect ◽  
Three Dimensional ◽  
Harmonic Oscillation ◽  
Coulomb Field ◽  
Central Field ◽  
Small Perturbations ◽  
Classical Analog ◽  
Small Force ◽  
The Given

In the central field, the energy and angular momentum are conserved. It allows for the reduction of this problem to the problem of the motion of the particle in the effective one-dimensional field. Here the motion of a particle in Coulomb field or in the field of the isotropic harmonic oscillation with small perturbations are the most important ones. The authors discuss how the motion of a particle in the given central field can be described qualitatively for different values of the angular momentum and of the energy. Several problems deal with the motion of a particle in the Coulomb field under influence of weak constant uniform electric or magnetic fields (the classical analog of the Stark or Zeeman effect). In addition, the authors consider the motion of a charged particle in the field of the magnetic monopole and magnetic dipole. The motion of the Earth–Moon system in the field of the Sun is considered in some approximation. The displacement of the Coulomb orbit under the influence of a small force of radiation damping.

Motion of a particle in three-dimensional fields

2020 ◽  
pp. 6-11
Author(s):  
Gleb L. Kotkin ◽  
Valeriy G. Serbo
Keyword(s):  
Angular Momentum ◽  
Zeeman Effect ◽  
Three Dimensional ◽  
Harmonic Oscillation ◽  
Coulomb Field ◽  
Central Field ◽  
Small Perturbations ◽  
Classical Analog ◽  
Small Force ◽  
The Given

In the central field, the energy and angular momentum are conserved. It allows for the reduction of this problem to the problem of the motion of the particle in the effective one-dimensional field. Here the motion of a particle in Coulomb field or in the field of the isotropic harmonic oscillation with small perturbations are the most important ones. The authors discuss how the motion of a particle in the given central field can be described qualitatively for different values of the angular momentum and of the energy. Several problems deal with the motion of a particle in the Coulomb field under influence of weak constant uniform electric or magnetic fields (the classical analog of the Stark or Zeeman effect). In addition, the authors consider the motion of a charged particle in the field of the magnetic monopole and magnetic dipole. The motion of the Earth–Moon system in the field of the Sun is considered in some approximation. The displacement of the Coulomb orbit under the influence of a small force of radiation damping.

Relativistische Verallgemeinerung des Lenzschen Vektors (I) / Relativistic generalisation of the Lenz vector (I)

1984 ◽  
Vol 39 (8) ◽  
pp. 720-732
Author(s):  
Eberhard Kern
Keyword(s):  
Angular Momentum ◽  
Kepler Problem ◽  
Coulomb Field ◽  
Equation Of Motion ◽  
Central Field ◽  
Relativistic Energy ◽  
Geometric Meaning ◽  
Type Curves ◽  
Relativistic Problem ◽  
Charged Nucleus

The non-relativistic motion of a particle in a central field with 1/r potential, e.g. the motion of an electron in the Coulomb field of a charged nucleus at rest, is described by the equation of motion (non-relativistic Kepler problem) m x″ = α · x /r3 with α = ez e (product of the charges of the central body ez and the electron e). From this equation of motion, three statements of conservation can be derived: in respect of the energy E, of the angular momentum L and of the Lenz vector Λ = m {x′× L + α ·x/r}. The geometric meaning of Λ is that of a vector pointing in the direction of the perihelion of the particle orbits (conic sections). It will be demonstrated that also at the relativistic Kepler problem, which is based on the equation of motion an analogous Lenz vector exists. It represents a quantity of conservation - in the same way as the relativistic energy and the relativistic angular momentum. For the transitional case → ∞, where the relativistic problem turns into the non-relativistic problem, the relativistic Lenz vector also turns into the non-relativistic Lenz vector. The generalised (relativistic) Lenz vector has also a geometric meaning. Its direction coincides with the oriented axis of symmetry of the orbits (rosettes, spirals, hyperbola-type curves etc.). The quantity of conservation Λ occupies a special position in respect of the quantities of conservation energy and angular momentum. Whereas the energy and the angular momentum correspond with a symmetry of time and space, the Lenz quantity of conservation corresponds with a symmetry of the orbits. The fact that the Lenz vector can relativistically be generalised touches thereby on principal aspects.

The relation between the Wentzel-Kramers-Brillouin and the Thomas-Fermi approximations

1956 ◽  
Vol 235 (1202) ◽  
pp. 419-431 ◽  
Keyword(s):  
Three Dimensional ◽  
Coulomb Field ◽  
Central Field ◽  
Dimensional Problem ◽  
Field Problem ◽  
One Dimensional ◽  
Thomas Fermi ◽  
Energy Formula ◽  
The Individual ◽  
Made In

It is shown that two essential approximations are made in using the customary Thomas-Fermi formula for the sum of the eigenvalues in any one-dimensional problem. The first is to start from the Wentzel-Kramers-Brillouin formula for the individual eigenvalues, and the second is to replace the summation by an integration. The three-dimensional central field problem is then considered, and by similar arguments, though with an additional approximation, the usual Thomas-Fermi energy formula is again obtained. Possible ways of correcting the errors introduced by integrating instead of summing are discussed and illustrative examples given. In the three-dimensional case particular attention is given to the Coulomb field problem. Finally, brief reference is made to the possibility of correcting for the errors of the Wentzel-Kramers-Brillouin formula.

scholarly journals Vortex states in an acoustic Weyl crystal with a topological lattice defect

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Qiang Wang ◽  
Yong Ge ◽  
Hong-xiang Sun ◽  
Haoran Xue ◽  
Ding Jia ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Orbital Angular Momentum ◽  
Lattice Defect ◽  
Three Dimensional ◽  
Surface States ◽  
Real Space ◽  
Lattice Defects ◽  
One Dimensional ◽  
Topological Features ◽  
Topological Lattice

AbstractCrystalline materials can host topological lattice defects that are robust against local deformations, and such defects can interact in interesting ways with the topological features of the underlying band structure. We design and implement a three dimensional acoustic Weyl metamaterial hosting robust modes bound to a one-dimensional topological lattice defect. The modes are related to topological features of the bulk bands, and carry nonzero orbital angular momentum locked to the direction of propagation. They span a range of axial wavenumbers defined by the projections of two bulk Weyl points to a one-dimensional subspace, in a manner analogous to the formation of Fermi arc surface states. We use acoustic experiments to probe their dispersion relation, orbital angular momentum locked waveguiding, and ability to emit acoustic vortices into free space. These results point to new possibilities for creating and exploiting topological modes in three-dimensional structures through the interplay between band topology in momentum space and topological lattice defects in real space.

Kinematical evolution of Globular Clusters

2019 ◽  
Vol 14 (S351) ◽  
pp. 524-527
Author(s):  
Maria A. Tiongco ◽  
Enrico Vesperini ◽  
Anna Lisa Varri
Keyword(s):  
Angular Momentum ◽  
Structural Properties ◽  
Globular Clusters ◽  
Velocity Dispersion ◽  
Three Dimensional ◽  
Stellar Populations ◽  
Velocity Space ◽  
Fundamental Understanding

AbstractWe present several results of the study of the evolution of globular clusters’ internal kinematics, as driven by two-body relaxation and the interplay between internal angular momentum and the external Galactic tidal field. Via a large suite of N-body simulations, we explored the three-dimensional velocity space of tidally perturbed clusters, by characterizing their degree of velocity dispersion anisotropy and their rotational properties. These studies have shown that a cluster’s kinematical properties contain distinct imprints of the cluster’s initial structural properties, dynamical history, and tidal environment. Building on this fundamental understanding, we then studied the dynamics of multiple stellar populations in globular clusters, with attention to the largely unexplored role of angular momentum.

scholarly journals Dynamical and Physical Processes Leading to Tropical Cyclone Intensification under Upper-Level Trough Forcing

2013 ◽  
Vol 70 (8) ◽  
pp. 2547-2565 ◽  
Author(s):  
Marie-Dominique Leroux ◽  
Matthieu Plu ◽  
David Barbary ◽  
Frank Roux ◽  
Philippe Arbogast
Keyword(s):  
Angular Momentum ◽  
Tropical Cyclone ◽  
Potential Vorticity ◽  
Weather Prediction ◽  
Three Dimensional ◽  
Vertical Wind Shear ◽  
Thick Layer ◽  
Limited Area ◽  
Southwest Indian Ocean ◽  
Upper Level

Abstract The rapid intensification of Tropical Cyclone (TC) Dora (2007, southwest Indian Ocean) under upper-level trough forcing is investigated. TC–trough interaction is simulated using a limited-area operational numerical weather prediction model. The interaction between the storm and the trough involves a coupled evolution of vertical wind shear and binary vortex interaction in the horizontal and vertical dimensions. The three-dimensional potential vorticity structure associated with the trough undergoes strong deformation as it approaches the storm. Potential vorticity (PV) is advected toward the tropical cyclone core over a thick layer from 200 to 500 hPa while the TC upper-level flow turns cyclonic from the continuous import of angular momentum. It is found that vortex intensification first occurs inside the eyewall and results from PV superposition in the thick aforementioned layer. The main pathway to further storm intensification is associated with secondary eyewall formation triggered by external forcing. Eddy angular momentum convergence and eddy PV fluxes are responsible for spinning up an outer eyewall over the entire troposphere, while spindown is observed within the primary eyewall. The 8-km-resolution model is able to reproduce the main features of the eyewall replacement cycle observed for TC Dora. The outer eyewall intensifies further through mean vertical advection under dynamically forced upward motion. The processes are illustrated and quantified using various diagnostics.

An Inverse Inviscid Method for the Design of Quasi-Three-Dimensional Turbomachinery Cascades

1993 ◽  
Vol 115 (1) ◽  
pp. 121-127 ◽  
Author(s):  
E. Bonataki ◽  
P. Chaviaropoulos ◽  
K. D. Papailiou
Keyword(s):  
Differential Equation ◽  
Three Dimensional ◽  
Test Cases ◽  
Elliptic Type ◽  
Flow Angle ◽  
Blade Shape ◽  
Closure Conditions ◽  
Type Boundary ◽  
Absolute Frame ◽  
The Given

The calculation of the blade shape, when the desired velocity distribution is imposed, has been the object of numerous investigations in the past. The object of this paper is to present a new method suitable for the design of turbomachinery stator and rotor blade sections, lying on an arbitrary axisymmetric stream-surface with varying streamtube width. The flow is considered irrotational in the absolute frame of reference and compressible. The given data are the streamtube geometry, the number of blades, the inlet flow conditions and the suction and pressure side velocity distributions as functions of the normalized arc-length. The output of the computation is the blade shape that satisfies the above data. The method solves an elliptic type partial differential equation for the velocity modulus with Dirichlet and periodic type boundary conditions on the (potential function, stream function)-plane (Φ, Ψ). The flow angle field is subsequently calculated solving an ordinary differential equation along the iso-Φ or iso-Ψ lines. The blade coordinates are, finally, computed by numerical integration. A set of closure conditions has been developed and discussed in the paper. The method is validated on several test cases and a discussion is held concerning its application and limitations.

AGN torus threaded by large scale magnetic field

2018 ◽  
Vol 27 (10) ◽  
pp. 1844006
Author(s):  
A. Dorodnitsyn ◽  
T. Kallman
Keyword(s):  
Magnetic Field ◽  
Angular Momentum ◽  
Accretion Disk ◽  
Large Scale ◽  
Three Dimensional ◽  
X Ray ◽  
Radiative Feedback ◽  
Large Scale Magnetic Field ◽  
Three Dimensional Simulations

Large scale magnetic field can be easily dragged from galactic scales toward AGN along with accreting gas. There, it can contribute to both the formation of AGN “torus” and help to remove angular momentum from the gas which fuels AGN accretion disk. However the dynamics of such gas is also strongly influenced by the radiative feedback from the inner accretion disk. Here we present results from the three-dimensional simulations of pc-scale accretion which is exposed to intense X-ray heating.

Short Helical Combustor: Flow Control in a Combustion System With Angular Air Supply

2017 ◽  
Author(s):  
Behdad Ariatabar ◽  
Rainer Koch ◽  
Hans-Jörg Bauer
Keyword(s):  
Temperature Field ◽  
Angular Momentum ◽  
Three Dimensional ◽  
The Novel ◽  
Combustion System ◽  
Flow Angle ◽  
Air Supply ◽  
Inhomogeneous Flow ◽  
Generic Design ◽  
Industrial Optimization

The concept of the novel Short Helical Combustor (SHC) was investigated in our previous work [1, 2]. Based on the insight gained from these previous investigations, we propose a generic design improvement to address the tremendous loss of initial angular momentum as well as inhomogeneous flow and temperature field at the outlet of the SHC. In the present paper, the main features of this design are introduced. It is shown that a three-dimensional shaping of the sidewalls, the dome, and the liners can effectively counteract the suboptimal interaction of the swirl flames with these surrounding walls. As a result, the flow at the outlet of the combustor features a high angular momentum and exhibits a uniform flow angle and temperature field. The insight gained from these generic investigations, and the resulting design optimization provides a useful framework for further industrial optimization of the SHC.

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