5.—Adiabatic Motion in a Charged Particle Trap

1972 ◽  
Vol 70 ◽  
pp. 47-61 ◽  
Author(s):  
J. Byrne
Keyword(s):  
Charged Particle ◽  
Electromagnetic Fields ◽  
Jacobi Equation ◽  
Charged Particles ◽  
Adiabatic Invariants ◽  
Reflection Symmetry ◽  
Adiabatic Conditions ◽  
Adiabatic Motion ◽  
Particle Trap ◽  
Hamilton Jacobi Equation

SynopsisThe adiabatic invariants associated with the motion of charged particles, trapped in electromagnetic fields with rotational and reflection symmetry, have been studied using classical methods based on the Hamilton-Jacobi equation. It has been shown that results, valid for trapping in purely magnetic configurations, may be applied in the analysis of electromagnetic charged particle traps, provided that suitably modified expressions are used for the angular frequencies in the various dynamical modes. Attention is drawn to circumstances in which the adiabatic conditions may be violated because of cancellation of electric and magnetic terms in the equations.

Radial motion of charged particles along axis of the Kerr–Newman– (anti-)de sitter space-time

2018 ◽  
Vol 6 (2) ◽  
pp. 19
Author(s):  
Kameshwar Nath Mishra ◽  
Varull Mishra
Keyword(s):  
Charged Particle ◽  
Effective Potential ◽  
Equatorial Plane ◽  
Jacobi Equation ◽  
Charged Particles ◽  
De Sitter Space ◽  
Space Time ◽  
Radial Motion ◽  
De Sitter ◽  
Hamilton Jacobi Equation

The motion of uncharged particles in the Kerr–Newman–(anti-)de Sitter space-time has been studied by using the Hamilton –Jacobi equation. We have considered both charged particle Q and non-vanishing cosmological constant Λ , focusing on the equatorial plane and space time of the axis. For the study of radial of radial motion on the axis of space time an effective potential have been developto describe the turning points  

Research of the Ionosphere Using Equipment, Placed on Nano-Satellites. Part 1. Modeling a Vacuum Transducer

ANRI ◽  
2020 ◽  
pp. 53-63
Author(s):  
Valeriy Dreyzin ◽  
Ali Nuri Al' Kadimi
Keyword(s):  
Charged Particle ◽  
Charged Particles ◽  
Upper Atmosphere ◽  
Ion Composition ◽  
Vacuum Gauge ◽  
Approximate Design ◽  
Ionization Processes ◽  
Particle Trap ◽  
Positively Charged

The urgency of the task of studying the density and composition of the upper layers of the atmosphere with the help of tools placed in micro- and nano-satellites vehicles is substantiated. A brief description of the structure of the atmosphere is carried out, the relevance and problems of instrumental studies of the density and composition of the upper atmosphere (ionosphere) are shown. A solution to these problems is proposed by developing a combined density and ion composition sensor for the upper atmosphere layers placed on nanosatellites. An approximate design of a compact inverse-magnetron vacuum gauge transducer is proposed, on the basis of which a combined transducer of density and ion composition of the upper atmosphere is constructed by combining it with a charged particle trap. This trap not only ensures the accuracy of its readings, but also allows you to determine the concentration of negatively and positively charged particles. The simulation of ionization processes in the working area of a compact inverse magnetron vacuum gauge transducer is carried out.

The Hamilton-Jacobi Equation for a Charged Particle in a Combined Gravitational and Electromagnetic Field

1963 ◽  
Vol 6 (3) ◽  
pp. 351-358
Author(s):  
D. K. Sen
Keyword(s):  
Electromagnetic Field ◽  
Charged Particle ◽  
Jacobi Equation ◽  
Jacobi Form ◽  
Equation Of Motion ◽  
Planetary Motion ◽  
Schwarzschild Metric ◽  
Special Case ◽  
Hamilton Jacobi Equation

The equation of motion of a charged particle in a combined gravitational and electromagnetic field is cast in the classical Hamilton-Jacobi form and then applied to the special case of a Schwarzschild metric, leading to the well established equation of planetary motion.

Center of Mass Theorem and the Separation of Variables for Two-Body System on a Three-Dimensional Sphere

2020 ◽  
Vol 23 (3) ◽  
pp. 306-311
Author(s):  
Yu. Kurochkin ◽  
Dz. Shoukavy ◽  
I. Boyarina
Keyword(s):  
Constant Curvature ◽  
Jacobi Equation ◽  
Separation Of Variables ◽  
Center Of Mass ◽  
Three Dimensional ◽  
Relative Distance ◽  
Dimensional Sphere ◽  
Body System ◽  
Spaces Of Constant Curvature ◽  
Hamilton Jacobi Equation

The immobility of the center of mass in spaces of constant curvature is postulated based on its definition obtained in [1]. The system of two particles which interact through a potential depending only on the distance between particles on a three-dimensional sphere is considered. The Hamilton-Jacobi equation is formulated and its solutions and trajectory equations are found. It was established that the reduced mass of the system depends on the relative distance.

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

scholarly journals Classification of extinction profiles for a one-dimensional diffusive Hamilton–Jacobi equation with critical absorption

2018 ◽  
Vol 148 (3) ◽  
pp. 559-574
Author(s):  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Jacobi Equation ◽  
Threshold Value ◽  
Monotonicity Property ◽  
Fast Diffusion ◽  
Extinction Time ◽  
One Step ◽  
Hamilton Jacobi Equation

A classification of the behaviour of the solutions f(·, a) to the ordinary differential equation (|f′|p-2f′)′ + f - |f′|p-1 = 0 in (0,∞) with initial condition f(0, a) = a and f′(0, a) = 0 is provided, according to the value of the parameter a > 0 when the exponent p takes values in (1, 2). There is a threshold value a* that separates different behaviours of f(·, a): if a > a*, then f(·, a) vanishes at least once in (0,∞) and takes negative values, while f(·, a) is positive in (0,∞) and decays algebraically to zero as r→∞ if a ∊ (0, a*). At the threshold value, f(·, a*) is also positive in (0,∞) but decays exponentially fast to zero as r→∞. The proof of these results relies on a transformation to a first-order ordinary differential equation and a monotonicity property with respect to a > 0. This classification is one step in the description of the dynamics near the extinction time of a diffusive Hamilton–Jacobi equation with critical gradient absorption and fast diffusion.

scholarly journals Multiplicity dependence of (multi-)strange hadron production in proton-proton collisions at $$\sqrt{s}$$ = 13 TeV

2020 ◽  
Vol 80 (2) ◽  
Author(s):  
S. Acharya ◽  
◽  
D. Adamová ◽  
S. P. Adhya ◽  
A. Adler ◽  
...  
Keyword(s):  
Charged Particle ◽  
Charged Particles ◽  
General Purpose ◽  
Particle Multiplicity ◽  
Charged Particle Multiplicity ◽  
Production Rates ◽  
Proton Proton ◽  
Proton Collisions ◽  
Strange Hadron ◽  
Proton Proton Collisions

Abstract The production rates and the transverse momentum distribution of strange hadrons at mid-rapidity ($$\left| y\right| < 0.5$$y<0.5) are measured in proton-proton collisions at $$\sqrt{s}$$s = 13 TeV as a function of the charged particle multiplicity, using the ALICE detector at the LHC. The production rates of $$\mathrm{K}^{0}_{S}$$KS0, $$\Lambda $$Λ, $$\Xi $$Ξ, and $$\Omega $$Ω increase with the multiplicity faster than what is reported for inclusive charged particles. The increase is found to be more pronounced for hadrons with a larger strangeness content. Possible auto-correlations between the charged particles and the strange hadrons are evaluated by measuring the event-activity with charged particle multiplicity estimators covering different pseudorapidity regions. When comparing to lower energy results, the yields of strange hadrons are found to depend only on the mid-rapidity charged particle multiplicity. Several features of the data are reproduced qualitatively by general purpose QCD Monte Carlo models that take into account the effect of densely-packed QCD strings in high multiplicity collisions. However, none of the tested models reproduce the data quantitatively. This work corroborates and extends the ALICE findings on strangeness production in proton-proton collisions at 7 TeV.

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