S^1 - index theory for the Lorentz force equation

In the present section, we shall develop methods, employing ideas contained in some of L.A. Lyusternik's work, which allow us to establish the existence of a denumerable number of stable critical values of an even functional – they do not disappear under small perturbations by odd functionals. M. A. Krasnosel'skii, Topological methods in the theory nonlinear integral equations , 1956. This chapter is devoted to the study of the symmetric MPT and its subsequent extensions. It is a multiplicity result asserting the existence of multiple critical points, when the functional is invariant under the action of a group of symmetries. It has been stated in the same time as the classical MPT by Ambrosetti and Rabinowitz [50]. This theorem can be seen as an extension of older multiplicity results of Ljusternik Schnirelman type. We will also review two other ways of obtaining multiplicity results; a procedure that inductively uses the (classical) MPT and does not pass by any Index theory, and a generalization of the symmetric MPT, the fountain theorem of Bartsch and its dual form by Bartsch and Willem. Some basic references for the material presented here include [93, 734, 748, 882] and of course [50]. The lecture notes [93] by Bartsch discuss very nicely and exhaustively the role of symmetry in variational methods.

A method for determining steady-state particle density distribution resulting from an incident standing-wave field is discussed. A brief review of photonic crystals relates their established properties and their symmetry along with spectral relationships using 1D eigenvalue equations. Electrostatic interactions between neutral but polarizable particles are derived from Maxwell's equations and the Lorentz force equation. The role of the stress tensor in electromagnetic propagation is discussed. Macroscopic van der Waals forces are described in terms of the Maxwell stress tensor, and equations for the microscopic van der Waals forces between two parallel dipoles are given for in- phase and out-of-phase cases. Spatial distribution of the electric field magnitude for the two cases are illustrated. Possible applications are described.© (1997) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Electron beam-wave interaction using FDTD and Lorentz force equation is used to analyze interaction between electron beam and propagation modes of a slow wave structure supported inside an air-core cylindrical waveguide. Based on this model a THz signal generator at frequency ≈ 400GHz is proposed.

A solid, two-component, quantum luminiferous aether is proposed to exist. Simple postulates are hypothesized, along with some physical laws and assignments. Derivations then lead to the equations of electrodynamics (Maxwell’s equations and the Lorentz force equation), Newton’s law of universal gravitation, and to two field-masses. The theory is shown to successfully meet the classic tests of general relativity: calculations for the advance of the perihelia, the Shapiro effect, and the gravitational redshift agree with experiment, and the experimental result concerning the bending of light in gravitational fields is also understood. Additionally, gravitational waves are understood, and the first of the field-masses allows for an understanding of what is presently known as dark matter. A new approach to analyzing dense objects such as white dwarfs and neutron stars is discussed, and since the theory has no singularity, a replacement for black holes is suggested. Replacing relativity with an absolute, realist, and physical model returns us to a flat Euclidean space and a separate time. Absolute simultaneity enables understanding of quantum mechanics. The underlying philosophical grounding is discussed.

Electromagnetic gating in ion channels

We theoretically study the resonant radiation force (RF) exerted on interacting quantum dots (QDs). For this purpose, the analytical expression of RF is derived based on the nonlocal theory and the Lorentz force equation. This explicitly shows how RF on QDs behaves under optical excitation of photomediated coupled‐states of polaritons in respective QDs confining excitons. In particular, if three QDs are aligned perpendicular to the optical axis, the negative dissipative (scattering and absorbing) force, whose direction is opposite to the light propagation direction, is exerted on the middle QD due to the phase inversion of the induced polarization in it. Furthermore, indirect attractive and repulsive forces arise between QDs at both ends via the middle QD. These effects would enable us to control the spatial dynamics of a QD‐assembly by exciting their collective modes. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A mathematical technique for solving Maxwell's equations and the Lorentz force equation with no approximations except the small-signal approximation is presented. A finite dc magnetic field parallel to the dc velocity of the charges is included. Polarization variables are used, and the boundary conditions include ac surface charge density and surface current density. The advantages of the method are that both fast waves and slow waves are included without a quasi-static approximation, and only the determinantal equation requires computer solution. The partial differential equations are solved directly and need not be solved by computer.

Charges and flelds in a rotating non-magnetic conducting sphere under stationary conditions are investigated by using Minkowski's electrodynamics of moving media and the Lorentz force equation, taking into account the electric permittivity of the sphere. Starting from the assumption that the magnetic fleld inside the sphere is constant, exact solutions of the corresponding fleld equations are obtained in a flrst-order theory. However, it is found that there is a range of values of the sphere's net charge for which the physical interpretation of the results is di-cult within a continuum model. Outside that range, our solution to the classic electromagnetostatic problem appears plausible.

In this chapter the forces that act upon static and moving charged particles in electric and magnetic fields are collected together as the Lorentz force equation. The motion of a free charged particle and a charged particle subject to a strong centrally directed force are explored as examples of these forces. The Larmor theorem is introduced, which describes in general the rotation that is superimposed on motion that is caused by a strong centrally directed force, when an additional magnetic field is applied. The relativistic approach to the understanding of magnetic fields is briefly discussed and contrasted with the approach taken in this book.

We obtain infinitely many nontrivial periodic solutions for a class of damped vibration problems, where nonlinearities are superlinear at infinity and primitive functions of nonlinearities are allowed to be sign-changing. By using some weaker conditions, our results extend and improve some existing results in the literature. Besides, some examples are given to illuminate our results.

In this article, we study the multiplicity and minimality of periodic solutions to difference systems, which are globally superquadratic or subquadratic. A new technique is given to detect the minimal period of periodic solutions to autonomous systems. Some weaker conditions than a globally subquadratic condition are obtained to guarantee the existence of periodic solutions with prescribed minimal period to autonomous system.

The Periodic Solutions of the Oregonator and Verification of Results. The Oregonator is a numerically ill-conditional mathematical model in chemical kinetics involving nonlinear highly stiff ODEs. For the (stiffly coupled) “simplified Oregonator”, periodic solutions in the phase plane can be confined to an annular closed strip S whose lateral extension can be made negligibly small within graphical accuracy. This result of the “Karlsruhe enclosure methods” has been obtained with a simultaneous verification of the existence of periodic solutions in S, making use of index theory and th Poincaré-Bendixson theory.KeywordsPeriodic SolutionPeriodic OrbitStationary PointIndex TheoryClosed PathThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Long-term deorbiting control for an electrodynamic tether system exploiting periodic solutions

We consider periodic solutions of the following nonlinear system associated with the fractional Laplacian $$(−∂_{xx})^su(x)+∇F(u(x))=0 \quad {\rm in} \quad \mathbb{R},$$ where $F$ : $\mathbb{R}^2$→$\mathbb{R}$ is a smooth double-well potential. For the case that $F$ is even in its two variables we obtain the existence of more and more periodic solutions with large period, by using Clark’s theorem. For the case that $F$ is only even in its the second variable and the origin is a saddle critical point of $F$, we give two periodic solutions by using Morse theory.

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of 2 n th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of 2 n th-order discrete systems.