Using a Lagrangian mechanics approach, we construct a framework to study the dissipative properties of systems composed of two components one of which is highly lossy and the other is lossless. We have shown in our previous work that for such a composite system the modes split into two distinct classes, high-loss and low-loss, according to their dissipative behavior. A principal result of this paper is that for any such dissipative Lagrangian system, with losses accounted by a Rayleigh dissipative function, a rather universal phenomenon occurs, namely, selective overdamping: The high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal number of low-loss modes, but the rest of the low-loss modes remain oscillatory each with an extremely high quality factor that actually increases as the loss of the lossy component increases. We prove this result using a new time dynamical characterization of overdamping in terms of a virial theorem for dissipative systems and the breaking of an equipartition of energy.

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## Scholarly Works (19 results)

One of important problems in mathematical physics concerns derivation of point dynamics from field equations. The most common approach to this problem is based on WKB method. Here we describe a different method based on the concept of trajectory of concentration. When we applied this method to nonlinear Klein-Gordon equation, we derived relativistic Newton's law and Einstein's formula for inertial mass. Here we apply the same approach to nonlinear Schrodinger equation and derive non-relativistic Newton's law for the trajectory of concentration.

We advance here our neoclassical theory of elementary charges by integrating into it the concept of spin of 1/2. The developed spinorial version of our theory has many important features identical to those of the Dirac theory such as the gyromagnetic ratio, expressions for currents including the spin current, and antimatter states. In our theory, the concepts of charge and anticharge relate naturally to their "spin" in its rest frame in two opposite directions. An important difference with the Dirac theory is that both the charge and anticharge energies are positive whereas their frequencies have opposite signs.

Wave propagation in spatially periodic media, such as photonic crystals, can
be qualitatively different from any uniform substance. The differences are
particularly pronounced when the electromagnetic wavelength is comparable to
the primitive translation of the periodic structure. In such a case, the
periodic medium cannot be assigned any meaningful refractive index. Still, such
features as negative refraction and/or opposite phase and group velocities for
certain directions of light propagation can be found in almost any photonic
crystal. The only reservation is that unlike hypothetical uniform left-handed
media, photonic crystals are essentially anisotropic at frequency range of
interest. Consider now a plane wave incident on a semi-infinite photonic
crystal. One can assume, for instance, that in the case of positive refraction,
the normal components of the group and the phase velocities of the transmitted
Bloch wave have the same sign, while in the case of negative refraction, those
components have opposite signs. What happens if the normal component of the
transmitted wave group velocity vanishes? Let us call it a "zero-refraction"
case. At first sight, zero normal component of the transmitted wave group
velocity implies total reflection of the incident wave. But we demonstrate that
total reflection is not the only possibility. Instead, the transmitted wave can
appear in the form of an abnormal grazing mode with huge amplitude and nearly
tangential group velocity. This spectacular phenomenon is extremely sensitive
to the frequency and direction of propagation of the incident plane wave. These
features can be very attractive in numerous applications, such as higher
harmonic generation and wave mixing, light amplification and lasing, highly
efficient superprizms, etc.

Magnetic Faraday rotation is widely used in optics. In natural transparent
materials, this effect is very weak. One way to enhance it is to incorporate
the magnetic material into a periodic layered structure displaying a high-Q
resonance. One problem with such magneto-optical resonators is that a
significant enhancement of Faraday rotation is inevitably accompanied by strong
ellipticity of the transmitted light. More importantly, along with the Faraday
rotation, the resonator also enhances linear birefringence and absorption
associated with the magnetic material. The latter side effect can put severe
limitations on the device performance. From this perspective, we carry out a
comparative analysis of optical microcavity and a slow wave resonator. We show
that slow wave resonator has a fundamental advantage when it comes to Faraday
rotation enhancement in lossy magnetic materials.

We apply the idea of giant slow wave resonance associated with a degenerate
photonic band edge to gain enhancement of active media. This approach allows to
dramatically reduce the size of slow wave resonator while improving its
performance as gain enhancer for light amplification and lasing. It also allows
to reduce the lasing threshold of the slow wave optical resonator by at least
an order of magnitude.

The physical phenomenon of amplification in traveling wave tubes can be understood mathematically as a result of system instability or exponentially growing eigenmodes. We study here instability in the uncoupled multi-stream electron beam model, focusing primarily on properties and solutions of the characteristic equation associated with the beam. In particular, we show that in general the zeroes of the characteristic function are distinct. Then, in the two-stream case, we construct a series representation of frequency-dependent solutions to the electron beam characteristic equation near the frequency at which these solutions transition from non-real (unstable eigenmodes) to real (oscillatory eigenmodes).

It is well known that the concept of a point charge interacting with the electromagnetic (EM) field has a problem. To address that problem we introduce the concept of wave-corpuscle to describe spinless elementary charges interacting with the classical EM field. Every charge interacts only with the EM field and is described by a complex valued wave function over the 4-dimensional space time continuum. A system of many charges interacting with the EM field is defined by a local, gauge and Lorentz invariant Lagrangian with a key ingredient—a nonlinear self-interaction term providing for a cohesive force assigned to every charge. An ideal wave-corpuscle is an exact solution to the Euler-Lagrange equations describing both free and accelerated motions. It carries explicitly features of a point charge and the de Broglie wave. Our analysis shows that a system of well separated charges moving with nonrelativistic velocities are represented accurately as wave-corpuscles governed by the Newton equations of motion for point charges interacting with the Lorentz forces. In this regime the nonlinearities are “stealthy” and don’t show explicitly anywhere, but they provide for the binding forces that keep localized every individual charge. The theory can also be applied to closely interacting charges as in hydrogen atom where it produces discrete energy spectrum.

A boundary value problem is commonly associated with constraints imposed on a system at its boundary. We advance here an alternative point of view treating the system as interacting "boundary" and "interior" subsystems. This view is implemented through a Lagrangian framework that allows to account for (i) a variety of forces including dissipative acting at the boundary; (ii) a multitude of features of interactions between the boundary and the interior fields when the boundary fields may differ from the boundary limit of the interior fields; (iii) detailed pictures of the energy distribution and its flow; and (iv) linear and nonlinear effects. We provide a number of elucidating examples of the structured boundary and its interactions with the system interior. We also show that the proposed approach covers the well known boundary value problems.

Using a Lagrangian framework, we study overdamping phenomena in gyroscopic
systems composed of two components, one of which is highly lossy and the other
is lossless. The losses are accounted by a Rayleigh dissipative function. As we
have shown previously, for such a composite system the modes split into two
distinct classes, high-loss and low-loss, according to their dissipative
behavior. A principal result of this paper is that for any such system a rather
universal phenomenon of selective overdamping occurs. Namely, first of all the
high-loss modes are all overdamped, i.e., non-oscillatory, as are an equal
number of low-loss modes. Second of all, the rest of the low-loss modes remain
oscillatory (i.e., the underdamped modes) each with an extremely high quality
factor (Q-factor) that actually increases as the loss of the lossy component
increases. We prove that selective overdamping is a generic phenomenon in
Lagrangian systems with gyroscopic forces and give an analysis of the
overdamping phenomena in such systems. Moreover, using perturbation theory, we
derive explicit formulas for upper bound estimates on the amount of loss
required in the lossy component of the composite system for the selective
overdamping to occur in the generic case, and give Q-factor estimates for the
underdamped modes. Central to the analysis is the introduction of the notion of
a "dual" Lagrangian system and this yields significant improvements on some
results on modal dichotomy and overdamping. The effectiveness of the theory
developed here is demonstrated by applying it to an electric circuit with a
gyrator element and a high-loss resistor.