Related Conservation Laws Research Articles

Euler–Lagrange equation for gradient-type Lagrangian and related conservation laws

AbstractVariational calculus with gradient-type variations has often been neglected, although it proves to be suitable for certain concrete problems governed by several evolution variables. These kinds of variations lead to Euler–Lagrange partial differential equations controlled by the right-hand member. In this context, we also introduce anti-trace Euler–Lagrange partial differential equations that are suitable for some innovative ideas. Also, some applications are provided for the theoretical results derived in the paper.

Nonlocal symmetries, nonlocally related systems, similarity solutions and conservation laws of the Tzitzéica-Dodd-Bullogh equation

In this paper, a methodical procedure is utilized for the identification of nonlocal symmetries of the (1+1)-dimensional Tzitzéica-Dodd-Bullogh equation. Firstly, by introducing a set of canonical coordinates corresponding to the local Lie point symmetries, the considered partial differential eq. (PDE) is mapped to an invertibly equivalent PDE system. Furthermore, nonlocal symmetries are obtained from the inverse potential system of the invertibly equivalent PDE system. The exact solutions for the aforementioned PDE are acquired with the help of the extended generalized Kudryashov method corresponding to the admitted symmetries. In addition, the derivation of local conservation laws for the Tzitzéica-Dodd-Bullogh equation is obtained through the multiplier method. Moreover, using a symmetry-based technique and local conservation principles, a complete tree of nonlocally related PDE systems has been constructed.

An asymptotic analysis and stability for a class of focusing Sobolev critical nonlinear Schrödinger equations

In this paper, we consider a class of focusing nonlinear Schrödinger equations involving power-type nonlinearity with critical Sobolev exponent{(i∂∂t+Δ)u+|u|4N−2su=0,inR+×RN,u=u0(x)∈H1(RN)∩L2(RN,|x|2dx),fort=0,x∈RN, where 2N−2s=:p⋆ be the critical Sobolev exponent with s<N2. For dimension N≥1, the initial data u0 belongs to the energy space and |⋅|u0∈L2(RN) and the power index s satisfies [0,1]∋s≡sc=N2−1p⋆, we prove that the problem is non-global existence in H1(RN) (here, finite-time blow-up occurs) with the energy of initial data E[u0] is negative. Moreover, we establish the stability for the solutions with the lower bound and the global a priori upper bound in dimension N≥2 related conservation laws. The motivation for this paper is inspired by the mass critical case with s=0 of the celebrated result of B. Dodson [9] and the work of R. Killip and M. Visan [18] represented with energy critical case for s=1. Our new results mention to nonlinear Schrödinger equation for interpolating between mass critical or mass super-critical (s≥0) and energy sub-critical or energy critical (s≤1).

Tests of Charge–Parity Symmetry and Lepton Flavor Conservation in the Top Quark Sector

The Standard Model (SM) of particle physics is the most general renormalizable theory which is built on a few general principles and fundamental symmetries with the given particle content. However, multiple symmetries are not built into the model and are simply consequences of renormalizabilty, gauge invariance, and particle content of the theory. It is crucial to test the validity of these types of symmetries and related conservation laws experimentally. The CERN LHC provides the highest sensitivity for testing the SM symmetries at high energy scales involving heavy particles such as the top quark. In this article, we are going to review the recent experimental searches of charge–parity and charged-lepton flavor violation in the top quark sector.

High harmonic generation by co-rotating bi-circular laser fields interaction with relativistic plasmas

We present an investigation of high harmonic generation (HHG) from relativistic plasmas driven by two-color co-rotating circularly polarized (CP) laser fields at various frequency ratios ω2/ω1. The theoretical analysis and the particle-in-cell simulation results reveal that when ω2/ω1≥4, CP harmonics with opposite helicities can appear simultaneously due to the relativistic mirror effect and the related conservation laws. Meanwhile, the harmonic spectra features (including the HHG efficiency, the allowed harmonic orders, and their polarization states) can be tuned through adjusting the frequency and intensity ratio of the two driving field components. This finding might be helpful for a consummate understanding of the bi-circular HHG technique.

High-order vortex harmonics generation by bi-circular Laguerre-Gaussian laser fields with relativistic plasmas

Vortex beams with ultra-high brilliance can greatly enrich the light and matter interaction process and even shed light on the unexpected information in relativistic nonlinear optics. Thus, we propose a scheme for relativistic intense vortex harmonic radiation by use of bi-circular Laguerre–Gaussian lasers irradiating relativistic plasmas. Three-dimensional particle-in-cell simulation results show that the emitted harmonics own controllable spin and orbital angular momentum simultaneously, which can be attributed to the vortex mirror model and the related conservation laws. Meanwhile, the conversion efficiency of harmonic generation can be tuned through adjusting the intensity ratio of the two driving field components.

From Electronegativity towards Reactivity-Searching for a Measure of Atomic Reactivity.

Pauling introduced the concept of electronegativity of an atom which has played an important role in understanding the polarity and ionic character of bonds between atoms. We set out to define a related concept of atomic reactivity in such a way that it can be quantified and used to predict the stability of covalent bonds in molecules. Guided by the early definition of electronegativity by Mulliken in terms of first ionization energies and Pauling in terms of bond energies, we propose corresponding definitions of atomic reactivity. The main goal of clearly distinguishing the inert gas atoms as nonreactive is fulfilled by three different proposed measures of atomic reactivity. The measure likely to be found most useful is based on the bond energies in atomic hydrides, which are related to atomic reactivities by a geometric average. The origin of the atomic reactivity is found in the symmetry of the atomic environment and related conservation laws which are also the origin of the shell structure of atoms and the periodic table. The reactive atoms are characterized by degenerate or nearly degenerate (several states of the same or nearly the same energy) ground states, while the inert atoms have nondegenerate ground states and no near-degeneracies. We show how to extend the use of the Aufbau model of atomic structure to qualitatively describe atomic reactivity in terms of ground state degeneracy. The symmetry and related conservation laws of atomic electron structures produce a strain (energy increase) in the structure, which we estimate by use of the Thomas-Fermi form of DFT implemented approximately with and without the symmetry and conservation constraints. This simplified and approximate analysis indicates that the total strain energy of an atom correlates strongly with the corresponding atomic reactivity measures but antibonding mechanisms prevent full conversion of strain relaxation to bonding.

Perturbation, symmetry analysis, Bäcklund and reciprocal transformation for the extended Boussinesq equation in fluid mechanics

In this work, we study a generalized double dispersion Boussinesq equation that plays a significant role in fluid mechanics, scientific fields, and ocean engineering. This equation will be reduced to the Korteweg–de Vries equation via using the perturbation analysis. We derive the corresponding vectors, symmetry reduction and explicit solutions for this equation. We readily obtain Bäcklund transformation associated with truncated Painlevé expansion. We also examine the related conservation laws of this equation via using the multiplier method. Moreover, we investigate the reciprocal Bäcklund transformations of the derived conservation laws for the first time.

Optical Helicity and Optical Chirality in Free Space and in the Presence of Matter

The inherently weak nature of chiral light–matter interactions can be enhanced by orders of magnitude utilizing artificially-engineered nanophotonic structures. These structures enable high spatial concentration of electromagnetic fields with controlled helicity and chirality. However, the effective design and optimization of nanostructures requires defining physical observables which quantify the degree of electromagnetic helicity and chirality. In this perspective, we discuss optical helicity, optical chirality, and their related conservation laws, describing situations in which each provides the most meaningful physical information in free space and in the context of chiral light–matter interactions. First, an instructive comparison is drawn to the concepts of momentum, force, and energy in classical mechanics. In free space, optical helicity closely parallels momentum, whereas optical chirality parallels force. In the presence of macroscopic matter, the optical helicity finds its optimal physical application in the case of lossless, dual-symmetric media, while, in contrast, the optical chirality provides physically observable information in the presence of lossy, dispersive media. Finally, based on numerical simulations of a gold and silicon nanosphere, we discuss how metallic and dielectric nanostructures can generate chiral electromagnetic fields upon interaction with chiral light, offering guidelines for the rational design of nanostructure-enhanced electromagnetic chirality.

Illustrations of Maxwell's term and the four conservation laws of electromagnetism

We illustrate the use of Maxwell's term in Ampere's law and also the conservation laws of electromagnetic theory. First, we outline the conservation laws for energy and linear momentum which often appear in textbooks of electromagnetism, and then we show how these conservation laws can be extended to the conservation law for angular momentum and to the relativistic conservation law involving the center of energy. Next we treat the traditional example of charging a parallel-plate capacitor but explore a number of energy-conservation aspects and variations which are not usually found in textbooks. Finally, we discuss the charging of spherical conductors where we can illustrate the use of Maxwell's term and can provide non-trivial examples of three of the four conservation laws.

Duality Occurrences: Physical Origin of Wave Functions

The quantum mystery began with the probabilistic interpretation of the wave function. However, this usefulness is definitive in Quantum Mechanics while the suspense continues. The present paper aims to investigate the origin of such a mystery. Hence, considering a relativistic charged particle in quantum vacuum, it appeared that: (i) from a classical association, this electromagnetismself-consistent derives from the usual wave function, which corresponds to the scalar nature in addition to the vector one. (ii) Such duplicity is only justifiable when the related gauge fields describe fermions, in accordance with the previous theory of duality field-matter. This occurrence then corresponds to the appearance of bosons at cell intersections in vacuum lattice, whatever is the field. (iii) From the related gauge couplings, the scalar function must have an unknown vector companion. Both appear as originating the related conservation laws on one side. On the other side, specific variations of the field front would explain their physical origin. (iv) Moreover, both define an original gauge field to which that of law conservation is sensible. (v) Due to definition validity in any reference system, their possible quantization should lead to that of scalar and vector fields of stationary states. At last, the results highlight the connection between waves and fields associable to any object, emphasizing the field unification framework.

A convergent finite volume method for the Kuramoto equation and related nonlocal conservation laws

Abstract We derive and study a Lax–Friedrichs-type finite volume method for a large class of nonlocal continuity equations in multiple dimensions. We prove that the method converges weakly to the measure-valued solution and converges strongly if the initial data is of bounded variation. Several numerical examples for the kinetic Kuramoto equation are provided, demonstrating that the method works well for both regular and singular data.

Spectral control of high harmonics from relativistic plasmas using bicircular fields.

We introduce two-color counterrotating circularly polarized laser fields as a way to spectrally control high harmonic generation (HHG) from relativistic plasma mirrors. Through particle-in-cell simulations, we show that only a selected group of harmonic orders can appear owing to the symmetry of the laser fields and the related conservation laws. By adjusting the intensity ratio of the two driving field components, we demonstrate the overall HHG efficiency, the relative intensity of allowed neighboring harmonic orders, and that the polarization state of the harmonic source can be tuned. The HHG efficiency of this scheme can be as high as that driven by a linearly polarized laser field.

Conserved norms and related conservation laws for multi-peakon equations

All nonlinear dispersive wave equations in the general class are known to possess multi-peakon weak solutions. A classification is presented for families of multi-peakon equations in this class that possess conserved momentum; conserved H1 norm of u; conserved H2 norm of u; conserved L2 norm of m; related conservation laws. The results yield, among others, two interesting wide families of equations: for which the H1 norm of u is conserved; for which the L2 norm of m is conserved. The overlap of these two families yields a singular equation which is nevertheless found to possess both smooth solitary wave solutions and peakon travelling wave solutions.

Variational integrators for reduced magnetohydrodynamics

Reduced magnetohydrodynamics is a simplified set of magnetohydrodynamics equations with applications to both fusion and astrophysical plasmas, possessing a noncanonical Hamiltonian structure and consequently a number of conserved functionals. We propose a new discretisation strategy for these equations based on a discrete variational principle applied to a formal Lagrangian. The resulting integrator preserves important quantities like the total energy, magnetic helicity and cross helicity exactly (up to machine precision). As the integrator is free of numerical resistivity, spurious reconnection along current sheets is absent in the ideal case. If effects of electron inertia are added, reconnection of magnetic field lines is allowed, although the resulting model still possesses a noncanonical Hamiltonian structure. After reviewing the conservation laws of the model equations, the adopted variational principle with the related conservation laws is described both at the continuous and discrete level. We verify the favourable properties of the variational integrator in particular with respect to the preservation of the invariants of the models under consideration and compare with results from the literature and those of a pseudo-spectral code.

Interaction of a magnet and a point charge: Unrecognized internal electromagnetic momentum

Whereas nonrelativistic mechanics always connects the total momentum of a system to the motion of the center of mass, relativistic systems, such as interacting electromagnetic charges, can have internal linear momentum in the absence of motion of the system's center of energy. This internal linear momentum of a system is related to the controversial concept of “hidden momentum.” We suggest that the term “hidden momentum” be abandoned. Here, we use the relativistic conservation law for the center of energy to give an unambiguous definition of the “internal momentum of a system,” and then we exhibit this internal momentum for the system of a magnet (modeled as a circular ring of moving charges) and a distant static point charge. The calculations provide clear illustrations of this system for three cases: (a) the moving charges of the magnet are assumed to continue in their unperturbed motion; (b) the moving charges of the magnet are free to accelerate but have no mutual interactions; and (c) the moving charges of the magnet are free to accelerate and also interact with each other. When the current-carrying charges of the magnet are allowed to interact, the magnet itself will contain internal electromagnetic linear momentum, something that has not been described clearly in the research and teaching literature.

Relabeling symmetry in relativistic fluids and plasmas

The conservation of the recently formulated relativistic canonical helicity (Yoshida et al 2014 J. Math. Phys. 55 043101) is derived from Noetherʼs theorem by constructing an action principle on the relativistic Lagrangian coordinates (we obtain general cross helicities that include the helicity of the canonical vorticity). The conservation law is, then, explained by the relabeling symmetry pertinent to the Lagrangian label of fluid elements. Upon Eulerianizing the Noether current, the purely spatial volume integral on the Lagrangian coordinates is mapped to a space-time mixed three-dimensional integral on the four-dimensional Eulerian coordinates. The relativistic conservation law in the Eulerian coordinates is no longer represented by any divergence-free current; hence, it is not adequate to regard the relativistic helicity (represented by the Eulerian variables) as a Noether charge, and this stands the reason why the ‘conventional helicity’ is no longer a constant of motion. We have also formulated a relativistic action principle of magnetohydrodynamics (MHD) on the Lagrangian coordinates, and have derived the relativistic MHD cross helicity.

Case studies in space charge and plasma acceleration of charged beams

Plasma acceleration with electron or proton driver beams is a challenging opportunity for high-energy physics. An energy doubling experiment with electron drivers was successfully performed at SLAC and a key experiment AWAKE with proton drivers is on schedule at CERN. Simulations play an important role in choosing the best experimental conditions and in interpreting the results. The Vlasov equation is the theoretical tool to describe the interaction of a driver particle beam or a driver laser pulse with a plasma. Collective effects, such as tune shift and mismatch instabilities, appear in high intensity standard accelerators and are described by the Poisson–Vlasov equation. In the paper, we review the Vlasov equation in the electrostatic and fully electromagnetic cases. The general framework of variational principles is used to derive the equation, the local form of the balance equations and related conservation laws. In the electrostatic case, we remind the analytic Kapchinskij–Vladimirskij (K–V) model and we propose an extension of the adiabatic theory for Hamiltonian systems, which ensures stability for perturbation of size ϵ on times of order 1/ϵ. The variational framework is used to derive the Maxwell–Vlasov equations and related conservation laws and to briefly sketch the particle-in-cell (PIC) approximation schemes. Finally, the proton-driven acceleration is examined in the linear and quasi-linear regime. A PIC simulation with the code ALaDyn developed at Bologna University is presented to illustrate the longitudinal and transverse fields evolution which allow a witness electron bunch to be accelerated with a gradient of a few GeV/m. We also present some remarks on future perspectives.

The relation between relativistic and non-relativistic continuum thermodynamics

We consider the relativistic theory of irreversible processes with the aim to answer the following questions: (1) Under which conditions is this theory a relativistic generalization of the non-relativistic theory of irreversible processes (in particular, this implies to ask for the conditions under which the first law of thermodynamics can be recovered from the relativistic conservation law of total energy), and (2) how do the relativistic corrections look like? To this end, we perform a low-energy approximation for the balance equations underlying the theory, i.e., for the balances of the particle number, the energy-momentum and the entropy. It is shown that, going up to the 3rd order in the expansion series of the balances, the non-relativistic theory can be derived when one assumes that the 4-current of the particle flow is purely convective and the product of the 3-dimensional acceleration and velocity is equal to zero. Afterwards, the higher-order terms are discussed. Since our discussion mainly makes use of those balance equations that lie on the basis of most versions of continuum thermodynamics, the results do not only refer to early TIP presented by Eckart (Phys Rev 58:919, 1940) and Landau and Lifshitz (Fluid mechanics. Pergamon Press, Oxford, 1940), but also to its extended and/or general-relativistic versions.

Momentum in general relativity: local versus quasilocal conservation laws

We construct a general relativistic conservation law for linear and angular momentum for matter and gravitational fields in a finite volume of space that does not rely on any spacetime symmetries. This work builds on our previous construction of a general relativistic energy conservation law with the same features (McGrath et al 2012 Class. Quantum Grav. 29 215012). Our approach uses the Brown and York (1993 Phys. Rev. D 47 1407–19) quasilocal stress–energy–momentum tensor for matter and gravitational fields, plus the concept of a rigid quasilocal frame (RQF) introduced in (Epp et al 2009 Class. Quantum Grav. 26 035015; 2012 Classical and Quantum Gravity: Theory, Analysis, and Applications (Nova Science)). The RQF approach allows us to construct, in a generic spacetime, frames of reference whose boundaries are rigid (their shape and size do not change with time), and that have precisely the same six arbitrary time-dependent degrees of freedom as the accelerating and tumbling rigid frames we are familiar with in Newtonian mechanics. These RQFs, in turn, give rise to a completely general conservation law for the six components of momentum (three linear and three angular) of a finite system of matter and gravitational fields. We compare in detail this quasilocal RQF approach to constructing conservation laws with the usual local one based on spacetime symmetries, and discuss the shortcomings of the latter. These RQF conservation laws lead to a deeper understanding of physics in the form of simple, exact, operational definitions of gravitational energy and momentum fluxes, which in turn reveal, for the first time, the exact, detailed mechanisms of gravitational energy and momentum transfer taking place in a wide variety of physical phenomena, including a simple falling apple. As a concrete example, we derive a general relativistic version of Archimedes’ law that we apply to understand electrostatic weight and buoyant force in the context of a Reissner–Nordström black hole.