Research

34 pages, 466 KiB

Open AccessArticle

by Antonio Masiello

Symmetry 2021, 13(8), 1422; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13081422 - 04 Aug 2021

Cited by 2 | Viewed by 1984

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of [...] Read more.

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

17 pages, 356 KiB

Open AccessEditor’s ChoiceArticle

Symmetric Ground States for Doubly Nonlocal Equations with Mass Constraint

by Silvia Cingolani, Marco Gallo and Kazunaga Tanaka

Symmetry 2021, 13(7), 1199; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071199 - 02 Jul 2021

Cited by 17 | Viewed by 2216

Abstract

We prove the existence of a spherically symmetric solution for a Schrödinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in [...] Read more.

We prove the existence of a spherically symmetric solution for a Schrödinger equation with a nonlocal nonlinearity of Choquard type. This term is assumed to be subcritical and satisfy almost optimal assumptions. The mass of of the solution, described by its norm in the Lebesgue space, is prescribed in advance. The approach to this constrained problem relies on a Lagrange formulation and new deformation arguments. In addition, we prove that the obtained solution is also a ground state, which means that it realizes minimal energy among all the possible solutions to the problem. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

21 pages, 403 KiB

Open AccessArticle

On Families of Wigner Functions for N-Level Quantum Systems

by Vahagn Abgaryan and Arsen Khvedelidze

Symmetry 2021, 13(6), 1013; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061013 - 04 Jun 2021

Cited by 7 | Viewed by 1909

Abstract

A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for [...] Read more.

A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a

N = (2 j + 1)

-level system, one can always identify the representative that realizes the so-called

S U (2)

-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

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9 pages, 270 KiB

Open AccessArticle

On the Integrable Deformations of the Maximally Superintegrable Systems

by Cristian Lăzureanu

Symmetry 2021, 13(6), 1000; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13061000 - 03 Jun 2021

Viewed by 1451

Abstract

In this paper, we present the integrable deformations method for a maximally superintegrable system. We alter the constants of motion, and using these new functions, we construct a new system which is an integrable deformation of the initial system. In this manner, new [...] Read more.

In this paper, we present the integrable deformations method for a maximally superintegrable system. We alter the constants of motion, and using these new functions, we construct a new system which is an integrable deformation of the initial system. In this manner, new maximally superintegrable systems are obtained. We also consider the particular case of Hamiltonian mechanical systems. In addition, we use this method to construct some deformations of an arbitrary system of first-order autonomous differential equations. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

21 pages, 285 KiB

Open AccessArticle

Multiple Critical Points for Symmetric Functionals without upper Growth Condition on the Principal Part

by Marco Degiovanni and Marco Marzocchi

Symmetry 2021, 13(5), 898; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050898 - 18 May 2021

Cited by 2 | Viewed by 1297

Abstract

This paper is concerned with variational methods applied to functionals of the calculus of variations in a multi-dimensional case. We prove the existence of multiple critical points for a symmetric functional whose principal part is not subjected to any upper growth condition. For [...] Read more.

This paper is concerned with variational methods applied to functionals of the calculus of variations in a multi-dimensional case. We prove the existence of multiple critical points for a symmetric functional whose principal part is not subjected to any upper growth condition. For this purpose, nonsmooth variational methods are applied. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

29 pages, 351 KiB

Open AccessArticle

A Model for the Maxwell Equations Coupled with Matter Based on Solitons

by Vieri Benci

Symmetry 2021, 13(5), 760; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13050760 - 27 Apr 2021

Viewed by 965

Abstract

We present a simple model of interaction of the Maxwell equations with a matter field defined by the Klein–Gordon equation. A simple linear interaction and a nonlinear perurbation produces solutions to the equations containing hylomorphic solitons, namely stable, solitary waves, whose existence is [...] Read more.

We present a simple model of interaction of the Maxwell equations with a matter field defined by the Klein–Gordon equation. A simple linear interaction and a nonlinear perurbation produces solutions to the equations containing hylomorphic solitons, namely stable, solitary waves, whose existence is related to the ratio energy/charge. These solitons, at low energy, behave as poinwise charged particles in an electromagnetic field. The basic points are the following ones: (i) the matter field is described by the Nonlinear Klein–Gordon equation with a suitable nonlinear term; (ii) the interaction is not described by the equivariant derivative, but by a very simple coupling which preseves the invariance under the Poincaré group; (iii) the existence of soliton can be proved using the tecniques of nonlinear analysis and, in particular, the Mountain Pass Theorem; (iv) a suitable choice of the parameters produces solitons with a prescribed electric charge and mass/energy; (v) thanks to the point (ii), the dynamics of these solitons at low energies is the same of classical charged particles. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

10 pages, 767 KiB

Open AccessArticle

Some Results on Ricci Almost Solitons

by Sharief Deshmukh, Hana Alsodais and Nasser Bin Turki

Symmetry 2021, 13(3), 430; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030430 - 07 Mar 2021

Cited by 9 | Viewed by 1348

Abstract

We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton

(M, g, w, σ)

to be a trivial Ricci soliton under the assumption that the soliton vector field

w

is a geodesic vector [...] Read more.

We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton

(M, g, w, σ)

to be a trivial Ricci soliton under the assumption that the soliton vector field

w

is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition

r^{2} \leq n σ r

on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field

w

is an eigen vector of the Ricci operator with constant eigenvalue

λ

satisfying

n^{2} λ^{2} \geq r^{2}

; the third result uses a suitable lower bound on the Ricci curvature

S (w, w)

. Finally, we show that an n-dimensional connected Ricci almost soliton

(M, g, w, σ)

with soliton vector field

w

is a geodesic vector field with a trivial Ricci soliton, if and only if,

n σ - r

is a constant along integral curves of

w

and the Ricci curvature

S (w, w)

has a suitable lower bound. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

17 pages, 302 KiB

Open AccessArticle

Brownian Swarm Dynamics and Burgers’ Equation with Higher Order Dispersion

by Max-Olivier Hongler

Symmetry 2021, 13(1), 57; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13010057 - 31 Dec 2020

Cited by 1 | Viewed by 1127

Abstract

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for [...] Read more.

The concept of ranked order probability distribution unveils natural probabilistic interpretations for the kink waves (and hence the solitons) solving higher order dispersive Burgers’ type PDEs. Thanks to this underlying structure, it is possible to propose a systematic derivation of exact solutions for PDEs with a quadratic nonlinearity of the Burgers’ type but with arbitrary dispersive orders. As illustrations, we revisit the dissipative Kotrweg de Vries, Kuramoto-Sivashinski, and Kawahara equations (involving third, fourth, and fifth order dispersion dynamics), which in this context appear to be nothing but the simplest special cases of this infinitely rich class of nonlinear evolutions. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

28 pages, 407 KiB

Open AccessArticle

A New Approach for Euler-Lagrange Orbits on Compact Manifolds with Boundary

by Dario Corona and Fabio Giannoni

Symmetry 2020, 12(11), 1917; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111917 - 20 Nov 2020

Cited by 1 | Viewed by 1349

Abstract

Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on [...] Read more.

Consider a compact manifold with boundary, homeomorphic to the N-dimensional disk, and a Tonelli Lagrangian function defined on the tangent bundle. In this paper, we study the multiplicity problem for Euler-Lagrange orbits that satisfy the conormal boundary conditions and that lay on the boundary only in their extreme points. In particular, for suitable values of the energy function and under mild hypotheses, if the Tonelli Lagrangian is reversible then the minimal number of Euler-Lagrange orbits with prescribed energy that satisfies the conormal boundary conditions is N. If L is not reversible, then this number is two. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

13 pages, 346 KiB

Open AccessArticle

Combining 3-Momentum and Kinetic Energy on Galilei/Newton Spacetime

by Christian Y. Cardall

Symmetry 2020, 12(11), 1775; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12111775 - 26 Oct 2020

Cited by 3 | Viewed by 1298

Abstract

Without the mass-energy equivalence available on Minkowski spacetime

M

, it is not possible on 4-dimensional non-relativistic Galilei/Newton spacetime

G

to combine 3-momentum and total mass-energy in a single tensor object. However, given a fiducial frame, it is possible to combine 3-momentum and [...] Read more.

Without the mass-energy equivalence available on Minkowski spacetime

M

, it is not possible on 4-dimensional non-relativistic Galilei/Newton spacetime

G

to combine 3-momentum and total mass-energy in a single tensor object. However, given a fiducial frame, it is possible to combine 3-momentum and kinetic energy into a linear form (particle) or

(1, 1)

tensor (continuum) in a manner that exhibits increased unity of classical mechanics on flat relativistic and non-relativistic spacetimes

M

and

G

. As on

M

, for a material continuum on

G

, the first law of thermodynamics can be considered a consequence of a unified dynamical law for energy-momentum rather than an independent postulate. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

0 pages, 470 KiB

Open AccessArticle

On Schwarzschild’s Interior Solution and Perfect Fluid Star Model

by Elisabetta Barletta, Sorin Dragomir and Francesco Esposito

Symmetry 2020, 12(10), 1669; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12101669 - 13 Oct 2020

Cited by 1 | Viewed by 1973

Abstract

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density

ρ

and pressure field

p (r)

located in a ball

r \leq r_{0}

. [...] Read more.

We solve the boundary value problem for Einstein’s gravitational field equations in the presence of matter in the form of an incompressible perfect fluid of density

ρ

and pressure field

p (r)

located in a ball

r \leq r_{0}

. We find a 1-parameter family of time-independent and radially symmetric solutions

\{(g_{a}, ρ_{a}, p_{a}) : - 2 m < a < a_{1}\}

satisfying the boundary conditions

g = g_{S}

and

p = 0

on

r = r_{0}

, where

g_{S}

is the exterior Schwarzschild solution (solving the gravitational field equations for a point mass M concentrated at

r = 0

) and containing (for

a = 0

) the interior Schwarzschild solution, i.e., the classical perfect fluid star model. We show that Schwarzschild’s requirement

r_{0} > 9 κ M / (4 c^{2})

identifies the “physical” (i.e., such that

p_{a} (r) \geq 0

and

p_{a} (r)

is bounded in

0 \leq r \leq r_{0}

) solutions

{p_{a} : a \in U_{0}}

for some neighbourhood

U_{0} \subset (- 2 m, + \infty)

of

a = 0

. For every star model

{g_{a} : a_{0} < a < a_{1}}

, we compute the volume

V (a)

of the region

r \leq r_{0}

in terms of abelian integrals of the first, second, and third kind in Legendre form. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

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21 pages, 2020 KiB

Open AccessArticle

Large Deflection Analysis of Axially Symmetric Deformation of Prestressed Circular Membranes under Uniform Lateral Loads

by Xue Li, Jun-Yi Sun, Zhi-Hang Zhao and Xiao-Ting He

Symmetry 2020, 12(8), 1343; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12081343 - 11 Aug 2020

Cited by 9 | Viewed by 2496

Abstract

In this study, the problem of axisymmetric deformation of peripherally fixed and uniformly laterally loaded circular membranes with arbitrary initial stress is solved analytically. This problem could be called the generalized Föppl–Hencky membrane problem as the case where the initial stress in the [...] Read more.

In this study, the problem of axisymmetric deformation of peripherally fixed and uniformly laterally loaded circular membranes with arbitrary initial stress is solved analytically. This problem could be called the generalized Föppl–Hencky membrane problem as the case where the initial stress in the membrane is equal to zero is the well-known Föppl–Hencky membrane problem. The problem can be mathematically modeled only in terms of radial coordinate owing to its axial symmetry, and in the present work, it is reformulated by considering an arbitrary initial stress (tensile, compressive, or zero) and by simultaneously improving the out-of-plane equilibrium equation and geometric equation, while the formulation was previously considered to fail to improve the geometric equation. The power-series method is used to solve the reformulated boundary value problem, and a new and more refined analytic solution of the problem is presented. This solution is actually observed to be able to regress into the well-known Hencky solution of zero initial stress, allowing the considered initial stress to be zero. Moreover, the numerical example conducted shows that the obtained power-series solutions for stress and deflection converge very well, and have higher computational accuracy in comparison with the existing solutions. Full article

(This article belongs to the Special Issue Recent Advance in Mathematical Physics)

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