Research

43 pages, 581 KiB

Open AccessArticle

Reparametrization Invariance and Some of the Key Properties of Physical Systems

by Vesselin G. Gueorguiev and Andre Maeder

Symmetry 2021, 13(3), 522; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030522 - 23 Mar 2021

Cited by 6 | Viewed by 2824

In this paper, we argue in favor of first-order homogeneous Lagrangians in the velocities. The relevant form of such Lagrangians is discussed and justified physically and geometrically. Such Lagrangian systems possess Reparametrization Invariance (RI) and explain the observed common Arrow of Time as [...] Read more.

In this paper, we argue in favor of first-order homogeneous Lagrangians in the velocities. The relevant form of such Lagrangians is discussed and justified physically and geometrically. Such Lagrangian systems possess Reparametrization Invariance (RI) and explain the observed common Arrow of Time as related to the non-negative mass for physical particles. The extended Hamiltonian formulation, which is generally covariant and applicable to reparametrization-invariant systems, is emphasized. The connection between the explicit form of the extended Hamiltonian

H

and the meaning of the process parameter

λ

is illustrated. The corresponding extended Hamiltonian

H

defines the classical phase space-time of the system via the Hamiltonian constraint

H = 0

and guarantees that the Classical Hamiltonian H corresponds to

p_{0}

—the energy of the particle when the coordinate time parametrization is chosen. The Schrödinger’s equation and the principle of superposition of quantum states emerge naturally. A connection is demonstrated between the positivity of the energy

E = c p_{0} > 0

and the normalizability of the wave function by using the extended Hamiltonian that is relevant for the proper-time parametrization. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

22 pages, 719 KiB

Open AccessArticle

Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

by Jehad Alzabut, A. George Maria Selvam, Rami A. El-Nabulsi, Vignesh Dhakshinamoorthy and Mohammad E. Samei

Symmetry 2021, 13(3), 473; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030473 - 13 Mar 2021

Cited by 56 | Viewed by 2269

Abstract

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form [...] Read more.

Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form

Δ_{*}^{β} [k] (t) = w (t + β, k (t + β), k (λ (t + β)))

, with condition

k (0) = p [k]

for

t \in N_{1 - β}

,

0 < β \leq 1

,

λ \in (0, 1)

and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

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7 pages, 748 KiB

Open AccessArticle

Stability Analysis of an LTI System with Diagonal Norm Bounded Linear Differential Inclusions

by Mutti-Ur Rehman, Sohail Iqbal, Jehad Alzabut and Rami Ahmad El-Nabulsi

Symmetry 2021, 13(1), 152; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13010152 - 19 Jan 2021

Cited by 1 | Viewed by 1758

Abstract

In this article, we present a stability analysis of linear time-invariant systems in control theory. The linear time-invariant systems under consideration involve the diagonal norm bounded linear differential inclusions. We propose a methodology based on low-rank ordinary differential equations. We construct an equivalent [...] Read more.

In this article, we present a stability analysis of linear time-invariant systems in control theory. The linear time-invariant systems under consideration involve the diagonal norm bounded linear differential inclusions. We propose a methodology based on low-rank ordinary differential equations. We construct an equivalent time-invariant system (linear) and use it to acquire an optimization problem whose solutions are given in terms of a system of differential equations. An iterative method is then used to solve the system of differential equations. The stability of linear time-invariant systems with diagonal norm bounded differential inclusion is studied by analyzing the Spectrum of equivalent systems. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

10 pages, 235 KiB

Open AccessArticle

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

by Xiao-Qun Cao, Ya-Nan Guo, Shi-Cheng Hou, Cheng-Zhuo Zhang and Ke-Cheng Peng

Symmetry 2020, 12(5), 850; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12050850 - 22 May 2020

Cited by 16 | Viewed by 2043

Abstract

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of [...] Read more.

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

8 pages, 235 KiB

Open AccessArticle

Oscillatory Behavior of Fourth-Order Differential Equations with Neutral Delay

by Osama Moaaz, Rami Ahmad El-Nabulsi and Omar Bazighifan

Symmetry 2020, 12(3), 371; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12030371 - 02 Mar 2020

Cited by 32 | Viewed by 2237

Abstract

In this paper, new sufficient conditions for oscillation of fourth-order neutral differential equations are established. One objective of our paper is to further improve and complement some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these [...] Read more.

In this paper, new sufficient conditions for oscillation of fourth-order neutral differential equations are established. One objective of our paper is to further improve and complement some well-known results which were published recently in the literature. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. An example is given to illustrate the importance of our results. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

11 pages, 277 KiB

Open AccessArticle

A Note on Ricci Solitons

by Sharief Deshmukh and Hana Alsodais

Symmetry 2020, 12(2), 289; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12020289 - 17 Feb 2020

Cited by 13 | Viewed by 3503

Abstract

In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of [...] Read more.

In this paper, we characterize trivial Ricci solitons. We observe the important role of the energy function f of a Ricci soliton (half the squared length of the potential vector field) in the charectrization of trivial Ricci solitons. We find three characterizations of connected trivial Ricci solitons by imposing different restrictions on the energy function. We also use Hessian of the potential function to characterize compact trivial Ricci solitons. Finally, we show that a solution of a Poisson equation is the energy function f of a compact Ricci soliton if and only if the Ricci soliton is trivial. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

12 pages, 299 KiB

Open AccessArticle

New Results for Oscillatory Behavior of Fourth-Order Differential Equations

by Rami Ahmad El-Nabulsi, Osama Moaaz and Omar Bazighifan

Symmetry 2020, 12(1), 136; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12010136 - 09 Jan 2020

Cited by 31 | Viewed by 2570

Abstract

Our aim in the present paper is to employ the Riccatti transformation which differs from those reported in some literature and comparison principles with the second-order differential equations, to establish some new conditions for the oscillation of all solutions of fourth-order differential equations. [...] Read more.

Our aim in the present paper is to employ the Riccatti transformation which differs from those reported in some literature and comparison principles with the second-order differential equations, to establish some new conditions for the oscillation of all solutions of fourth-order differential equations. Moreover, we establish some new criterion for oscillation by using an integral averages condition of Philos-type, also Hille and Nehari-type. Some examples are provided to illustrate the main results. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

9 pages, 262 KiB

Open AccessArticle

Quantum Correction for Newton’s Law of Motion

by Timur F. Kamalov

Symmetry 2020, 12(1), 63; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12010063 - 27 Dec 2019

Cited by 14 | Viewed by 4924

Abstract

A description of the motion in noninertial reference frames by means of the inclusion of high time derivatives is studied. Incompleteness of the description of physical reality is a problem of any theory, both in quantum mechanics and classical physics. The “stability principle” [...] Read more.

A description of the motion in noninertial reference frames by means of the inclusion of high time derivatives is studied. Incompleteness of the description of physical reality is a problem of any theory, both in quantum mechanics and classical physics. The “stability principle” is put forward. We also provide macroscopic examples of noninertial mechanics and verify the use of high-order derivatives as nonlocal hidden variables on the basis of the equivalence principle when acceleration is equal to the gravitational field. Acceleration in this case is a function of high derivatives with respect to time. The definition of dark metrics for matter and energy is presented to replace the standard notions of dark matter and dark energy. In the Conclusion section, problem symmetry is noted for noninertial mechanics. Full article

(This article belongs to the Special Issue Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics)

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