Special Issue "Non-Standard Lagrangians and Hamiltonians in Theoretical Physics and Applied Mathematics"

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics and Symmetry".

Deadline for manuscript submissions: closed (28 February 2021).

Special Issue Editor

Prof. Dr. Rami Ahmad El-Nabulsi
E-Mail Website
Guest Editor
Mathematics and Physics Divisions, Athens Institute for Education and Research, 10671 Athens, Greece
Interests: geometrical dynamics; quantum mechanics; nonlinearity; fractal dynamics; geometrical physics; general relativity and gravitation; operators theory; quantum field theory; plasma MHD and planetary dynamics; chaos and bifurcations; reactor physics and nuclear sciences; solid state physics and magnetism; quantum electronics and nanostructures
Special Issues and Collections in MDPI journals

Special Issue Information

Dear Colleagues,

“Non-standard Lagrangians” (NSLs), which involve neither the ordinary kinetic term nor the classical potential function, form an interesting field in theoretical physics and applied mathematics despite their anomalous or irregular physical forms. They were introduced in 1978 by Arnold in his classic book “Mathematical Methods of Classical Mechanics”. Nevertheless, their real implications for theoretical physics date back to 1984 when Alekseev and Arbuzov used them to describe large distances interactions in the region of applicability of classical theory, a problem which is related to the color confinement issue. Regardless of their strange properties, NSLs play a significant role in the theory of nonlinear differential equations, dissipative dynamical systems, earthquake physics, plasma physics, astrophysics, quantum mechanics, and quantum field theory, among others. They are an emerging phenomenon. The main aim of this Special Issue is to discuss new implications of NSLs in different fields of theoretical physics and applied mathematics, in particular classical and quantum mechanics, quantum hydrodynamics, kinetic theory, solid state physics, classical and quantum electrodynamics, nuclear physics, astrophysics, and cosmology.

Prof. Rami Ahmad El-Nabulsi
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • non-standard Lagrangians and Hamiltonians
  • non-standard fractional Lagrangians
  • dynamical systems and nonlinear differential equations
  • Noether's symmetries and dissipative systems
  • calculus of variations and the inverse problem
  • non-standard Lagrangians on time-scales
  • non-standard Lagrangians in theoretical physics and applied mathematics
  • applications (astrophysics, solid state physics, nuclear physics, quantum mechanics, quantum field theory, astrophysics).

Published Papers (8 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

Article
Reparametrization Invariance and Some of the Key Properties of Physical Systems
Symmetry 2021, 13(3), 522; https://doi.org/10.3390/sym13030522 - 23 Mar 2021
Viewed by 461
Abstract
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 p0—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=cp0>0 and the normalizability of the wave function by using the extended Hamiltonian that is relevant for the proper-time parametrization. Full article
Article
Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions
Symmetry 2021, 13(3), 473; https://doi.org/10.3390/sym13030473 - 13 Mar 2021
Cited by 1 | Viewed by 441
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 Δβ[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for tN1β, 0<β1, λ(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
Show Figures

Figure 1

Article
Stability Analysis of an LTI System with Diagonal Norm Bounded Linear Differential Inclusions
Symmetry 2021, 13(1), 152; https://doi.org/10.3390/sym13010152 - 19 Jan 2021
Viewed by 459
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
Article
Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water
Symmetry 2020, 12(5), 850; https://doi.org/10.3390/sym12050850 - 22 May 2020
Viewed by 620
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
Article
Oscillatory Behavior of Fourth-Order Differential Equations with Neutral Delay
Symmetry 2020, 12(3), 371; https://doi.org/10.3390/sym12030371 - 02 Mar 2020
Cited by 9 | Viewed by 769
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
Article
A Note on Ricci Solitons
Symmetry 2020, 12(2), 289; https://doi.org/10.3390/sym12020289 - 17 Feb 2020
Cited by 4 | Viewed by 496
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
Article
New Results for Oscillatory Behavior of Fourth-Order Differential Equations
Symmetry 2020, 12(1), 136; https://doi.org/10.3390/sym12010136 - 09 Jan 2020
Cited by 21 | Viewed by 1024
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
Article
Quantum Correction for Newton’s Law of Motion
Symmetry 2020, 12(1), 63; https://doi.org/10.3390/sym12010063 - 27 Dec 2019
Cited by 6 | Viewed by 1192
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
Back to TopTop