Search Results (108)

The concept of symmetry under various transformations of quantities describing basic natural phenomena is one of the fundamental principles in the mathematical formulation of physical laws. Starting with Noether’s theorems, we highlight some well–known examples of global symmetries and symmetry breaking on the particle level, such as the separation of strong and electroweak interactions and the Higgs mechanism, which gives mass to leptons and quarks. The relation between symmetry energy and charge symmetry breaking at both the nuclear level (under the interchange of protons and neutrons) and the particle level (under the interchange of u and d quarks) forms the main subject of this work. We trace the concept of symmetry energy from its introduction in the simple semi-empirical mass formula and liquid drop models to the most sophisticated non-relativistic, relativistic, and ab initio models. Methods used to extract symmetry energy attributes, utilizing the most significant combined terrestrial and astrophysical data and theoretical predictions, are reviewed. This includes properties of finite nuclei, heavy-ion collisions, neutron stars, gravitational waves, and parity–violating electron scattering experiments such as CREX and PREX, for which selected examples are provided. Finally, future approaches to investigation of the symmetry energy and its properties are discussed. Full article

This study investigates the geometric linearization of constraint Hamiltonian systems using the Jacobi metric and the Eisenhart lift. We establish a connection between linearization and maximally symmetric spacetimes, focusing on the Noether symmetries admitted by the constraint Hamiltonian systems. Specifically, for systems derived from the singular Lagrangian $L\left(N,q^k,{\dot{q}}^k\right)={\displaystyle \frac{1}{2N}}{g}_{ij}{\dot{q}}^i{\dot{q}}^j-NV\left(q^k\right),$ where N and $q^i$ are dependent variables and $dim{g}_{ij}=n$, the existence of ${\displaystyle \frac{n\left(n+1\right)}{2}}$ Noether symmetries is shown to be equivalent to the linearization of the equations of motion. The application of these results is demonstrated through various examples of special interest. This approach opens new directions in the study of differential equation linearization. Full article

In this paper, Lie symmetries and Noether symmetries along with the corresponding conservation laws are derived for weakly nonlinear dispersive magnetohydrodynamic wave equations, also known as the triple degenerate derivative nonlinear Schrödinger equations. The main goal of this study is to obtain Noether symmetries of the second-order Lagrangian density for these equations using the Noether symmetry approach with a gauge term. For this Lagrangian density, we compute the conserved densities and fluxes corresponding to the Noether symmetries with a gauge term, which differ from the conserved densities obtained using Lie symmetries in Webb et al. (J. Plasma Phys. 1995, 54, 201–244; J. Phys. A Math. Gen. 1996, 29, 5209–5240). Furthermore, we find some new Lie symmetries of the dispersive triple degenerate derivative nonlinear Schrödinger equations for non-vanishing integration functions $K_i\left(t\right)$ ($i=1,2,3$). Full article

This research aims to investigate the Noether symmetry and conserved quantity for the fractional Lagrange system with nonholonomic constraints, which are based on the Herglotz principle. Firstly, the fractional-order Herglotz principle is given, and the Herglotz-type fractional-order differential equations of motion for the fractional Lagrange system with nonholonomic constraints are derived. Secondly, by introducing infinitesimal generating functions of space and time, the Noether symmetry of the Herglotz type is defined, along with its criteria, and the conserved quantity of the Herglotz type is given. Finally, to demonstrate how to use this method, two examples are provided. Full article

We examine the key aspects of gravitational theories that incorporate non-local terms, particularly in the context of cosmology and spherical symmetry. We thus explore various extensions of General Relativity, including non-local effects in the action through the function $F(R,{\square }^{-1}R)$, where R denotes the Ricci curvature scalar and the operator ${\square }^{-1}$ introduces non-locality. By selecting the functional forms using Noether Symmetries, we identify exact solutions within a cosmological framework. We can thus reduce the dynamics of these chosen models and obtain analytical solutions for the equations of motion. Therefore, we study the capability of the selected models in matching cosmological observations by evaluating the phase space and the fixed points; this allows one to further constrain the non-local model selected by symmetry considerations. Furthermore, we also investigate gravitational non-local effects on astrophysical scales. In this context, we seek symmetries within the framework of $f(R,{\square }^{-1}R)$ gravity and place constraints on the free parameters. Specifically, we analyze the impact of non-locality on the orbits of the S2 star orbiting SgrA*. Full article

A geometric approach to the integrability and reduction of dynamical systems, both when dealing with systems of differential equations and in classical physics, is developed from a modern perspective. The main ingredients of this analysis are infinitesimal symmetries and tensor fields that are invariant under the given dynamics. A particular emphasis is placed on the existence of alternative invariant volume forms and the associated Jacobi multiplier theory, and then the Hojman symmetry theory is developed as a complement to the Noether theorem and non-Noether constants of motion. We also recall the geometric approach to Sundman infinitesimal time-reparametrisation for autonomous systems of first-order differential equations and some of its applications to integrability, and an analysis of how to define Sundman transformations for autonomous systems of second-order differential equations is proposed, which shows the necessity of considering alternative tangent bundle structures. A short description of alternative tangent structures is provided, and an application to integrability, namely, the linearisability of scalar second-order differential equations under generalised Sundman transformations, is developed. Full article

A semilinear quadratic equation of the form $A_{ij}\left(x\right)u^{ij}=B_i(x,u)u^i+F(x,u)$ defines a metric $A_{ij}$; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries of this metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative of the metric wrt the Lie symmetry generator, and the other set containing the quantities $B_i(x,u),F(x,u).$ From the first set, it follows that the generators of Lie point symmetries are elements of the conformal algebra of the metric $A_{ij}$, while the second set serves as constraint equations, which select elements from the conformal algebra of $A_{ij}.$ Therefore, it is possible to determine the Lie point symmetries using a geometric approach based on the computation of the conformal Killing vectors of the metric $A_{ij}$. In the present article, the nonlinear Poisson equation ${\Delta }_{g}u-f\left(u\right)=0$ is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector ${\partial }_t$. It is a conformally flat metric, which admits 10 conformal Killing vectors. We determine the conformal Killing vectors of this metric using a general geometric method, which computes the conformal Killing vectors of a general $1+(n-1)$ decomposable metric in a systematic way. It is found that the nonlinear Poisson equation ${\Delta }_{g}u-f\left(u\right)=0$ admits Lie point symmetries only when $f\left(u\right)=ku$, and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Lie point symmetries. This approach/method can be used to study the Lie point symmetries of more complex equations and with more degrees of freedom. Full article

The study of multi-time-delay dynamical systems has highlighted many challenges, especially regarding the solution and analysis of multi-time-delay equations. The symmetry and conserved quantity are two important and effective essential properties for understanding complex dynamical behavior. In this study, a multi-time-delay non-conservative mechanical system is investigated. Firstly, the multi-time-delay Hamilton principle is proposed. Then, multi-time-delay non-conservative dynamical equations are deduced. Secondly, depending on the infinitesimal group transformations, the invariance of the multi-time-delay Hamilton action is studied, and Noether symmetry, Noether quasi-symmetry, and generalized Noether quasi-symmetry are discussed. Finally, Noether-type conserved quantities for a multi-time-delay Lagrangian system and a multi-time-delay non-conservative mechanical system are obtained. Two examples in terms of a multi-time-delay non-conservative mechanical system and a multi-time-delay Lagrangian system are given. Full article

This article analyzes the symmetry of two-dimensional stationary gas dynamics equations in Lagrangian coordinates, including the search for equivalence transformations, the group classification of equations, the derivation of group foliations, and the construction of conservation laws. The consideration of equations in Lagrangian coordinates significantly simplifies the procedure for obtaining conservation laws, which are derived using the Noether theorem. The final part of the work is devoted to group foliations of the gas dynamics equations, including for the nonstationary isentropic case. The group foliations approach is usually employed for equations that admit infinite-dimensional groups of transformations (which is exactly the case for the gas dynamics equations in Lagrangian coordinates) and may make it possible to simplify their further analysis. The results obtained in this regard generalize previously known results for the two-dimensional shallow water equations in Lagrangian coordinates. Full article

We investigate the Noether symmetries of the Klein–Gordon Lagrangian for Bianchi I spacetime. This is accomplished using a set of new Noether symmetry relations for the Klein–Gordon Lagrangian of Bianchi I spacetime, which reduces to the wave equation in a special case. A detailed Noether symmetry analysis of the Klein–Gordon and the wave equations for Bianchi I spacetime is presented, and the corresponding conservation laws are derived. Full article

Explicit and spontaneous breaking of spacetime symmetry under diffeomorphisms, local translations, and local Lorentz transformations due to the presence of fixed background fields is examined in Einstein–Cartan theory. In particular, the roles of torsion and violation of local translation invariance are highlighted. The nature of the types of background fields that can arise and how they cause spacetime symmetry breaking is discussed. With explicit breaking, potential no-go results are known to exist, which if not evaded lead to inconsistencies between the Bianchi identities, Noether identities, and the equations of motion. These are examined in detail, and the effects of nondynamical backgrounds and explicit breaking on the energy–momentum tensor when torsion is present are discussed as well. Examples illustrating various features of both explicit and spontaneous breaking of local translations are presented and compared to the case of diffeomorphism breaking. Full article

This review paper gives an overview of the method of multipliers for partial differential equations (PDEs). This method has made possible a lot of solutions to PDEs that are of interest in many areas such as applied mathematics, mathematical physics, engineering, etc. Looking at the history of the method and synthesizing the newest developments, we hope to give it the attention that it deserves to help develop the vast amount of work still needed to understand it and make the best use of it. It is also an interesting and a relevant method in itself that could possibly give interesting results in areas of mathematics such as modern algebra, group theory, topology, etc. The paper will be structured in such a manner that the last review known for this method will be presented to understand the theoretical framework of the method and then later work done will be presented. The information of four recent papers further developing the method will be synthesized and presented in such a manner that anyone interested in learning this method will have the most relevant information available and have all details cited for checking. Full article

Classical forbidden processes paved the way for the description of mechanical systems with the help of complex Hamiltonians. Fractional integrals of complex order appear as a natural generalization of those of real order. We propose the complex fractional Euler-Lagrange equation, obtained by finding the stationary values associated with the fractional integral of complex order. The complex Hamiltonian obtained from the Lagrangian is suitable for describing nonconservative systems. We conclude by presenting the conserved quantities attached to Noether symmetries corresponding to complex systems. We illustrate the theory with the aid of the damped oscillatory system. Full article

A rigorous analysis is undertaken based on the analysis of both Galilean and Lorentz (Poincaré) invariance in field theories in general in order to (i) identify a unique analytical scheme for canonical pairs of Lagrangians, one of them having Galilean, the other one Poincaré invariance; and (ii) to obtain transition conditions for the state function purely from Hamilton’s principle and extended Noether’s theorem applied to the aforementioned symmetries. The general analysis is applied on Schrödinger and Klein–Gordon theory, identifying them as a canonical pair in the sense of (i) and exemplified for the scattering problem for both theories for a particle beam against a potential step in order to show that the transition conditions that result according to (ii) in a ‘natural’ way reproduce the well-known ‘methodical’ continuity conditions commonly found in the literature, at least in relevant cases, closing a relevant argumentation gap in quantum mechanical scattering problems. Full article