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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. [ABSTRACT FROM AUTHOR]

Highly important is a three-dimensional nonlinear partial differential equation because for many physical systems, one can, subject to suitable idealizations, formulate a differential equation that describes how the system changes in time. Thus, this article comprehensively reveals the investigation carried out on a (3+1)-dimensional generalized fifth-order Zakharov–Kuznetsov equation with power-law as well as dual power-law nonlinearities analytically, where the fifth-order term involved is regarded as a dispersion perturbation term. We utilize the well-celebrated Noether's theorem to comprehensively construct conserved currents of the underlying equation. A detailed Lie group analysis of the understudied model consisting of power-law nonlinearities is further performed. This involves performing reductions of the underlying models using their Lie point symmetries. In consequence, various invariants are found. In addition, the equation reduces to diverse ordinary differential equations using its point symmetries and consequently diverse solutions of interest were achieved. Moreover, we derive some solitary wave solutions by invoking the newly introduced logistic function technique for some particular cases of the equation under consideration. In consequence, we achieve some exponential function solutions. In addition, the physical meaning of the results is put on the front burner by revealing the wave dynamics of these solutions via graphical depictions. Finally, the significance of the robust and detailed findings in the work are further corroborated with various real-world applications. [ABSTRACT FROM AUTHOR]

There is a deep link existing between symmetry and nature that may be best apprehended by using group theory. In physics, giving a symmetry group is entirely equivalent to fix natural laws (Noether’s theorem). Within such a framework, it follows that all fundamental mechanical laws may be given by defining a group of symmetry operations belonging to the Gal (3,1) symmetry group having ten generators (3 + 3 + 3 + 1 = 10). The move to special relativity amounts simply to substitute another group ISO (3,1) to Gal (3,1) and having the same number of generators. In both cases, one immediately perceives that we live in universe with 3 dimensions of space and 1 dimension of time. A further progress could be made by looking for the symmetry group that leaves Maxwell’s equations invariant. The answer to this fundamental question is the group ISO (4,2) ⊗U (2) ⊗U (2) characterized by 23 generators: 15 = 10 + 5 for the conformal group ISO(4,2), 4 for U(2) and hence 8 = 4 + 4 for the compact part U(2) ⊗U(2) that could be related to the existence of quantum physics besides relativity. As ISO (3,1) is a sub-group of ISO (4,2), the two numbers 4 and 2 suggests that, taking into account the existence of light besides matter, we live in an universe having 4 spacelike dimensions and 2 time-like dimensions. Here, it is proposed that, as music is characterized by a scale-invariant symmetry operation (transposition by octaves) and that as time may be perceived differently in shamanic trance for instance, the two “new” dimensions of our universe may be related to music and consciousness. Related to the fifth dimension of music, the notion of diapason linking physical frequencies to a musical note is revisited. It is thus proposed to adopt a new A4 = 429.62 Hz value based on universal physical constants of our universe and on the mass of the water molecule H₂O, "The most abundant heteronuclear molecule in the entire universe". Demonstration is done that this new diapason is superior to other diapasons based on frequencies of 440 Hz or even 432 Hz. [ABSTRACT FROM AUTHOR]

In this paper, the Lie symmetry analysis was executed for the nonlinear fractional-order conduction-diffusion Buckmaster model (BM), which involves the Riemann-Liouville (R-L) derivative of fractional-order ‘β’. In the study of groundwater flow and oil reservoir engineering where fluid flow through porous materials is crucial, BM played an important role. The Lie point infinitesimal generators and Lie algebra were constructed for the equation. The Lie symmetries were acquired for the ordinary fractional-order BM. The power series solution and its convergence were also analyzed with the application of the implicit theorem. Noether’s theorem was employed to ensure the consistency of a system by deriving the conservation laws of its physical model. [ABSTRACT FROM AUTHOR]

Lateral momentum conservation is typically kept in a non-absorptive rotationally symmetric system through mirror symmetry via Noether's theorem when illuminated by a homogeneous light wave. Therefore, it is still very challenging to break the mirror symmetry and generate a lateral optical force (LOF) in the rotationally symmetric system. Here, we report a general dynamic action in the SO(2) rotationally symmetric system, originating from the polarization-tuned mirror symmetry breaking (MSB) of the light scattering. We demonstrate theoretically and experimentally that MSB can be generally applied to the SO(2) rotationally symmetric system and tuned sinusoidally by polarization orientation, leading to a highly tunable and highly efficient LOF (9.22 pN/mW/μm−2) perpendicular to the propagation direction. The proposed MSB mechanism and LOF not only complete the sets of MSB of light-matter interaction and non-conservative force only using a plane wave but also provide extra polarization manipulation freedom. In a SO(2) rotationally symmetric system, a general polarization-tuned mirror symmetry breaking mechanism was reported to highly efficiently generate a lateral optical force, enriching the optical manipulation mechanism for mirror-symmetric objects. [ABSTRACT FROM AUTHOR]

In this paper, we present a covariant approach that utilizes Noether's second theorem to derive a symmetric stress tensor from the grand thermodynamic potential functional. We focus on the practical case where the density of the grand thermodynamic potential is dependent on the first and second coordinate derivatives of the scalar order parameters. Our approach is applied to several models of inhomogeneous ionic liquids that consider electrostatic correlations of ions or short-range correlations related to packing effects. Specifically, we derive analytical expressions for the symmetric stress tensors of the Cahn–Hilliard-like model, Bazant–Storey–Kornyshev model, and Maggs–Podgornik–Blossey model. All of these expressions are found to be consistent with respective self-consistent field equations. [ABSTRACT FROM AUTHOR]

We investigate variational, Lie and Killing symmetries of the Lagrangian of an essential class of four-dimensional (pseudo-)Riemannian manifolds, i.e., non-reductive homogeneous spaces. Moreover, a Lie algebra analysis is shown. With the help of a result of the Noether's theorem, we have presented the expressions for conservation laws corresponding to all variational symmetries. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of the time fractional Black–Scholes equation derived by the fractional Brownian motion. Some exact solutions are obtained, the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, a new conservation theorem and a generalization of the Noether operators are developed to construct the conservation laws for the time fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

In the real world, individuals may conceal some of their real decision information to their neighbors due to competition. It is a challenge to explore the distributed Nash equilibrium when individuals play the noncooperative game with partial decision information in complex networks. In this paper, we investigate the distributed Nash equilibrium seeking problem with partial decision information of neighbors. Specifically, we construct a two-layer network model, where players in the first layer engage in game interactions and players in the second layer exchange estimations of real actions with each other. We also consider the case where the actions of some players remain unchanged due to the cost of updating or personal reluctance. By means of the Lyapunov function method and LaSalle's invariance principle, we obtain the sufficient conditions in which the consensus of individual actions and estimations can be achieved and the population actions can converge to the Nash equilibrium point. Furthermore, we investigate the case with switched topologies and derive the sufficient conditions for the convergence of individual actions to Nash equilibrium by the average dwell time method. Finally, we give numerical examples for cases of fixed and switched topologies to verify our theoretical results. [ABSTRACT FROM AUTHOR]

An earlier model of the electron distribution function accounts for the pressure anisotropy that develops in the inflow regions during anti‐parallel magnetic reconnection. However, the model is not applicable to the electron diffusion region (EDR), where the electron magnetic moments break as an adiabatic invariant. The earlier model is here generalized through the use of the current‐sheet adiabatic invariant of a 1D current layer, to become applicable also to the EDR. The new generalized model reproduces the main features of the EDR and its vicinity observed in a fully kinetic simulation, formally proving the 1D equilibrium relationship between the upstream electron pressure anisotropy and the EDR electron jets. Plain Language Summary: Ever since 1953 when Dungey introduced the concept of magnetic reconnection in the Earth's magnetotail, the process has been investigated as the potential cause of the Aurora phenomenon. During reconnection the magnetic field lines break and reform with changed connectivity as they pass through a small electron diffusion region (EDR). In the magnetotail the EDRs responsible for the Aurora often forms about 20 Earths radii away from Earth, spectacularly breaking the standard rules of plasma physics in ways not fully understood. In the present letter we show how the structure of the EDR can largely be accounted for through a relatively simple kinetic model. The model relies on an adiabatic invariant of the electron motion perpendicular to the magnetotail current sheet, and it accounts for the electron pressure anisotropy which builds at the edge of the EDR. In turn, we show how the thermal forces associated with this pressure anisotropy are responsible for driving the strong electron jets observed within in the EDR. The model is validated against a state‐of‐the‐art supercomputer simulation of the reconnection process. Key Points: During anti‐parallel magnetic reconnection the electron diffusion region (EDR) has a largely 1D structureThe cross‐section of the 1D structure is governed by the invariance of the electron current sheet action integral, Jz=∮vzdz ${\mathcal{J}}_{z}=\oint {v}_{z}dz$Electron pressure anisotropy forms upstream of the EDR, couples to the Speiser orbits, and drives the electron jets of the EDR [ABSTRACT FROM AUTHOR]

In this work, a master–slave system composed by a pair of damped Duffing oscillators with variable coefficients and nonlinear coupling is investigated. An integral of motion for the system is obtained using a symmetry transformation and Noether’s theorem. Some numerical examples are presented for different cases of damping and oscillation frequency, for a varying coupling constant. The system dynamics is studied by means of space-time surfaces, time series and phase portraits. For a constant oscillation frequency, the slave presents envelopes that tend to become chaotic as the coupling constant increases. Meanwhile, as the frequency increases with time, the slave has higher amplitudes and speeds than the master oscillator. [ABSTRACT FROM AUTHOR]

The present paper introduces an innovative strain energy function (SEF) for incompressible anisotropic fiber-reinforced materials. This SEF is specifically designed to understand the mechanical behavior of carbon fiber-woven fabric. The considered model combines polyconvex invariants forming an integrity basisin polynomial form, which is inspired by the application of Noether's theorem. A single solution can be obtained during the identification because of the relationship between the SEF we have constructed and the material parameters, which are linearly dependent. The six material parameters were precisely determined through a comparison between the closed-form solutions from our model and the corresponding tensile experimental data with different stretching ratios, with determination coefficients consistently reaching a remarkable value of 0.99. When considering only uniaxial tensile tests, our model can be simplified from a quadratic polynomial to a linear polynomial, thereby reducing the number of material parameters required from six to four, while the fidelity of the model's predictive accuracy remains unaltered. The comparison between the results of numerical calculations and experiments proves the efficiency and accuracy of the method. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We give details and derivations for the Noether invariance theory that characterizes the spatial equilibrium structure of inhomogeneous classical many-body systems, as recently proposed and investigated for bulk systems (Sammüller et al 2023 Phys. Rev. Lett. 130 268203). Thereby an intrinsic thermal symmetry against a local shifting transformation on phase space is exploited on the basis of the Noether theorem for invariant variations. We consider the consequences of the shifting that emerge at second order in the displacement field that parameterizes the transformation. In a natural way the standard two-body density distribution is generated. Its second spatial derivative is thereby balanced by two further and different two-body correlation functions, which respectively introduce thermally averaged force correlations and force gradients in a systematic and microscopically sharp way into the framework. Separate exact self and distinct sum rules express this balance. We exemplify the validity of the theory on the basis of computer simulations for the Lennard–Jones gas, liquid, and crystal, the Weeks–Chandler–Andersen fluid, monatomic Molinero–Moore water at ambient conditions, a three-body-interacting colloidal gel former, the Yukawa and soft-sphere dipolar fluids, and for isotropic and nematic phases of Gay–Berne particles. We describe explicitly the derivation of the sum rules based on Noether's theorem and also give more elementary proofs based on partial phase space integration following Yvon's theorem. [ABSTRACT FROM AUTHOR]

All low‐order conservation laws are found for a general class of nonlinear wave equations in one dimension with linear damping which is allowed to be time‐dependent. Such equations arise in numerous physical applications and have attracted much attention in analysis. The conservation laws describe generalized momentum and boost momentum, conformal momentum, generalized energy, dilational energy, and light cone energies. Both the conformal momentum and dilational energy have no counterparts for nonlinear undamped wave equations in one dimension. All of the conservation laws are obtainable through Noether's theorem, which is applicable because the damping term can be transformed into a time‐dependent self‐interaction term by a change of dependent variable. For several of the conservation laws, the corresponding variational symmetries have a novel form which is different than any of the well‐known variational symmetries admitted by nonlinear undamped wave equations in one dimension. [ABSTRACT FROM AUTHOR]

This article focuses on three main contributions. Firstly, we provide an in-depth overview of the nonlocal Lagrangian formalism. Secondly, we introduce an extended version of the second Noether's theorem tailored for nonlocal Lagrangians. Finally, we apply both the formalism and the extended theorem to the context of non-commutative U(1) gauge theory, including its Hamiltonian and quantization, showcasing their practical utility. [ABSTRACT FROM AUTHOR]

In this paper, we investigate various types of symmetries and their mutual relationships in Hamiltonian systems defined on manifolds with different geometric structures: symplectic, cosymplectic, contact and cocontact. In each case, we pay special attention to non-standard (non-canonical) symmetries, in particular scaling symmetries and canonoid transformations, as they provide new interesting tools for the qualitative study of these systems. Our main results are the characterizations of these non-standard symmetries and the analysis of their relation with conserved (or dissipated) quantities. [ABSTRACT FROM AUTHOR]

Noether invariance in statistical mechanics provides fundamental connections between the symmetries of a physical system and its conservation laws and sum rules. The latter are exact identities that involve statistically averaged forces and force correlations and they are derived from statistical mechanical functionals. However, the implications for more general observables and order parameters are unclear. Here, we demonstrate that thermally averaged classical phase space functions are associated with exact hyperforce sum rules that follow from translational Noether invariance. Both global and locally resolved identities hold and they relate the mean gradient of a phase-space function to its negative mean product with the total force. Similar to Hirschfelder's hypervirial theorem, the hyperforce sum rules apply to arbitrary observables in equilibrium. Exact hierarchies of higher-order sum rules follow iteratively. As applications we investigate via computer simulations the emerging one-body force fluctuation profiles in confined liquids. These local correlators quantify spatially inhomogeneous self-organization and their measurement allows for the development of stringent convergence tests and enhanced sampling schemes in complex systems. Noether's theorem relates the symmetries of a statistical mechanical system to conservation laws and to constraints of thermal averages, but the extension to more general observables is still elusive. The authors demonstrate that any statistical observable is associated with a hierarchy of exact identities emerging from translational Noether invariance. [ABSTRACT FROM AUTHOR]

In this paper, we investigate relations or differences among various conserved quantities which involve the matter Energy Momentum Tensor (EMT) in general relativity. These quantities include the energy with Einstein's pseudo EMT, the generalized Komar integral, or the ADM energy, all of which can be derived from Noether's second theorem, as well as an extra conserved charge recently proposed in general relativity. For detailed analyses, we apply definitions of these charges to a system of free massive particles. We employ the post-Newtonian (PN) expansion to make physical interpretations. We find that the generalized Komar integral is not conserved at the first non-trivial order in the PN expansion due to non-zero contributions at spatial boundaries, while the energy with Einstein's pseudo EMT at this order agrees with a total energy of massive particles with gravitational interactions through the Newtonian potential, and thus is conserved. In addition, this total energy is shown to be identical to the ADM energy not only at this order but also all orders in the PN expansion. We next calculate an extra conserved charge for the system of massive particles, at all orders in the PN expansion, which turns out to be a total number of particles. We call it a gravitational charge, since it is clearly different from the total energy. We finally discuss an implication from a fact that there exist two conserved quantities, energy and gravitational charge, in general relativity. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Today Noether's principal theorem occupies a prominent place in theoretical physics, though for a long time its significance was largely overlooked. Even now, relatively few physicists realize that Emmy Noether's original paper from 1918 contains two fundamental theorems. Moreover, both theorems are essential for understanding her original motivation, namely to distinguish between proper and improper conservation laws in physics. [ABSTRACT FROM AUTHOR]

The anomaly cancellation in superstring theory is known to hold at leading order in the curvature for the gauge groups S O (3 2) and E 8 × E 8 . The coefficients of the next-to-leading order terms may be evaluated, and a mechanism for cancellation is described, which would remain valid at higher genus when there exists a global splitting of the coordinates of supermoduli space. Since the spin- 1 2 fields transform under the adjoint representation in these models, compactification of the heterotic string over G 2 / S U (3) or a fundamental 12-dimensional theory, from which superstrings are produced through an elliptic fibration, over G 2 × S U (2) × U (1) S U (3) × U (1) ′ × U (1) ′ ′ , provides another phenomenologically viable theory at lower energies compatible with the standard description of the elementary particle interactions. The 96 spin- 1 2 fields now would transform under the fundamental representation of G 2 × S U (2) × U (1) and the spin-one gauge fields would belong to the adjoint representation. The sum of the anomaly polynomials for the particle content of the G 2 × S U (2) × U (1) model vanishes at n 1 2 ≃ 1 0 3. 3 4 2 7 1 9 2 4. The contributions to the gravitational anomaly from the particles and antiparticles cancel by the CPT theorem and the duality transformations of polynomials of degree 6 in the curvature and the field strength. The existence of the interaction of a spin-2 charge, which is conserved only over a finite time interval, can be traced to nonlocal terms in the reduction of the string field theory to the gravitational sector. The source of the global gravitational anomaly cancellation in the modular form equations derived from an elliptic fibration of a 12-dimensional theory would restrict the compactifications and provide a method for preserving the absence of anomalies in four dimensions. [ABSTRACT FROM AUTHOR]

In this paper, it has been proven that locally the inertial frames and gauge states of the electromagnetic field are equivalent. This proof is valid for Einstein–Maxwell theories in four-dimensional Lorentzian spacetimes. Theorems proved in a previous paper will be used. These theorems state that locally the group of electromagnetic gauge transformations is isomorphic to the local group of Lorentz transformations of a special set of tetrad vectors. The tetrad that locally and covariantly diagonalizes any non-null electromagnetic stress–energy tensor. There are in total two isomorphisms, one for each orthogonal plane of stress–energy eigenvectors. We discuss the opposite problem in this paper. What happens with local electromagnetic gauge when the test object under study is boosted by any mechanical means? We will prove that boosting matter is indistinguishable from introducing an appropriate local electromagnetic gauge transformation. [ABSTRACT FROM AUTHOR]

The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by a Hamilton equation with respect to the Lie–Poisson bracket. The Nambu bracket manifestly represents the Lie–Poisson structure in terms of derivatives of the Casimir invariants. We construct a compact Nambu bracket representation for the 3D ideal MHD equations with the use of three Casimirs for the second Hamiltonians, the total entropy, and the magnetic and cross-helicities, whose coefficients are all constant. The Lie–Poisson bracket induced by this Nambu bracket does not coincide with the original one, but is supplemented by terms with an auxiliary variable. The supplemented Lie–Poisson bracket qualifies the cross-helicity as the Casimir. By appealing to Noether's theorem, we show that the cross-helicity is an integral invariant associated with the particle-relabeling symmetry. Employing a Lagrange label function as the independent variable in the variational framework facilitates implementation of the relabeling transformation. By incorporating the divergence symmetry, other known topological invariants are put on the same ground as Noether's theorem. [ABSTRACT FROM AUTHOR]

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the Lie symmetry analysis of bi-fractional Black–Scholes equation derived by the fractional G-Brownian motion. Then some exact solutions are obtained and the figures of which are presented to illustrate the characteristics with different values of the parameters. In addition, the new conservation theorem and the generalization of the Noether operators are developed to construct the conservation laws for bi-fractional Black–Scholes equation. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

In this paper, the Noether theorems and their inverse theorems for generalized Chaplygin systems with two types of nonstandard Lagrangians, related to exponential and power-law Lagrangian, are explored and presented. The variational principles for the Chaplygin systems with nonstandard Lagrangian are derived, and the generalized Chaplygin equations for the corresponding systems are established, the Noether transformations are considered, from which the corresponding conserved quantities are deduced. And their inverse theorems for nonstandard generalized Chaplygin systems are given. Two examples show the validity of the results. [ABSTRACT FROM AUTHOR]

We propose a category of bundles in order to perform Lagrangian reduction by stages in covariant Field Theory. This category plays an analogous role to Lagrange–Poincaré bundles in Lagrangian reduction by stages in Mechanics and includes both jet bundles and reduced covariant configuration spaces. Furthermore, we analyze the resulting reconstruction condition and formulate the Noether theorem in this context. Finally, a model of a molecular strand with rotors is seen as an application of this theoretical frame. [ABSTRACT FROM AUTHOR]

This present paper investigates Lie symmetry analysis, one-dimensional optimal system, exact solutions and conservation laws of the (2 + 1)-dimensional time fractional thin liquid film equations (TFTLFE) with Riemann–Liouville fractional derivative. Explicitly, we obtain six vector fields and the one-dimensional optimal system admitted by TFTLFE. Then, we perform the symmetry reductions with the help of Erdélyi–Kober fractional differential operator and (2 + 1)-dimensional TFTLFE is reduced into (1 + 1)-dimensional fractional partial differential equations (FPDEs). Additionally, by means of compound variable transformation and the power series expansion method, the solution of reduced FPDEs is obtained and its convergence is verified. Moreover, we derive other solutions for the reduced equations taking advantage of the invariant subspace method. Furthermore, the conservation laws are also established utilizing generalized Noether's theorem. Finally, we construct the exact solution using the method of conservation laws. [ABSTRACT FROM AUTHOR]

The Zakharov–Kuznetsov (ZK) equation is extensively investigated in this study. This equation is a two-dimensional (2D) generalization of the well-known Korteweg–de Vries (KdV) equation, which is spatially limited as the 1D model of weakly nonlinear waves in shallow water. In this paper, we use three different methods to solve the conservation laws of it. First, the direct construction method is used to calculate the multipliers of the objective equation, and the conservation laws can be obtained by using the multipliers. Then its strict self-adjoint property is verified, and its conservation laws are solved by Ibragimov's method. Finally, the conservation laws of the target equation are solved by Noether's theorem. Then we calculate some exact solutions of the ZK equation by the extended Kudryashov method and seek a few solutions in terms of hyperbolic tangent functions and exp (ξ). Meanwhile, the conservation law analysis of the target equation is carried out, and its Lie point symmetries, reduced order form and invariant solution are determined. In the end, the Hamiltonian structure of the target equation, the generalized pre-symplectic that maps symmetries into adjoint-symmetries and some of its soliton solutions are calculated. [ABSTRACT FROM AUTHOR]

In a seminal paper in the Philosophical Transactions of the Royal Society (A244, 87–112). Eshelby (Eshelby 1951 Phil. Trans. R. Soc. Lond. A 244, 87–112. (doi:10.1098/rsta.1951.0016)) introduced the concept of 'the force on an elastic singularity' and suggested that the extensions to dynamics include the application of the momentum flux. In this direction, it is shown that, in the non-uniformly motion of a dislocation there is a loss in quasi-momentum (or pseudo-momentum) across the scales ε to ε2, which induces an effective mass of the dislocation, and a loss in kinetic energy across the scales. It is shown through Noether's theorem that the rate of change of quasi-momentum in the volume enclosing the dislocation is equal to the flux of it through the surface minus a quasi-force, which is the dynamic J integral. The connection between the variation in the Hamiltonian and in the Lagrangian relates the quasi-force to the energy-release rate, yielding the same effective mass, while providing physical meaning both through momentum and energy. The effective mass of a dislocation is important in relating the energetics of defects from the micro to the macro scales, and the loss of quasi-momentum can have wider applications in continua. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'. [ABSTRACT FROM AUTHOR]

We look at a mechanical dissipation inequality differing from the standard one by what we call a relative power, a notion that is appropriate in the presence of material mutations. We prove that a requirement of structural invariance for such an inequality under the action of diffeomorphism-based changes of observers (covariance) implies (i) the representation of contact actions in terms of the first Piola–Kirchhoff stress, (ii) local balances of standard and configurational actions, (iii) a priori constitutive restrictions in terms of free energy, and (iv) a representation of viscous-type stress components. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'. [ABSTRACT FROM AUTHOR]

This paper offers a number of examples showing that in the case of two independent variables the uniform ellipticity of a linear system of differential equations with partial derivatives of the second order, which fulfills condition (3), do not always cause the normal solvability of formulated exterior elliptic problems in the sense of Noether. Nevertheless, from the system of differential equations with partial derivatives of elliptic type it is possible to choose, under certain additional conditions, classes which are normally solvable in the sense of Noether. This paper also shows that for the so-called decomposed system of differential equations, with partial derivatives of an elliptic type in the case of exterior regions, the Noether theorems are valid. [ABSTRACT FROM AUTHOR]

The role of symmetries and first integrals are well known mechanisms for the reduction of ordinary differential equations (odes) and, used in conjunction, lead to double reductions of the odes. In this article, we attempt to construct the first integrals of a large class of the well known second-order Painlevé equations. In some cases, variational and/or gauge symmetries have additional applications following a known Lagrangian in which case the first integral is obtained by Noether's theorem. Sometimes, it is more convenient to adopt the 'multiplier' approach to find the first integrals. In a number of cases, we can conclude that the class is linearizable. [ABSTRACT FROM AUTHOR]

We review an algorithm, in the context of gauge and gravity theories described by differential forms, to read off the symmetries of a physical system out of its action, which was originally proposed in [C. Corral and Y. Bonder, Symmetry algebra in gauge theories of gravity, Class. Quantum Grav. 36 (2019) 045002]. In particular, we study the interplay between gauge symmetries and a gauge-covariant version of diffeomorphisms dubbed local translations. We focus on the role of boundary terms and conserved charges that were not discussed in the original proposal. We provide a review of the formalism and analyze several examples of classical theories, including gauge theories coupled to matter fields in flat space and gravitational theories in different dimensions. [ABSTRACT FROM AUTHOR]

The propagation of a cylindrical shock wave in rotating medium with azimuthal magnetic field under the action of monochromatic radiation using a method of group invariance is investigated. To derive similarity solutions as well as exact solutions, the group invariance technique is used. All classes of the solutions depending on the absorption coefficient are discussed by considering absorption coefficient to be variable or constant. A similarity solution is obtained, when the absorption coefficient is assumed to be variable. Two cases of solutions with a power law shock path are obtained by the different choices of arbitrary constants involving in the infinitesimal generators of the Lie group of transformations. To obtain the similarity solution in the case of the power law shock path, the density, magnetic field, axial and azimuthal velocity components are assumed to be varying and obeying power laws in the undisturbed medium. It is observed that with increase in the values of Alfven Mach number, adiabatic exponent and rotational parameter, shock strength decreases. The effects of variation of magnetic field strength, adiabatic exponent, rotational parameter and initial magnetic field variation index on the flow variables and on shock waves are analyzed graphically. Also, all classes of exact solutions are obtained by considering a constant absorption coefficient. [ABSTRACT FROM AUTHOR]

In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Gordan and Noether proved in their fundamental theorem that an hypersurface X = V (F) ⊆ P n with n ≤ 3 is a cone if and only if F has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if n ≥ 4 , by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein K -algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra R = K [ x 0 , ⋯ , x 4 ] / J with J generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds. [ABSTRACT FROM AUTHOR]

In this present paper, we concern a non-smooth higher-order extension of Noether's symmetry theorem for variational isoperimetric problems with delayed arguments. The result is proven to be valid in the class of Lipschitz functions, as long as the delayed higher-order Euler–Lagrange extremals are restricted to those that satisfy the delayed higher-order DuBois-Reymond necessary optimality condition. The important case of delayed isoperimetric optimal control problems is considered as well. [ABSTRACT FROM AUTHOR]

The vanishing phase space generator of the full four-dimensional diffeomorphism-related symmetry group in the context of the Barbero–Immirz–Holst Lagrangian is derived directly, for the first time, from Noether's second theorem. Its applicability in the construction of classical diffeomorphism invariants is reviewed. [ABSTRACT FROM AUTHOR]

A coupled massive Thirring model of two interacting Dirac spinors in 1 + 1 dimensions with fields taking values in a Grassmann algebra is introduced, which is closely related to a SU(1) version of the Grassmannian Thirring model also introduced in this work. The Lax pair for the system is constructed, and its equations of motion are obtained from a zero curvature condition. It is shown that the system possesses several infinite hierarchies of conserved quantities, which strongly confirms its integrability. The model admits a canonical formulation and is invariant under space-time translations, Lorentz boosts and global U(1) gauge transformations, as well as discrete symmetries like parity and time reversal. The conserved quantities associated to the continuous symmetries are derived using Noether's theorem, and their relation to the lower-order integrals of motion is spelled out. New nonlocal integrable models are constructed through consistent nonlocal reductions between the field components of the general model. The Lagrangian, the Hamiltonian, the Lax pair and several infinite hierarchies of conserved quantities for each of these nonlocal models are obtained substituting its reduction in the expressions of the analogous quantities for the general model. It is shown that, although the Lorentz symmetry of the general model breaks down for its nonlocal reductions, these reductions remain invariant under parity, time reversal, global U(1) gauge transformations and space-time translations. [ABSTRACT FROM AUTHOR]

We study the work fluctuations in ergotropic heat engines, namely two-stroke quantum Otto engines where the work stroke is designed to extract the ergotropy (the maximum amount of work by a cyclic unitary evolution) from a couple of quantum systems at canonical equilibrium at two different temperatures, whereas the heat stroke thermalizes back the systems to their respective reservoirs. We provide an exhaustive study for the case of two qutrits whose energy levels are equally spaced at two different frequencies by deriving the complete work statistics. By varying the values of temperatures and frequencies, only three kinds of optimal unitary strokes are found: the swap operator U 1 , an idle swap U 2 (where one of the qutrits is regarded as an effective qubit), and a non-trivial permutation of energy eigenstates U 3 , which indeed corresponds to the composition of the two previous unitaries, namely U 3 = U 2 U 1 . While U 1 and U 2 are Hermitian (and hence involutions), U 3 is not. This point has an impact on the thermodynamic uncertainty relations (TURs), which bound the signal-to-noise ratio of the extracted work in terms of the entropy production. In fact, we show that all TURs derived from a strong detailed fluctuation theorem are violated by the transformation U 3 . [ABSTRACT FROM AUTHOR]