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We prove a time scales version of the Noether theorem relating group of symmetries and conservation laws in the framework of the shifted and nonshifted Δ calculus of variations. Our result extends the continuous version of the Noether theorem as well as the discrete one and corrects a previous statement of Bartosiewicz and Torres [“Noether’s theorem on time scales,” J. Math. Anal. Appl. 342(2), 1220–1226 (2008)]. This result implies also that the second Euler–Lagrange equation on time scales is derived by Bartosiewicz, Martins, and Torres [“The second Euler–Lagrange equation of variational calculus on time scales,” Eur. J. Control 17(1), 9–18 (2011)]. Using the Caputo duality principle introduced by Caputo, [“Time scales: From Nabla calculus to delta calculus and vice versa via duality, Int. J. Differ. Equations 5, 25–40, (2010)], we provide the corresponding Noether theorem on time scales in the framework of the shifted and nonshifted ∇ calculus of variations. All our results are illustrated with numerous examples supported by numerical simulations. [ABSTRACT FROM AUTHOR]

The standard Λ CDM model of cosmology is formulated as a simple modified gravity coupled to a single scalar field ("darkon") possessing a nontrivial hidden nonlinear Noether symmetry. The main ingredient in the construction is the use of the formalism of non-Riemannian spacetime volume-elements. The associated Noether conserved current produces stress–energy tensor consisting of two additive parts — dynamically generated dark energy and dark matter components noninteracting among themselves. Noether symmetry breaking via an additional scalar "darkon" potential introduces naturally an interaction between dark energy and dark matter. The correspondence between the Λ CDM model and the present "darkon" Noether symmetry is exhibited up to linear order with respect to gravity-matter perturbations. With the Cosmic Chronometers (CC) and the Redshift Space Distortion (RSD) datasets, we study an example for the "darkon" potential that breaks the Noether symmetry and we show that the preservation of this symmetry yields a better fit. [ABSTRACT FROM AUTHOR]

We discuss nonminimally coupled cosmologies involving different geometric invariants. Specifically, actions containing a nonminimally coupled scalar field to gravity described, in turn, by curvature, torsion and Gauss–Bonnet scalars are considered. We show that couplings, potentials and kinetic terms are determined by the existence of Noether symmetries which, moreover, allows to reduce and solve dynamics. The main finding of the paper is that different nonminimally coupled theories, presenting the same Noether symmetries, are dynamically equivalent. In other words, Noether symmetries are a selection criterion to compare different theories of gravity. [ABSTRACT FROM AUTHOR]

A complete geometric classification of symmetries of autonomous Hamiltonian systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied. [ABSTRACT FROM AUTHOR]

The present paper analysed the model of a pre-cancerous cell population which describes early carcinogenesis from the view point of invariant solutions associated with conservation laws. The conservation laws related with symmetries of the nonlinear heat equations are obtained and used for constructing analytic solutions. The determination of conservation laws is obtained by mean of Noether's theorem. This result is used to construct a linear Lagrangian of the given pre-cancerous model. The reduction of a three- dimensional system into a corresponding one dimensional system serves to obtained a nonlinear Lagrangian. L'application of Lie group theory to a reduced model produces the combinations of parameters in which the linearization of the system is established and provide the corresponding analytical solutions. [ABSTRACT FROM AUTHOR]

• A fractional nonlocal time-space viscoelasticity theory incorporating the spatio-temporal nonlocality is proposed. • The fractional Taylor expansion series is used to derive a fractional nonlocal time-space model. • The correlation between the intrinsic length and time parameters is discussed. • The spatio-temporal effect becomes substantial when the external and internal spatio-temporal parameters interact to each other. To overcome the long wavelength and time limits of classical elastic theory, this paper presents a fractional nonlocal time-space viscoelasticity theory to incorporate the non-locality of both time and spatial location. The stress (strain) at a reference point and a specified time is assumed to depend on the past time history and the stress (strain) of all the points in the reference domain through nonlocal kernel operators. Based on an assumption of weak non-locality, the fractional Taylor expansion series is used to derive a fractional nonlocal time-space model. A fractional nonlocal Kevin–Voigt model is considered as the simplest fractional nonlocal time-space model and chosen to be applied for structural dynamics. The correlation between the intrinsic length and time parameters is discussed. The effective viscoelastic modulus is derived and, based on which, the tension and vibration of rods and the bending, buckling and vibration of beams are studied. Furthermore, in the context of Hamilton's principle, the governing equation and the boundary condition are derived for longitudinal dynamics of the rod in a more rigorous manner. It is found that when the external excitation frequency and the wavenumber interact with the intrinsic microstructures of materials and the intrinsic time parameter, the nonlocal space-time effect will become substantial, and therefore the viscoelastic structures are sensitive to both microstructures and time. [ABSTRACT FROM AUTHOR]

Based on the frame of elastic theory for unsaturated porous medium, considering the influence of the thermal effect, the effects of the tortuosity of fluid phases on the phase velocity of Rayleigh wave in unsaturated porous media are studied. Firstly, the thermoelastic wave equations for three-phase porous media are established by taking into account the effect of the tortuosity of fluid phases. Secondly, through a theoretical derivation, the dispersion equation of Rayleigh wave for unsaturated porothermoelastic media is given by introducing the potential functions. Finally, the variations of the phase velocity of Rayleigh wave are analyzed with numerical examples. The results demonstrate that the phase velocity of Rayleigh wave is not just related to the frequency but also affected by the tortuosity of pore-fluid. The effect of the tortuosity of pore-liquid on the phase velocity of Rayleigh wave is more significant than that of the tortuosity of pore-gas on the phase velocity of Rayleigh wave. [ABSTRACT FROM AUTHOR]

In this paper, we research Noether's theorems of fractional generalized Birkhoffian systems in terms of classical and combined Caputo derivatives. First, the generalized Pfaff–Birkhoff principle and Birkhoff's equations with classical and combined Caputo derivatives are given. Then, in the case of without and with transforming time, respectively, we obtain two kinds of Noether symmetry and their conserved quantities by the method of time re-parameterization. Finally, by taking case of generalized Birkhoffian systems, we study the relationship between Noether symmetry and Mei symmetry. It is concluded that the generalized Birkhoff equations, Noether identity and Noether conserved quantity obtained by using the generalized Pfaff–Birkhoff principle based on the action functional with dynamical functions after infinitesimal transformation are completely consistent with the criterion equation, structural equation and Mei conserved quantity of the fractional generalized Birkhoffian system, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the surface roughness-dependence of buckling of beam-nanostructures. A new variational formulation of buckling of Euler-Bernoulli rough beams is developed based on the Hamilton's principle. The equation of motion of the beam is obtained with a coupling term that depends on the beam surface roughness. Exact solutions are derived for the buckling configurations and the pre-buckling and postbuckling vibrations of simply supported structures. The derived solutions are used to comprehensively explain influence of surface roughness on buckling characteristics of micro/nano-beams. We reveal that the buckling configurations and postbuckling mode shapes are distorted due to surface roughness. Thus, the beam with a rough surface may exhibit a localized buckling configuration where the buckling energy is confined over a small portion of the beam. In addition, the postbuckling mode shape of simply supported beams with rough surfaces is applied load-dependent. We report a value of the applied axial load at which the postbuckling mode shape is completely distorted, and the beam exhibits a mode shape that is identical to its buckling configuration but with a different amplitude. For the first time, we reveal a postbuckling mode inversion due to surface roughness. We demonstrate that the postbuckling mode shape starts to exhibit an inverted form of its original shape at high load values. The findings presented in this study highlight new insights into buckling characteristics of micro/nano-beams. [ABSTRACT FROM AUTHOR]

A state of matter with a multi-component order parameter can give rise to vestigial order. In the vestigial phase, the primary order is only partially melted, leaving a remaining symmetry breaking behind, an effect driven by strong classical or quantum fluctuations. Vestigial states due to primary spin and charge-density-wave order have been discussed in iron-based and cuprate materials. Here we present the observation of a partially melted superconductivity in which pairing fluctuations condense at a separate phase transition and form a nematic state with broken Z3, i.e., three-state Potts-model symmetry. Thermal expansion, specific heat and magnetization measurements of the doped topological insulators NbxBi2Se3 and CuxBi2Se3 reveal that this symmetry breaking occurs at T nem ≃ 3.8 K above T c ≃ 3.25 K , along with an onset of superconducting fluctuations. Thus, before Cooper pairs establish long-range coherence at Tc, they fluctuate in a way that breaks the rotational invariance at Tnem and induces a crystalline distortion. When an order parameter has multiple components, fluctuations can suppress the ordering partially, leaving behind vestigial order. Here the authors show that nematic superconductivity in electron-doped Bi2Se3 gives way to a vestigial nematic phase driven by fluctuating Cooper pairs. [ABSTRACT FROM AUTHOR]

Lithium-ion rechargeable batteries are widely used in various electronic products and equipment due to their immense benefits in power supplying. The exact remaining useful life (RUL) prediction of lithium-ion batteries has shown excellent achievements in preventing severe economic and security consequences incurred in failing to provide necessary power levels. Recently, the nonlinear-drifted fractional Brownian motion made quite a splash in RUL prediction, since its first hitting time distribution can be approximated by weak convergence theorem and time-space transformation. However, the previous RUL prediction methods based on fractional Brownian motion only considered current state measurement. In this paper, a prediction framework based on nonlinear-drifted fractional Brownian motion with multiple hidden state variables is put forward to estimate RUL. Specifically, all the parameters of nonlinear function are defined as specific hidden state variables of lithium-ion battery degradation model, and all the state measurements are used to posteriorly estimate the distribution of the multiple hidden state variables by unscented particle filter algorithm. Four sets of lithium-ion battery degradation data provided by NASA Ames Research Center are used to validate the proposed prediction framework. According to comparison study with other methods, the proposed prediction framework demonstrates greater precision in the RUL prediction. [ABSTRACT FROM AUTHOR]

Teleparallel gravity has been formulated in the symplectic-geometrical fashion. For that purpose, the symplectic potential and the pre-symplectic structure on the manifold of all solutions of the field equation of teleparallel gravity have been firstly constructed from the Lagrangian density of the theory. That the obtained pre-symplectic structure is a symplectic structure also has been proved. It has been shown that the symplectic potential obtained from the Lagrangian density up to a factor is equal to the superpotential of teleparallel gravity. The Hamiltonian formulation of teleparallel gravity has been derived. Furthermore, the Poisson bracket between arbitrary two classical observables (i.e. real-valued functional) defined on the symplectic manifold of all solutions of the field equation of teleparallel gravity has constructed. The Hamiltonian vector field associated to an observable has been obtained. The formulation of Noether theorem in the symplectic formulation of teleparallel gravity has been investigated. [ABSTRACT FROM AUTHOR]

We investigate the dynamic quantities: momentum, spin and orbital angular momenta (SAM and OAM), and their conversion relationship in the structured optical fields at subwavelength scales, where the spin–orbit interaction (SOI) plays a key role and determines the behaviors of light. Specifically, we examine a nanostructure of a Ag nanoparticle (Ag NP) attached on a cylindrical Ag nanowire (Ag NW) under illumination of elliptically polarized light. These dynamic quantities obey the Noether theorem, i.e., for the Ag nanoparticle with spherical symmetry, the total angular momentum consisting of SAM and OAM conserves; for the Ag NW with translational symmetry, the orbital momentum conserves. Meanwhile, the spin-to-orbital angular momentum conversion is mediated by SOI arising from the spatial variation of the optical potential. In this nanostructure, the conservation of momentum imposes a strict restriction on the propagation direction of the surface plasmon polaritons along the Ag NW. Meanwhile, the orbital momentum is determined by the polarized properties of the excitation light and the topography of the Ag NP. Our work offers insights to comprehend the light behaviors in the structured optical fields in terms of the dynamic quantities and benefits to the design of optical nano-devices based on interactions between spin and orbital degrees of freedom. [ABSTRACT FROM AUTHOR]

Noether's theorem reads: 'A symmetry implies a conservation law'. From a single relativistic energy equation of fluid motion, two conservation equations are obtained in the non-relativistic limit: mass conservation and energy conservation of traditional form. We are concerned with the mass conservation equation and investigate what symmetry implies the mass conservation, and conversely what symmetry the mass conservation implies. This study is guided by the general representation of rotational flows of an ideal compressible fluid satisfying Euler's equation derived by Kambe in 2013, giving a hint of existence of a set of gauge fields. Thus our physical system is a combined system consisting of a fluid flow and a background field described with a set of gauge fields. The gauge invariance of the latter fields assures the law of mass conservation. Conversely as far as the mass conservation law is valid, the gauge invariance is assured for the action representing interaction between the two components of the combined fields. The proposed background field has wavy nature. Present combined system describes wave excitation by fluid flows, both irrotational and rotational, and enables rederivation of the Lighhthill's equation of aerodynamic sound with a correction term naturally. Rotational waves are also excited in shear flows with their phase speed different from the sound speed. [ABSTRACT FROM AUTHOR]

Continuity as appears to us immediately by intuition (in the flow of time and in motion) differs from its current formalization, the arithmetical continuum or equivalently the set of real numbers used in modern mathematical analysis. Motivated by the known mathematical and physical problems arising from this formalization of the continuum, our aim in this paper is twofold. Firstly, by interpreting Chaitin's variant of Gödel's first incompleteness theorem as an inherent uncertainty or fuzziness of the arithmetical continuum, a formal set-theoretic entropy is assigned to the arithmetical continuum. Secondly, by analyzing Noether's theorem on symmetries and conserved quantities, we argue that whenever the four dimensional space-time continuum containing a single, stationary, asymptotically flat black hole is modeled by the arithmetical continuum in the mathematical formulation of general relativity, the hidden set-theoretic entropy of this latter structure reveals itself as the entropy of the black hole (proportional to the area of its "instantaneous" event horizon), indicating that this apparently physical quantity might have a pure set-theoretic origin, too. [ABSTRACT FROM AUTHOR]

In recent years, several gauge-symmetric particle-in-cell (PIC) methods have been developed whose simulations of particles and electromagnetic fields exactly conserve charge. While it is rightly observed that these methods' gauge symmetry gives rise to their charge conservation, this causal relationship has generally been asserted via ad hoc derivations of the associated conservation laws. In this work, we develop a comprehensive theoretical grounding for charge conservation in gauge-symmetric Lagrangian and Hamiltonian PIC algorithms. For Lagrangian variational PIC methods, we apply Noether's second theorem to demonstrate that gauge symmetry gives rise to a local charge conservation law as an off-shell identity. For Hamiltonian splitting methods, we show that the momentum map establishes their charge conservation laws. We define a new class of algorithms – gauge-compatible splitting methods – that exactly preserve the momentum map associated with a Hamiltonian system's gauge symmetry – even after time discretization. This class of algorithms affords splitting schemes a decided advantage over alternative Hamiltonian integrators. We apply this general technique to design a novel, explicit, symplectic, gauge-compatible splitting PIC method, whose momentum map yields an exact local charge conservation law. Our study clarifies the appropriate initial conditions for such schemes and examines their symplectic reduction. [ABSTRACT FROM AUTHOR]

A phase-space version of the ideal magnetohydrodynamic (MHD) Lagrangian is derived from first principles and shown to give a relabeling transformation when a cross-helicity constraint is added in Hamilton's Action Principle. A new formulation of time-dependent "relaxed" magnetohydrodynamics is derived using microscopic conservation of mass and macroscopic constraints on total magnetic helicity, cross helicity, and entropy under variations of density, pressure, fluid velocity, and magnetic vector potential. This gives Euler–Lagrange equations consistent with previous work on both ideal and relaxed MHD equilibria with flow, but generalizes the relaxation concept from statics to dynamics. The application of the new dynamical formalism is illustrated for short-wavelength linear waves, and the interface connection conditions for Multiregion Relaxed MHD (MRxMHD) are derived. The issue of whether E + u × B = 0 should be a constraint is discussed. [ABSTRACT FROM AUTHOR]

In contrast to the symmetries of translation in space, rotation in space, and translation in time, the known laws of physics are not universally invariant under transformation of scale. However, a special case exists in which the action is scale invariant if it satisfies the following two constraints: 1) it must depend upon a scale-free Lagrangian, and 2) the Lagrangian must change under scale in the same way as the inverse time, 1t. Our contribution lies in the derivation of a generalised Lagrangian, in the form of a power series expansion, that satisfies these constraints. This generalised Lagrangian furnishes a normal form for dynamic causal models–state space models based upon differential equations–that can be used to distinguish scale symmetry from scale freeness in empirical data. We establish face validity with an analysis of simulated data, in which we show how scale symmetry can be identified and how the associated conserved quantities can be estimated in neuronal time series. Author summary: Considerations of the way in which a dynamical system changes under transformation of scale offer insight into its operational principles. Scale freeness is a paradigm that has been observed in a variety of physical and biological phenomena and describes a situation in which appropriately scaling the space and time coordinates of any evolution of the system yields another possible evolution. In the brain, scale freeness has drawn considerable attention, as it has been associated with optimal information transmission capabilities. Scale symmetry describes a special case of scale freeness, in which a system is perfectly unchanged under transformation of scale. Noether's theorem tells us that in a system that possesses such a symmetry, an associated conservation law must also exist. Here we show that scale symmetry can be identified, and the related conserved quantities measured, in both simulations and real-world data. We achieve this by deriving a generalised equation of motion that leaves the action invariant under spatiotemporal scale transformations and using a modified version of Noether's theorem to write the associated family of conservation laws. Our contribution allows for the first such statistical characterisation of the quantity that is conserved purely by virtue of scale symmetry. [ABSTRACT FROM AUTHOR]

Nonlinear WKB is a multiscale technique for studying locally plane-wave solutions of nonlinear partial differential equations (PDEs). Its application comprises two steps: (1) replacement of the original PDE with an extended system separating the large scales from the small and (2) reduction of the extended system to its slow manifold. In the context of variational fluid theories with particle relabeling symmetry, nonlinear WKB in the mean Eulerian frame is known to possess a variational structure. This much has been demonstrated using, for instance, the theoretical apparatus known as the generalized Lagrangian mean. On the other hand, the variational structure of nonlinear WKB in the conventional Eulerian frame remains mysterious. By exhibiting a variational principle for the extended equations from step (1) above, we demonstrate that nonlinear WKB in the Eulerian frame is in fact variational. Remarkably, the variational principle for the extended system admits loops of relabeling transformations as a symmetry group. Noether's theorem therefore implies that the extended Eulerian equations possess a family of circulation invariants parameterized by S1. As an illustrative example, we use our results to systematically deduce a variational model of high-frequency acoustic waves interacting with a larger-scale compressible isothermal flow. [ABSTRACT FROM AUTHOR]

This article investigates wave propagation behavior of a multi-phase nanocrystalline nanobeam subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton's principle is employed to derive the governing equations which are solved by applying an analytical method. The frequencies are compared with those of nonlocal and couple stress-based beams. It is showed that wave frequencies and phase velocities of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, magnetic field, surface effect and nonlocality. [ABSTRACT FROM AUTHOR]

Fundamental physics contains an important link between properties of elementary particles and continuous symmetries of particle systems. For example, properties such as mass and spin are said to be 'associated' with specific continuous symmetries. This link has played a key role in the discovery of various new particle kinds, but more importantly: it is thought to provide a deep insight into the nature of physical reality. The link between properties and symmetries has been said to call for a radical revision of perceived metaphysical orthodoxy. However, if we are to use claims about a link between properties and symmetries in the articulation of metaphysical views, we first need to develop a sufficiently precise understanding of the content of these claims. The goal of this paper is to do just that. • Physics suggests an important link between symmetries and physical properties. • This link has been used to make substantive metaphysical claims. • This paper explains this link by examining its standard mathematical description. • The main result of this paper is the notion of a fine-grained invariant. • Fine-grained invariance is narrower than standard property invariance. [ABSTRACT FROM AUTHOR]

These notes provide a detailed catalog of surface charge formulas for different classes of gravity theories. The present catalog reviews and extends the existing literature on the topic. Part of the focus is on reviewing the method to compute quasi-local surface charges for gauge theories in order to clarify conceptual issues and their range of applicability. Many surface charge formulas for gravity theories are expressed in metric, tetrads-connection, Chern–Simons connection, and even BF variables. For most of them, the language of differential forms is exploited and contrasted with the more popular metric components language. The gravity theory is coupled with matter fields as scalar, Maxwell, Skyrme, Yang–Mills, and spinors. Furthermore, three examples with ready-to-download notebook codes, show the method in full action. Several new results are highlighted through the notes. [ABSTRACT FROM AUTHOR]

In this paper, the generalized Noether's theorem for mechanical systems is extended to the classical fields with variable mass, i.e., to the corresponding continuous systems. Noether's theorem is based on the modified Lagrangian, which, besides time derivatives of the field function, contains its partial derivatives with respect to the space coordinates. The generalized Noether's theorem for the classical fields systems with variable mass enables us to find transformations of field functions and independent variables for which there are some integrals of motion. In the paper, Noether's theorem is adopted for non-conservative fields, and energy integrals in a broader sense are determined. In the case of non-conservative fields, a complementary approach to the problem is introduced by applying so-called pseudo-conservative fields. It has been demonstrated that the pseudo-conservative systems have the same energy laws as the non-conservative fields where the laws are obtained by means of this generalized Noether's theorem. As the special case, the natural classical fields with standard Lagrangian are considered. [ABSTRACT FROM AUTHOR]

By considering the closure property of a Lagrangian multiform as a conservation law, we use Noether's theorem to show that every variational symmetry of a Lagrangian leads to a Lagrangian multiform. In doing so, we provide a systematic method for constructing Lagrangian multiforms for which the closure property and the multiform Euler–Lagrange (EL) both hold. We present three examples, including the first known example of a continuous Lagrangian 3-form: a multiform for the Kadomtsev–Petviashvili equation. We also present a new proof of the multiform EL equations for a Lagrangian k-form for arbitrary k. [ABSTRACT FROM AUTHOR]

This paper considers pattern forming nonlinear models arising in the study of thermal convection and continuous media. A primary method for the deriva- tion of symmetries and conservation laws is Noether's theorem. However, in the ab- sence of a Lagrangian for the equations investigated, we propose the use of partial Lagrangians within the framework of calculating conservation laws. Additionally, a nonlinear Kuramoto-Sivashinsky equation is recast into an equation possessing a per- turbation term. To achieve this, the knowledge of approximate transformations on the admissible coefficient parameters is required. A perturbation parameter is suit- ably chosen to allow for the construction of nontrivial approximate symmetries. It is demonstrated that this selection provides approximate solutions. [ABSTRACT FROM AUTHOR]

Summary: Because of the existence of rotational boom motion, the load sway characteristics is more complex. In particular, when the sway presents double‐pendulum phenomenon, the design of the controller is more challenging. Furthermore, the uncertain parameters and external disturbances in crane system make it difficult for traditional control methods to obtain satisfactory control performance. Hence, this paper presents an adaptive nonlinear controller based on the dynamic model of double‐pendulum rotary crane. Unlike a traditional method, the proposed one does not need to linearize the crane system for controller design; therefore, the control performance can be guaranteed even if the system states are far away from the equilibrium point. By using Lyapunov technique and LaSalle's invariance theorem, it is strictly proved that the whole control system is asymptotically stable at the equilibrium point. The effectiveness of the presented controller is demonstrated via comparative simulations. [ABSTRACT FROM AUTHOR]

This paper systematically investigates the Lie symmetry analysis of the time-fractional Buckmaster equation in the sense of Riemann–Liouville fractional derivative. With the aid of infinitesimal symmetries, this equation is transformed into a nonlinear ordinary differential equation of fractional order (FODE), where the fractional derivatives are in Erdelyi–Kober sense. The reduced FODE is solved with the explicit power series method and some figures for the obtained power series solutions are also depicted. Finally, Ibragimov's method and Noether's theorem have been employed to conclude the conservation laws of this equation. [ABSTRACT FROM AUTHOR]

A group analysis of the second-order equation including the one-dimensional gas dynamics equations in Lagrangian coordinates as a particular case is performed. The use of Lagrangian coordinates makes it possible to consider the one-dimensional gas dynamics equations as a variational Euler-Lagrange equation with an appropriate Lagrangian. Conservation laws are derived with the use of the variational presentation and Noether's theorem. A complete group classification of the Euler-Lagrange equation is obtained; as a result, 18 different classes can be identified. [ABSTRACT FROM AUTHOR]

We propose two models of the Boltzmann equation (BGK and Fokker-Planck models) for rarefied flows of diatomic gases in vibrational non-equilibrium. These models take into account the discrete repartition of vibration energy modes, which is required for high temperature flows, like for atmospheric re-entry problems. We prove that these models satisfy conservation and entropy properties (H-theorem), and we derive their corresponding compressible Navier–Stokes asymptotics. [ABSTRACT FROM AUTHOR]

This paper deals with the symmetry analysis of the Einstein Cartan Theory [E. Cartan, C. R. Acad. Sci. (Paris)174, 593 (1922); E. Cartan, Ann. Sci. Ec. Norm. Super40, 325 (1923)] which is an extension of the general relativity and it accounts for the spacetime torsion [S. Basilakos et al., Phys. Rev. D88, 103526 (2013)]. We begin by applying Noether theorem [S. Capozziello, La Rivista del Nuovo Cimento (1978–1999) 19(4), 1 (1996)] to the Lagrangian of the FRW type cosmology with torsion and choose a point transformation: (a , ϕ , N) → (u , v , W) , such that one of the transformed variables is cyclic [S. Capozziello, M. De Laurentis and S. D. Odintsov, Eur. Phys. J. C72, 2068 (2012)] for the Lagrangian. Then using the conserved charge [S. Capozziello, M. De Laurentis and S. D. Odintsov, Eur. Phys. J. C72, 2068 (2012)], which is obtained by applying Noether theorem, and the constant of motion, we get the solutions and conclude that due to the presence of torsion, the FRW type cosmology is in the de Sitter phase [D. Kranas, C. G. Tsagas, J. D. Barrow and D. Iosifidis, Eur. Phys. J. C79, 341 (2019)]. [ABSTRACT FROM AUTHOR]

Piezoelectric materials can be controlled with current (or charge) as the electrical input, instead of voltage. The main purpose of this paper is to derive the governing equations for a current-controlled piezo-electric beam and to investigate stabilizability. The magnetic permeability in piezo-electric materials is generally neglected in models. However, it has a significant qualitative effect on properties of the control system such as stabilizability. Besides the consideration of current control, there are several new aspects to the model. Most importantly, a fully dynamic magnetic model is included. Also, electrical potential and magnetic vector potential are chosen to be quadratic-through thickness to include the induced effects of the electromagnetic field. Hamilton's principle is used to derive a boundary value problem that models a single piezo-electric beam actuated by a current (or charge) source at the electrodes. Two sets of decoupled system of partial differential equations are obtained; one for stretching of the beam and another one for bending motion. Since current (or charge) controller only affects the stretching motion, attention is focused on control of the stretching equations in this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations. A Coulomb type gauge condition is used. This gauge condition decouples the electrical potential equation from the equations of the magnetic potential. A semigroup approach is used to prove that the Cauchy problem is well-posed. Unlike voltage actuation, a bounded control operator in the natural energy space is obtained. The paper concludes with analysis of stabilizability and comparison with other actuation approaches and models. [ABSTRACT FROM AUTHOR]

Summary: In this paper, we explore how to get the information of input‐output coupling parameters (IOCPs) for a class of uncertain discrete‐time systems by using iterative learning technique. Firstly, by taking advantage of repetitiveness of control system and informative input and output data, we design an iterative learning scheme for unknown IOCPs. It is shown that we can get the exact values of IOCPs one by one through running the repetitive system T+1 times if the control system is with identical initial state and noise free. Secondly, we give the iterative learning scheme for unknown IOCPs in the presence of measurement noise, system noise, or initial state drift and analyze the influence factors on the performance of developed iterative learning scheme. Meanwhile, we introduce the maximum allowable control deviation into the iterative learning mechanism to minimize the negative impact of noise on the performance of learning scheme and to enhance the robust of iterative learning scheme. Thirdly, for a class of multiple‐input–multiple‐output systems, we also develop iterative learning mechanism for unknown input‐output coupling matrices. Finally, an illustrative example is given to demonstrate the effectiveness of proposed iterative learning scheme. [ABSTRACT FROM AUTHOR]

Hamilton's principle is applied to obtain the equations of motion for fully relativistic collision-free plasma. The variational treatment is presented in both the Eulerian and Lagrangian frameworks. A Clebsch representation of the plasma fluid equations shows the connection between the Lagrangian and Eulerian formulations, clarifying the meaning of the multiplier in Lin's constraint. The existence of a fully relativistic hydromagnetic Cauchy invariant is demonstrated. The Lagrangian approach allows a straightforward determination of the Hamiltonian density and energy integral. The definitions of momentum, stress, and energy densities allow one to write the conservation equations in a compact and covariant form. The conservation equations are also written in an integral form with an emphasis on a generalized virial theorem. The treatment of boundary conditions produces a general expression for energy density distribution in plasma fluid. [ABSTRACT FROM AUTHOR]

Under study in this paper is a time fractional generalized nonlinear diffusion equation which can be better to express diffusion phenomena than diffusion equation of integer order. Firstly, we apply the symmetry analysis method to find the symmetry of this considered equation. Then some conservation laws can also be constructed through the above obtained symmetry with the help of the Noether's theorem. Next, we reduce this equation into an ordinary differential equation of fractional order in the symmetry with Erdélyi–Kober fractional differential operator under one-dimensional subalgebras optimal system framework. Finally, some exact solutions contain the invariant solutions have found for this given equation. The results give us a new interpretation of this type diffusion process. [ABSTRACT FROM AUTHOR]

• Flows of one-dimensional continuum in Lagrangian coordinates are studied. • Lagrangian coordinates used to find new conservation laws. • Using Noether's theorem, conservation laws are found. The equations describing the flow of a one-dimensional continuum in Lagrangian coordinates are studied in this paper by the group analysis method. They are reduced to a single Euler–Lagrange equation which contains two undetermined functions (arbitrary elements). Particular choices of these arbitrary elements correspond to different forms of the shallow water equations, including those with both, a varying bottom and advective impulse transfer effect, and also some other motions of a continuum. A complete group classification of the equations with respect to the arbitrary elements is performed. One advantage of the Lagrangian coordinates consists of the presence of a Lagrangian, so that the equations studied become Euler–Lagrange equations. This allows us to apply Noether's theorem for constructing conservation laws in Lagrangian coordinates. Not every conservation law in Lagrangian coordinates has a counterpart in Eulerian coordinates, whereas the converse is true. Using Noether's theorem, conservation laws which can be obtained by the point symmetries are presented, and their analogs in Eulerian coordinates are given, where they exist. [ABSTRACT FROM AUTHOR]

The paper considers target tracking system with multiplicative noise. Gaussian white, colored and non-Gaussian multiplicative noise are taken into account. We present optimal filtering algorithms to settle tracking system contaminated by Gaussian white and colored multiplicative noise and a suboptimal filtering algorithm to tackle the non-Gaussian case. The proposed optimal filtering algorithms are derived by using projection theorem and the suboptimal filtering algorithm is deduced by applying variational Bayesian inference. Simulation results verify the effectiveness of these presented algorithms. [ABSTRACT FROM AUTHOR]

An edge e = uv of graph G = (V, E) is said to edge-vertex dominate vertices u and v as well as all vertices adjacent to u and v. A set S ⊆ E is called a double edge-vertex dominating set, if every vertex of V is edge-vertex dominated by at least two edges of S. The minimum cardinality of a double edge-vertex dominating set of G is the double edge-vertex domination number and is denoted γ dev (G). In this paper, we define double edge-vertex domination. Then we derive formulas for some special classes of graphs on double edge-vertex domination number. [ABSTRACT FROM AUTHOR]

In this paper, we study the (3+1)-dimensional negative-order Korteweg-de Vries equation, which was recently formulated by using the recursion operator of the Korteweg-de Vries equation. Firstly, we eliminate the integral appearing in the equation and use the Lie symmetries of the resultant partial differential equation to perform symmetry reductions. The obtained fourth-order ordinary differential equation is then solved by two methods and closed-form solutions are constructed. In addition, we compute the conserved vectors of the underlying equation by engaging Noether's theorem. [ABSTRACT FROM AUTHOR]

An important application of Lie group theory of differential equations is applied to study conservation laws of time-fractional Zakharov-Kuznetsov-Burgers (ZKB) equation with Riemann-Liouville and Caputo derivatives. This analysis is based on a modified version of Noether's theorem provided by Ibragimov to construct the conserved vectors of the equation. This is done by non-linearly self-adjointness of the equation which will be stated via a formal Lagrangian in the sequel. [ABSTRACT FROM AUTHOR]

While looking for additional integrals of motion of several minimally superintegrable systems in static electric and magnetic fields, we have realized that in some cases Lie point symmetries of Euler-Lagrange equations imply existence of explicitly time-dependent integrals of motion through Noether's theorem. These integrals can be combined to get an additional time-independent integral for some values of the parameters of the considered systems, thus implying maximal superintegrability. Even for values of the parameters for which the systems don't exhibit maximal superintegrability in the usual sense they allow a completely algebraic determination of the trajectories (including their time dependence). [ABSTRACT FROM AUTHOR]

The force virial is a construct combining information about the current configuration of a mechanical system in motion with information about the acting forces; its long-time average turns out to be proportional to the long-time average of the system's kinetic energy, that is, the virial theorem holds true; a version of this theorem is obtained when a connection between kinetic energy and temperature is established, as it happens in the kinetic theory of gases or in classical equilibrium statistical mechanics. In fact, a virial theorem holds in whatever mechanics. This paper is an exposition of its various formulations, from the simplest deterministic formulation for a single massy particle in Newtonian motion to the fairly more complicated statistical formulation, which makes the theorem a corollary of the equipartition theorem. [ABSTRACT FROM AUTHOR]

The present work considers the Lie group analysis of a system of linear wave type perturbed systems. The methodology is based on finding approximate symmetry operators of a given system. Approximate conservation laws are found via an approximate version of Noether's theorem. This is based on the modified Noether's method provided by Ibragimov. Finally a numerical method is applied to solve the considered system. [ABSTRACT FROM AUTHOR]