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In this paper, the connections between fundamental solutions and Lie symmetry groups for a class of conformable time fractional partial differential equations (PDEs) with variable coefficient are investigated for the first time. The group-invariant solutions to the considered equations are constructed by means of symmetry group method. Then, the corresponding fundamental solutions for these PDEs are established by taking the inverse Laplace transform of the group invariant solutions. In addition, some examples are introduced to illustrate the effectiveness of this approach. Furthermore, the conservation laws of these fractional PDEs are obtained making use of new Noether theorem. [ABSTRACT FROM AUTHOR]

A generalization of the KP equation involving higher-order dispersion is studied. This equation appears in several physical applications. As new results, the Lie point symmetries are obtained and used to derive conservation laws via Noether's theorem by introduction of a potential which gives a Lagrangian formulation for the equation. The meaning and properties of the symmetries and the conserved quantities are described. Explicit sech-type line wave solutions are found and their features are discussed. They are shown to describe dark solitary waves on a background which depends on a dispersion ratio and on the speed and direction of the waves. The zero-background case is explored. [ABSTRACT FROM AUTHOR]

Standard action principle and first Noether theorem are generalized to take into account general form of nonlocality in time (memory) and to describe dissipative and non-Lagrangian nonlinear systems. General fractional calculus (GFC) of the Luchko form is used to take into account a wide class of nonlocalities time compared to the usual fractional calculus. Nonlocality is described by a pair of operator kernels belonging to the Luchko set. Non-holonomic variation equations of Sedov type is used to take into account equations of motion for wide class of dissipative and non-Lagrangian systems. In addition, within the framework of the proposed approach, equations of motion are considered not only with general fractional derivatives, but also for general fractional integrals. Example of application is considered. • Standard action principle is generalized for nonlocal classical mechanics. • Non-holonomic variation equations are generalized for fractional dynamics. • Standard Noether theorem is generalized for nonlocal classical mechanics. [ABSTRACT FROM AUTHOR]

In this paper we study the fourth-order three-dimensional generalized potential Yu-Toda-Sasa-Fukuyama (gpYTSF) equation by first computing its Lie point symmetries and then performing symmetry reductions. The resulting ordinary differential equations are then solved using direct integration, and exact solutions of gpYTSF equation are obtained. The obtained group invariant solutions include the solution in terms of incomplete elliptic integral. Furthermore, conservation laws for the gpYTSF equation are derived using both the multiplier and Noether’s methods. The multiplier method provides eight conservation laws, while the Noether’s theorem supplies seven conservation laws. These conservation laws include the conservation of energy and mass.

The Philosophy and Physics of Noether's Theorems: A Centenary Volume, James Read and Nicholas J. Teh, eds. [ABSTRACT FROM AUTHOR]

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

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'.

We undertake a detailed analysis of the symmetry structures of the plane symmetric and the locally rotationally symmetric (LRS) Bianchi type I spacetimes in the f (R , T) gravity. In particular, we construct all the variational symmetries associated with its Lagrangian and, in some cases, construct the associated conservation laws using Noether's theorem. Giving a comparison between isometries and variational symmetries, we give symmetry structures of some well-known spacetimes. [ABSTRACT FROM AUTHOR]

Gauss integral theorems for electric and magnetic fields, Faraday's law of electromagnetic induction, magnetic field circulation theorem, theorems on the flux and circulation of vector potential, which are valid in curved spacetime, are presented in a covariant form. Covariant formulas for magnetic and electric fluxes, for electromotive force and circulation of the vector potential are provided. In particular, the electromotive force is expressed by a line integral over a closed curve, while in the integral, in addition to the vortex electric field strength, a determinant of the metric tensor also appears. Similarly, the magnetic flux is expressed by a surface integral from the product of magnetic field induction by the determinant of the metric tensor. A new physical quantity is introduced -- the integral scalar potential, the rate of change of which over time determines the flux of vector potential through a closed surface. It is shown that the commonly used four-dimensional Kelvin-Stokes theorem does not allow one to fully deduce the integral laws of the electromagnetic field and in the covariant notation requires the addition of determinant of the metric tensor, besides the fact that the validity of the Kelvin-Stokes theorem is limited to the cases when determinant of metric tensor and the contour area are independent from time. This disadvantage is not present in the approach that uses the divergence theorem and equation for the dual electromagnetic field tensor. The problem of interpreting the law of electromagnetic induction and magnetic field circulation theorem cannot be solved on the basis of the Lorentz force in the absence of charges, and therefore requires a more general approach, when transformation of the field components from the reference frame at rest into the moving reference frame is taken into account. A new effect is predicted, according to which the circulation of magnetic field can appear even in the absence of electric current and with a constant electric field through the contour, if the area of this contour would change. By analogy with electromagnetic induction, for the magnetic field circulation to appear it is important that electric field flux passing through the area of the contour would change over time. [ABSTRACT FROM AUTHOR]

This paper investigates the stability and nonplanar buckling of a current-carrying microwire immersed in a three-dimensional (3D) magnetic field based on modified coupled stress theory and Euler–Bernoulli beam assumptions. Utilizing Hamilton's principle, the partial differential governing equations for 3D vibrations of the current-carrying microwire are derived. These governing equations are then discretized via the Galerkin's approach and solved by a fourth-order Runge–Kutta method. Numerical results show that the slenderness ratio and magnetic field have obvious effect on the stability of the microwire. When the applied magnetic field is along the axial direction of the undeformed microwire, the shape of the postbuckling configuration of the microwire is very sensitive to initial conditions employed. If, however, the magnetic field is deviated from the axial direction of the undeformed microwire, it is interesting that the shape of the postbuckling configuration is independent of initial conditions even if the y- and/or z-component of the 3D magnetic field strength is sufficiently small. Another attractive feature of the results is that the lateral deflection of the microwire can be adjusted by changing the magnetic field strength in each direction, which may have potential application to the microelectromechanical systems (MEMS). Finally, the influence of a 3D magnetic field on the axial displacement of the microwire is analyzed, and it is demonstrated that in practice the axial motion of the microwire may be considered negligible. [ABSTRACT FROM AUTHOR]

By introducing a new nonlocal argument, the Lagrangian formulation of peridynamics is investigated. The peridynamic Euler–Lagrange equation is derived from Hamilton's principle, and Noether's theorem is extended into peridynamics. With the help of the peridynamic Noether's theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby integral are determined. The results show that the peridynamic conservation laws exist only in a spatial integral form rather than in a pointwise form due to nonlocality. In bond-based peridynamics, energy conservation requires that the influence function is independent of the relative displacement field, or energy dissipation will occur. In state-based peridynamics, the angular momentum conservation causes a constraint on the constitutive relation between the force vector-state and the deformation vector-state. The Eshelby integral of peridynamics is given, which can be used to judge nucleation of defects and to calculate the energy release rates caused by damage, fracture and phase transition. [ABSTRACT FROM AUTHOR]

The Noether Isomorphism Theorems and the Zassenhaus Lemma from group theory have a non-pointed version in a suitable categorical context first considered by W. Tholen in his PhD thesis. This article leads to a unification of these results with the ones in the pointed categorical context previously considered by O. Wyler, by working in the framework of star-regular categories introduced by M. Gran, Z. Janelidze and A. Ursini. Some concrete examples of categories where these results hold are examined in detail. [ABSTRACT FROM AUTHOR]

This essay proposes a new notion - the landing zone - in order to identify conceptual features that allow modelers to transfer mathematical tools across disciplinary borders. This discussion refers to the transferable models as 'templates'. Templates are functions, equations, or computational methods that are capable of being generalized from a particular subject matter. There are formal and conceptual prerequisites for the transfer of a template to a new domain. A landing zone is an ontology that contributes to the satisfaction of these conditions for successful transfer. This paper presents a case study on a model in chemistry - the Quantum Theory of Atoms in Molecules (QTAIM) - that makes use of transferred templates from physics - the virial theorem and the wave function. The landing zone in this case is a new ontological notion, that of the topological atom, which prepares ground for the use of the virial theorem and the wave function in chemistry. The virial theorem requires that there exists in-principle stability to the system that it represents, and the wave function requires transformation in its representation that is justified. The ontology of QTAIM - the landing zone for these templates - grounds the scientific use of these templates in the context of chemistry. [ABSTRACT FROM AUTHOR]

We develop a calculus of variations for functionals on a space of laws of continuous stochastic processes, which extends the classical one. We extend Hamilton's least action principle and Noether's theorem to this generalized framework. As an application we obtain, under mild conditions, a stochastic Euler − Lagrange condition and invariants for the critical points of recent problems in stochastic control, namely for semi-martingale optimal transportation problems. [ABSTRACT FROM AUTHOR]

Noetherian forms provide an abstract self-dual context in which one can establish homomorphism theorems (Noether isomorphism theorems and homological diagram lemmas) for groups, rings, modules and other group-like structures. In fact, any semi-abelian category in the sense of G. Janelidze, L. Márki and W. Tholen, as well as any exact category in the sense of M. Grandis (and hence, in particular, any abelian category), can be seen as an example of a noetherian form. In this paper we generalize the notion of a biproduct of objects in an abelian category to a noetherian form and apply it do develop commutator theory in noetherian forms. In the case of semi-abelian categories, biproducts give usual products of objects and our commutators coincide with the so-called Huq commutators (which in the case of groups are the usual commutators of subgroups). This paper thus shows that the structure of a noetherian form allows for a self-dual approach to products and commutators in semi-abelian categories, similarly as has been known for homomorphism theorems. [ABSTRACT FROM AUTHOR]

The dynamics of a spherical body with a non-uniform mass distribution rolling on a plane were discussed by Sergey Chaplygin, whose 150th birthday we celebrate this year. The Chaplygin top is a non-integrable system, with a colourful range of interesting motions. A special case of this system was studied by Edward Routh, who showed that it is integrable. The Routh sphere has a centre of mass offset from the geometric centre, but it has an axis of symmetry through both these points, and equal moments of inertia about all axes orthogonal to the symmetry axis. There are three constants of motion: the total energy and two quantities involving the angular momenta. It is straightforward to demonstrate that these quantities, known as the Jellett and Routh constants, are integrals of the motion. However, their physical significance has not been fully understood. In this paper, we show how the integrals of the Routh sphere arise from Emmy Noether's invariance identity. We derive expressions for the infinitesimal symmetry transformations associated with these constants. We find the finite version of these symmetries and provide their geometrical interpretation. As a further demonstration of the power and utility of this method, we find the Noetherian symmetries and corresponding integrals for a system introduced recently, the Chaplygin ball on a rotating turntable, confirming that the known integrals are directly obtained from Noether's theorem. [ABSTRACT FROM AUTHOR]

The influence of magnetic field on size sensitivity of nonlinear vibration of embedded nanobeams is studied based on von Kármán geometric nonlinearity and nonlocal Timoshenko beam theory. Adopting Hamilton's principle, governing equations for nonlinear vibration of nanobeams are derived and then solved by using Differential Quadrature method to obtain nonlinear responses under different boundary conditions. The results reveal that high magnetic flux restrains nonlinear vibration of nanobeams. At higher magnetic flux, the nonlocal effect could be ignored. For soft foundation, nonlinear vibration of embedded nanobeams is independent on its stiffness variation, and the increase of magnetic flux broadens this range. [ABSTRACT FROM AUTHOR]

Recently, the fractional Noether's theorem derived by G. Frederico and D.F.M. Torres in [10] was proved to be wrong by R.A.C. Ferreira and A.B. Malinowska in (see [7]) using a counterexample and doubts are stated about the validity of other Noether's type Theorem, in particular ([9], Theorem 32). However, the counterexample does not explain why and where the proof given in [10] does not work. In this paper, we make a detailed analysis of the proof proposed by G. Frederico and D.F.M. Torres in [9] which is based on a fractional generalization of a method proposed by J. Jost and X.Li-Jost in the classical case. This method is also used in [10]. We first detail this method and then its fractional version. Several points leading to difficulties are put in evidence, in particular the definition of variational symmetries and some properties of local group of transformations in the fractional case. These difficulties arise in several generalization of the Jost's method, in particular in the discrete setting. We then derive a fractional Noether's Theorem following this strategy, correcting the initial statement of Frederico and Torres in [9] and obtaining an alternative proof of the main result of Atanackovic and al. [3]. [ABSTRACT FROM AUTHOR]

Perturbation to Noether symmetry and adiabatic invariants for Birkhoffian system，Hamiltonian system and Lagrangian system with delta derivative are investigated，respectively. Firstly，the definition and some related properties of time scale calculus are listed simply as preliminaries. Secondly，the Birkhoffian system with delta derivative is studied. Based on the differential equation of motion as well as Noether symmetry and conserved quantity，perturbation to Noether symmetry and adiabatic invariant are investigated. Thirdly，adiabatic invariants for the Hamiltonian system and the Lagrangian system are presented through some transformations. And finally，an example is given to illustrate the methods and results. [ABSTRACT FROM AUTHOR]

In fact, the first law of conservation (that of mass) was found in chemistry and generalized to the conservation of energy in physics by means of Einstein’s famous “E=mc2”. Energy conservation is implied by the principle of least action from a variational viewpoint as in Emmy Noether’s theorems (1918): any chemical change in a conservative (i.e. “closed”) system can be accomplished only in the way conserving its total energy. Bohr’s innovation to found Mendeleev’s periodic table by quantum mechanics implies a certain generalization referring to the quantum leaps as if accomplished in all possible trajectories (according to Feynman’s interpretation) and therefore generalizing the principle of least action and needing a certain generalization of energy conservation as to any quantum change. The transition from the first to the second theorem of Emmy Noether represents well the necessary generalization: its chemical meaning is the generalization of any chemical reaction to be accomplished as if any possible course of time rather than in the standard evenly running time (and equivalent to energy conservation according to the first theorem). The problem: If any quantum change is accomplished in all possible “variations (i.e. “violations) of energy conservation” (by different probabilities), what (if any) is conserved? An answer: quantum information is what is conserved. Indeed, it can be particularly defined as the counterpart (e.g. in the sense of Emmy Noether’s theorems) to the physical quantity of action (e.g. as energy is the counterpart of time in them). It is valid in any course of time rather than in the evenly running one. That generalization implies a generalization of the periodic table including any continuous and smooth transformation between two chemical elements. [ABSTRACT FROM AUTHOR]

We give a presentation of the plane Cremona group over an algebraically closed field with respect to the generators given by the theorem of Noether and Castelnuovo. This presentation is particularly simple and can be used for explicit calculations. [ABSTRACT FROM AUTHOR]

We consider the variational formulation for vertical slice models introduced in Cotter and Holm (Proc Roy Soc, 2013). These models have a Kelvin circulation theorem that holds on all materially-transported closed loops, not just those loops on isosurfaces of potential temperature. Potential vorticity conservation can be derived directly from this circulation theorem. In this paper, we show that this property is due to these models having a relabelling symmetry for every single diffeomorphism of the vertical slice that preserves the density, not just those diffeomorphisms that preserve the potential temperature. This is developed using the methodology of Cotter and Holm (Foundations of Computational Mathematics, 2012). [ABSTRACT FROM AUTHOR]

The article profiles mathematician Amalie Emmy Noether. She was born in Franconian city of Erlangen in 1882, and daughter of algebraic geometer Max Noether. She was permitted to audit courses in modern languages and mathematics at the university of her father. Her dissertation certainly showed her ability to deal complicated algebraic calculations.

Two-dimensional gas dynamics equations in mass Lagrangian coordinates are studied in this paper. The equations describing these flows are reduced to two Euler–Lagrange equations. Using group classification and Noether's theorem, conservation laws are obtained. Their counterparts in Eulerian coordinates are given. Among these counterparts there are new conservation laws. • Lagrangian coordinates used to find new conservation laws. • Complete group classification of the two-dimensional gas dynamics equations of a polytropic gas is performed. • Using Noether's theorem, conservation laws are found. [ABSTRACT FROM AUTHOR]

In this paper, we construct and analyse the conservation laws of a model of a nonlinear Schrödinger equation involving a spatially extended system consisting of two coupled elements. In the first of two cases, we study the space–time system and, separately, the stationary model arising from a standard transformation. In the second case, we analyse a 'toy model PT-symmetric dimer' involving a 'gain-loss amplitude parameter' for which only approximate conserved forms exist. In this latter case, Noether's theorem is not applicable as it is not variational and the conserved forms are adapted in an 'approximate' and novel way due to the nature of the system. [ABSTRACT FROM AUTHOR]

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér's theorem and permits the standard formulation of the quantum Noether's theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner's approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr's thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr's picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics. [ABSTRACT FROM AUTHOR]

A connection between linearized Gauss–Bonnet gravity and classical electrodynamics is found by developing a procedure which can be used to derive completely gauge-invariant models. The procedure involves building the most general Lagrangian for a particular order of derivatives (N) and a rank of tensor potential (M), then solving such that the model is completely gauge-invariant (the Lagrangian density, equation of motion and energy–momentum tensor are all gauge-invariant). In the case of N = 1 order of derivatives and M = 1 rank of tensor potential, electrodynamics is uniquely derived from the procedure. In the case of N = 2 order of derivatives and M = 2 rank of symmetric tensor potential, linearized Gauss–Bonnet gravity is uniquely derived from the procedure. The natural outcome of the models for classical electrodynamics and linearized Gauss–Bonnet gravity from a common set of rules provides an interesting connection between two well-explored physical models. [ABSTRACT FROM AUTHOR]

Eddy current methods are commonly used in the inspection of riveted multilayer structures. In a practical scenario, steel rivets may be magnetized to varying levels which can distort the magnetic field and pose a major challenge since the distortion can result in false positives and negatives. This paper presents an invariance transformation procedure that renders the signal insensitive to changes in magnetization while retaining the sensitivity to defect information. The approach is first implemented on simulated data, where the signals from magnetized steel rivets can be generated under controlled variation of relative permeability. The algorithm is then validated on experimental data acquired using steel rivets with different residual magnetization levels. [ABSTRACT FROM AUTHOR]

Abstract Boundaries in gauge theories are a delicate issue. Arbitrary boundary choices enter the calculation of charges via Noether's second theorem, obstructing the assignment of unambiguous physical charges to local gauge symmetries. Replacing the arbitrary boundary choice with new degrees of freedom suggests itself. But, concretely, such boundary degrees of freedom are spurious—i.e. they are not part of the original field content of the theory—and have to disappear upon gluing. How should we fit them into what we know about field-theory? We resolve these issues in a unified and geometric manner, by introducing a connection 1-form, ϖ , in the field-space of Yang–Mills theory. Using this geometric tool, a modified version of symplectic geometry—here called 'horizontal'—is possible. Independently of boundary conditions, this formalism bestows to each region a physical notion of charge: the horizontal Noether charge. The horizontal gauge charges always vanish, while global charges still arise for reducible configurations characterized by global symmetries. The field-content itself is used as a reference frame to distinguish 'gauge' and 'physical'; no new degrees of freedom, such as group-valued edge modes, are required. Different choices of reference fields give different ϖ 's, which are cousins of gauge-fixings like the Higgs-unitary and Coulomb gauges. But the formalism extends well beyond gauge-fixings, for instance by avoiding the Gribov problem. For one choice of ϖ , would-be Goldstone modes arising from the condensation of matter degrees of freedom play precisely the role of the known group-valued edge modes, but here they arise as preferred coordinates in field space, rather than new fields. For another choice, in the Abelian case, ϖ recovers the Dirac dressing of the electron. [ABSTRACT FROM AUTHOR]

Highlights • Simulations reveal relaxation of keV electrons in metals toward equilibrium. • Faster electrons cause less radiation damage in femtosecond XFEL experiments. • Significant amount of secondary electrons generated near the Fermi surface. • Demonstrates quantum H-theorem in nonequilibrium statistical mechanics. Abstract Collisional relaxation of high-energy primary electrons (PEs) and generation of secondary electrons (SEs) in a metal are simulated self-consistently by the quantum Boltzmann equation taking into account dynamic screening. An efficient scheme to evaluate the multi-dimensional integral in the collision term is introduced, which enables one to compute time evolutions of nonequilibrium energy distribution functions over an energy range up to a few keV until the whole system reaches thermal equilibrium. Numerical results reveal that rapid energy loss of PEs by plasmon excitation occurs in an early stage, followed by a cascade excitation of SEs near the Fermi level. The overall energy distribution eventually converges to the equilibrium Fermi distribution of partially degenerate electron gas, corroborating the quantum-mechanical H -theorem; time evolutions of the corresponding nonequilibrium entropies are also simulated. The electron temperature and total number of SEs in the final state are consistent with thermodynamic values predicted from the initial energy density of PEs. The number of SEs excited during an early stage (initial 10 fs) turns out smaller for higher PE energies; this feature would bear relevance to an estimation of transient radiation damage in a metallic sample irradiated by an intense femtosecond x-ray laser pulse. [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]

• Variational formulation of growth mechanics and study of the inhomogeneities and conservation laws of a growing body. • Determination of a thermodynamically inspired, intrinsic time scale associated with growth, termed 'internal time'. • Solution of a proof of concept to determine the 'internal time' of an idealised growing tumour. Starting from the observation that the growth of a body breaks the time translation symmetry of the body's dynamics, we determine a scalar field, called internal time , that defines an indicator of the intrinsic time scale of the growth-related body's structural evolution. By recasting the theory of growth for monophasic media within a variational framework, we obtain the internal time as the solution of a partial differential equation descending from Noether's Theorem. We do this by considering two approaches, one formulated in terms of internal variables and one adopting the concept of augmented kinematics. [ABSTRACT FROM AUTHOR]

The relation between symmetries and local conservation laws, known as Noether's theorem, plays an important role in modern theoretical physics. As a discrete analog of the differentiable physical system, a good numerical scheme should admit the discrete local conservation laws and inherent mathematical structures. A class of variational schemes constructed for the hydrodynamic-electrodynamic model of lossless free-electron gas in a quasi-neutral background shows good properties in secular simulations of surface plasmon polaritons [Q. Chen et al., Phys. Rev. E 99, 023313 (2019)]. We show the discrete local conservation laws admitted by these schemes. Based on the gauge symmetry of the discrete action functional, a discrete charge conservation law is realized locally, which is consistent with the discrete Euler-Lagrange equations obtained from the variational schemes. Based on the discrete Euler-Lagrange equations, discrete local momentum and energy conservation laws are derived directly, which are rigorous in theory. The preservation of the discrete local conservation laws and Lagrangian symplectic structure ensure that the numerical scheme is correct in physics. [ABSTRACT FROM AUTHOR]

In this work we study the Boussinesq system, which is a natural model for the propagation of long waves on the surface of water with a small amplitude and is used to compute a complete set of local conservation laws of the model through the direct method. In this method, some local multipliers are found to construct the fluxes of the conservation law. These multipliers are used to find new conservation laws via another method such as Noether's theorem, Boyer's formulation, Homotopy operator method and Ibragimov's theorem. It is noteworthy that this paper reviews these methods to compare all obtained fluxes and local conservation laws. [ABSTRACT FROM AUTHOR]

This paper shows anthropic principle's predicting symmetric distribution matter strata, their physics laws, and verifications, concretely deduces characteristic time, energy, and temperature expressions at different scales, discovers four interesting invariant quantities, shows homeomorphic theorem of space map, and naturally presents a supersymmetric scale energy. We further discover that any infinitesimal space has the same proportional structure space; namely, they have renormalization group invariance. Consequently, this paper shows that the region of any nth level Plank-scope is from the nth level Planck scale to the (n+1)​th level Planck scale, where the different matters of the nthlevel Planck scale build up the (n+1)th level Planck scale matter. The branches of physics science for this region include the nth level Planck scale matter dynamics and the nth level Planck scale matter group dynamics. The nth level Planck scale matter group dynamics describe how the nth level Planck scale matter constructs the (n+1)th level Planck scale matter and how the different matters of the nth level Planck scale evolve in the group system. This paper discovers that the different matters below Planck scale can exist with our matter world at the same time and same place and may be some candidates for dark matter; furthermore, this paper shows a relative theorem of matter scale: for the world of any nth level, the matters' sizes are relative, not absolute. Evidently, the discoveries of both the symmetrical distribution scales and the relations among the corresponding different physics laws from infinitesimal to infinitely large scales give a scientific solid development platform for formation of new scientific branches and deeper development of old scientific branches, because we can precisely construct many kinds of scientific theories relevant to all the corresponding matter strata. All the branch sciences of different matter strata up to now naturally need to be included in the framework of the new scientific system of physics. [ABSTRACT FROM AUTHOR]

Abstract A manifold that contains small perturbations will induce a perturbed partial differential equation. The partial differential equation that we select is the Poisson equation – in order to explore the interplay between the geometry of the manifold and the perturbations. Specifically, we show how the problem of symmetry determination, for higher-order perturbations, can be elegantly expressed via geometric conditions. [ABSTRACT FROM AUTHOR]

Abstract Noether's theorem, that local gauge variations of gauge invariant actions are identically conserved (more tautologically, that gauge variations of gauge invariants vanish) was established a century ago. Its converse, in the geometric context: are all identically conserved local symmetric tensors variations of some coordinate invariant action? remains unsolved to this day. We survey its present state and discuss some of our concrete attempts at a solution, including a significant improvement. For notational simplicity, details are primarily given in D = 2 , but we discuss generic D as well. [ABSTRACT FROM AUTHOR]

We show that the standard Lorentz transformations admit an invariant mass (length) scale, such as the Planck scale. In other words, the frame independence of such a scale is built-in within those transformations, and one does not need to invoke the principle of relativity for their invariance. This automatically ensures the frame-independence of the spectrum of geometrical operators in quantum gravity. Furthermore, we show that the above predicts a small but measurable difference between the inertial and gravitational mass of any object, regardless of its size or whether it is elementary or composite. [ABSTRACT FROM AUTHOR]

Abstract The generalized variational principle of Herglotz type provides a variational method for describing nonconservative or dissipative processes. The purpose of this letter is to extend this variational principle to a first order linear nonholonomic system and study the conservation laws of the nonconservative nonholonomic system based on Herglotz variational problem. A new differential variational principle of the nonconservative nonholonomic system is proposed, which is based on Herglotz variational problem. And the differential equations of motion of the system are also obtained. Then, according to the condition for the invariance of the differential variational principle, the conservation theorem based on Herglotz variational problem for the nonconservative nonholonomic system are obtained. The theorem contains the conservation theorem of the nonconservative holonomic system as its special case, which can be reduced to the first Noether's theorem based on Herglotz variational problem under proper conditions. The inverse theorem of the conservation theorem is also provided and proved. An example is given to illustrate the application at the end of this letter. Highlights • Herglotz variational principle of a nonholonomic system is proposed firstly. • The conditions of invariance for a nonholonomic system of Herglotz type are given. • Conservation laws of holonomic and nonholonomic nonconservative systems are obtained. • The corresponding inverse theorem of the conservation theorem is established. • A special case is discussed to illustrate the application of the results we obtained. [ABSTRACT FROM AUTHOR]

Abstract I review and critically examine the four textbook arguments commonly taken to establish that gravitational waves (GWs) carry energy-momentum: 1. the increase in kinetic energy that a GW confers on a ring of test particles, 3.Bondi/Feynman's Sticky Bead Argument of a GW heating up a detector, 3. nonlinearities within perturbation theory, construed as the gravity's contribution to its own source, and 4. the Noether Theorems, linking symmetries and conserved quantities. As it stands, each argument is found to be either contentious, or incomplete in that it presupposes substantive assumptions which the standard exposition glosses over. I finally investigate the standard interpretation of binary systems, according to which orbital decay is explained by the system's energy being dissipated via GW energy-momentum transport. I contend that for the textbook treatment of binary systems an alternative interpretation, drawing only on the general-relativistic equations of motions and the Einstein Equations, is available. It's argued to be even preferable to the standard interpretation. Thereby an inference to the best explanation for GW energy-momentum is blocked. I conclude that a defence of the claim that GWs carry energy can't rest on the standard arguments. [ABSTRACT FROM AUTHOR]

We know from a result due to Noether–Lefschetz that a very general surface of degree at least 4 in ℙ 3 contains only curves which are complete intersections with other surfaces. The main goal of this paper is to construct an explicit and smooth compactification of a parameter space for surfaces in ℙ 3 of degree d , for all sufficiently large d , containing one conic and one line. The construction also applies to surfaces in ℙ 3 containing one plane curve and one line. As an application, we compute the degree of the locus of surfaces of degree d , containing one conic and one line. [ABSTRACT FROM AUTHOR]

Given a noether algebra with a noncommutative resolution, a general construction of new noncommutative resolutions is given. As an application, it is proved that any finite length module over a regular local or polynomial ring gives rise, via suitable syzygies, to a noncommutative resolution. [ABSTRACT FROM AUTHOR]

Rational curves on Hilbert schemes of points on K3 surfaces and generalised Kummer manifolds are constructed by using Brill-Noether theory on nodal curves on the underlying surface. It turns out that all wall divisors can be obtained, up to isometry, as dual divisors to such rational curves. The locus covered by the rational curves is then described, thus exhibiting algebraically coisotropic subvarieties. This provides strong evidence for a conjecture by Voisin concerning the Chow ring of irreducible holomorphic symplectic manifolds. Some general results concerning the birational geometry of irreducible holomorphic symplectic manifolds are also proved, such as a non-projective contractibility criterion for wall divisors. [ABSTRACT FROM AUTHOR]