Computer science, Mathematics and Mathematical physics

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A generalized Noether theorem for scaling symmetry


ABSTRACT

The recently discovered conserved quantity associated with Kepler rescaling is generalized by an extension of Noether’s theorem which involves the classical action integral as an additional term. For a free particle, the familiar Schrödinger dilations are recovered. A general pattern arises for homogeneous potentials. The associated conserved quantity allows us to derive the virial theorem. The relation to the Bargmann framework is explained and illustrated by exact plane gravitational waves.


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Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields


ABSTRACT

The integrals of the motion associated with conformal Killing vectors of a curved space–time with an additional electromagnetic background are studied for massive particles. They involve a new term which might be non-local. The difficulty disappears for pp-waves, for which explicit, local conserved charges are found. Alternatively, the mass can be taken into account by “distorting” the conformal Killing vectors. The relation of these non-point symmetries to the charges is analysed both in the Lagrangian and Hamiltonian approaches, as well as in the framework of Eisenhart–Duval lift.


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“Kepler Harmonies” and conformal symmetries


ABSTRACT

Kepler's rescaling becomes, when “Eisenhart-Duval lifted” to 5-dimensional “Bargmann” gravitational wave spacetime, an ordinary spacetime symmetry for motion along null geodesics, which are the lifts of Keplerian trajectories. The lifted rescaling generates a well-behaved conserved Noether charge upstairs, which takes an unconventional form when expressed in conventional terms. This conserved quantity seems to have escaped attention so far. Applications include the Virial Theorem and also Kepler's Third Law. The lifted Kepler rescaling is a Chrono-Projective transformation. The results extend to celestial mechanics and Newtonian Cosmology.


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Shifted Legendre polynomials algorithm used for the dynamic analysis of PMMA viscoelastic beam with an improved fractional model


ABSTRACT

In this paper, a fractional viscoelastic model is proposed to describe the physical behaviour of polymeric material. The material parameters in the model are characterized by the experimental data obtained in the dynamical mechanical analysis. The proposed model is integrated into the fractional governing equation of polymethyl methacrylate (PMMA) above its glass transition temperature. The numerical algorithm based on the shifted Legendre polynomials is retained to solve the fractional governing equations in the time-domain. The accuracy and effectiveness of the algorithm are verified according to the mathematical examples. The advantage of this method is that Laplace transform and the inverse Laplace transform commonly used in fractional calculus are avoided. The dynamical response of the viscoelastic PMMA beam is determined with several loading conditions (uniformly distributed load and harmonic load). The effects of the loading condition and the temperature on the dynamic response of the beam are investigated in the results. The proposed approach shows great potentials for the high-precision calculation in solving the fractional equations in the science and engineering.


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Numerical analysis of fractional viscoelastic column based on shifted Chebyshev wavelet function


ABSTRACT

An innovative numerical procedure for solving the viscoelastic column problem based on fractional rheological models, directly in the time domain, is investigated. Firstly, the governing equation is established according to the fractional constitutive relation. Secondly, the resulting equation is transformed into algebraic equation and solved by using the shifted Chebyshev wavelet function. Furthermore, the convergence analysis and the retained numerical benchmarks are carried out to validate the performance of the proposed method. A small value of the absolute error between numerical and accurate solution is obtained. Finally, the dynamic analysis of viscoelastic beam-column problems is investigated with different cross-section shape (circular and square) under various loading conditions (axial compressive force and harmonic load). The displacement, strain and stress of the viscoelastic column at different time and position are determined. The deformation and stress of the viscoelastic column of different materials under the same loading condition are compared. The results in the paper show the highly accuracy and efficiency of the proposed numerical algorithm in the dynamical stability analysis of the viscoelastic column.


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Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm


ABSTRACT

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.


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Ion traps and the memory effect for periodic gravitational waves


ABSTRACT

The Eisenhart lift of a Paul trap used to store ions in molecular physics is a linearly polarized periodic gravitational wave. A modified version of Dehmelt’s Penning trap is, in turn, related to circularly polarized periodic gravitational waves, sought in inflationary models. Similar equations also govern the Lagrange points in celestial mechanics. The explanation is provided by anisotropic oscillators.


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Standard model as the topological material


ABSTRACT

The study of the Weyl and Dirac topological materials (topological semimetals, insulators, superfluids and superconductors) opens the route for the investigation of the topological quantum vacua of relativistic fields. The symmetric phase of the standard model (SM), where both electroweak and chiral symmetry are not broken, represents the topological semimetal. The vacua of the SM (and its extensions) in the phases with broken electroweak symmetry represent the topological insulators of different types. We discuss in detail the topological invariants in both the symmetric and broken phases and establish their relation to the stability of vacuum.


Dr Erida Gjini

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Numerical simulation of singular conservation laws and related applications


ABSTRACT

We design a scheme for the Euler equations under gravitational fields based on our subcell hydrostatic reconstruction framework.

To give a proper definition of the nonconservative product terms due to the gravitational potential, we first separate the singularity to be an infinitely thin layer, on where  the potential is smoothed  by defining an intermediate potential without disturbing its monotonicity ; then the physical variables are extended and controlled to be consistent with the Rayleigh-Taylor stability, which contribute the positivity-preserving property to keep the nonnegativity of both gas density and pressure even with vacuum states. By using the hydrostatic equilibrium state variables the well-balanced property is obtained to maintain the steady state even with vacuum fronts.  In addition, we proved the full discrete discrete entropy inequality, which preserve the convergence of the solution to the physical solution, with an error term which tends to zero as the mesh size approaches to zero if the potential is Lipschitz continuous. The new scheme is very natural to understand and easy to implement.

The numerical experiments demonstrate the scheme's robustness to resolve the nonlinear waves and vacuum fronts.