Recent zbMATH articles in MSC 70Ghttps://zbmath.org/atom/cc/70G2022-07-25T18:03:43.254055ZWerkzeugContinuous and discrete Noether's fractional conserved quantities for restricted calculus of variationshttps://zbmath.org/1487.490272022-07-25T18:03:43.254055Z"Cresson, Jacky"https://zbmath.org/authors/?q=ai:cresson.jacky"Jiménez, Fernando"https://zbmath.org/authors/?q=ai:jimenez.fernando"Ober-Blöbaum, Sina"https://zbmath.org/authors/?q=ai:ober-blobaum.sinaSummary: We prove a Noether's theorem of the first kind for the so-called \textit{restricted fractional Euler-Lagrange equations} and their discrete counterpart, introduced in [\textit{F. Jiménez} and \textit{S. Ober-Blöbaum}, ``A fractional variational approach for modelling dissipative mechanical systems: continuous and discrete settings'', IFAC-PapersOnLine 51, No. 3, 50--55 (2018; \url{doi:10.1016/j.ifacol.2018.06.013}); J. Nonlinear Sci. 31, No. 2, Paper No. 46, 43 p. (2021; Zbl 1477.70031)], based in previous results [\textit{L. Bourdin} et al., Commun. Nonlinear Sci. Numer. Simul. 18, No. 4, 878--887 (2013; Zbl 1328.70013); \textit{F. Riewe}, ``Nonconservative Lagrangian and Hamiltonian mechanics'', Phys. Rev. E (3) 53, 1890--1899 (1996; \url{doi:10.1103/PhysRevE.53.1890})]. Prior, we compare the restricted fractional calculus of variations to the \textit{asymmetric fractional calculus of variations}, introduced in [\textit{J. Cresson} and \textit{P. Inizan}, J. Math. Anal. Appl. 385, No. 2, 975--997 (2012; Zbl 1250.49024)], and formulate the restricted calculus of variations using the \textit{discrete embedding} approach [\textit{L. Bourdin} et al., Appl. Numer. Math. 71, 14--23 (2013; Zbl 1284.65183); \textit{J. Cresson} and \textit{F. Pierret}, ``Continuous versus discete structures I: discrete embeddings and ordinary differential equations'', Preprint, \url{arXiv:1411.7117}]. The two theories are designed to provide a variational formulation of dissipative systems, and are based on modeling irreversbility by means of fractional derivatives. We explicit the role of time-reversed solutions and causality in the restricted fractional calculus of variations and we propose an alternative formulation. Finally, we implement our results for a particular example and provide simulations, actually showing the constant behaviour in time of the discrete conserved quantities outcoming the Noether's theorems.The geodesic flow on nilpotent Lie groups of steps two and threehttps://zbmath.org/1487.531092022-07-25T18:03:43.254055Z"Ovando, Gabriela P."https://zbmath.org/authors/?q=ai:ovando.gabriela-pThis work investigates the geodesic flow on \(k\)-step nilpotent Lie groups for \(k=2,3\). For such Lie groups in dimension at most five, the author obtains a left-invariant metric for which the geodesic flow is Liouville integrable. The main tools used to obtain Liouville integrability are Killing vector fields and symmetric Killing 2-tensors associated to quadratic invariant first integrals of the geodesic flow. The author obtains explicit conditions for a symmetric map to induce a first integral as well as conditions for invariant linear or quadratic polynomials to become first integrals. Several explicit involution formulas for first integrals are obtained. Moreover, they also investigate six-dimensional \(k\)-step nilpotent Lie groups for \(k=2,3\), and in most cases (all but two) show that these groups admit a left-invariant metric for which the geodesic flow is completely integrable.
Reviewer: Julie Rowlett (Gothenburg)Bifurcations of the levels of the first integrals of the problem of the motion of a heavy gyrostathttps://zbmath.org/1487.700062022-07-25T18:03:43.254055Z"Gashenenko, I. N."https://zbmath.org/authors/?q=ai:gashenenko.igor-nSummary: In the problem of the motion of a heavy gyrostat about a fixed point, the bifurcation set for three-dimensional surfaces of integral levels is studied. A special case is considered when the gyrostatic moment is directed along the axis passing through the center of gravity of the gyrostat. It is assumed that only the generators of the Staude cone can be the axes of the uniform rotations of the body. The direction curves of this cone are studied and the bifurcation diagrams on the plane of the constants of the first integrals are classified.The bifurcation diagram of a generalization of the fourth Appel'rot classhttps://zbmath.org/1487.700262022-07-25T18:03:43.254055Z"Kharlamov, M. P."https://zbmath.org/authors/?q=ai:kharlamov.mikhail-pTranslation from the Russian: This paper is a continuation of the author's paper [Mekh. Tverd. Tela 34, 47--58 (2004; Zbl 1487.70027)] in which he obtained generalizations of the Appel'rot classes of motions of the Kovalevskaya top for a double force field. Here he considers an analogue of the fourth Appel'rot class. The trajectories of this family fill a surface that is four-dimensional in a neighborhood of its points of general equilibrium. He finds two partial integrals that form the complete system. He obtains their bifurcation diagram and the range of values of the corresponding constants.The critical set and bifurcation diagram of the problem of the motion of the Kovalevskaya top in a double fieldhttps://zbmath.org/1487.700272022-07-25T18:03:43.254055Z"Kharlamov, M. P."https://zbmath.org/authors/?q=ai:kharlamov.mikhail-pTranslation from the Russian: For a completely integrable system with three degrees of freedom describing the motion of a rigid body in a double force field satisfying Kovalevskaya-type conditions \((A=B=2C\), the centers of force lie in the equatorial plane of the ellipsoid of inertia), the set of critical points of an integral mapping generated by three integrals in involution is found. It consists of invariant subsets on which the induced dynamical system is almost everywhere Hamiltonian with two degrees of freedom. With the critical set its image is associated -- the bifurcation diagram in the space of constants of the first integrals, which lies in the union of three surfaces, two of which are defined by explicit equations, and the third one by parametric equations, in which the role of parameters is played by a constant of one of the general integrals and the multiple root of a polynomial that generalizes the Euler resolvent of the second Kovalevskaya polynomial. The author draws an analogy with the Appel'rot classes in the problem of the motion of the Kovalevskaya top in a gravitational field.Isoenergy surfaces in the problem of the motion of a body with a fixed pointhttps://zbmath.org/1487.700342022-07-25T18:03:43.254055Z"Gashenenko, I. N."https://zbmath.org/authors/?q=ai:gashenenko.igor-nTranslation from the Russian: In the problem of the motion of a heavy rigid body about a fixed point, the bifurcations of three-dimensional integral manifolds (joint level surfaces of energy and area integrals) inside nondegenerate isoenergy surfaces homeomorphic to \(S^4\) and \(S^2\times S^2\) are studied. The critical values of the area constant, which separate the topologically nonequivalent integral manifolds, and also their main projections on the phase space are found. Special canonical variables are introduced. It is establish an analogy between different forms of the phase-space cross sections used in problems of the dynamics of a rigid body.Domains of existence of critical motions of the generalized Kovalevskaya top, and bifurcation diagramshttps://zbmath.org/1487.700392022-07-25T18:03:43.254055Z"Kharlamov, M. P."https://zbmath.org/authors/?q=ai:kharlamov.mikhail-pTranslation from the Russian: This paper completes a cycle of investigations of the bifurcation diagrams of a Hamiltonian system with three degrees of freedom, which describes the motion of an axisymmetric rigid body with Kovalevskaya-type conditions in a double force field. We obtain explicit inequalities that determine the set of critical values of first integrals on surfaces carrying the bifurcation diagram [the author, Mekh. Tverd. Tela 34, 47--58 (2004; Zbl 1487.70027)]. We construct all the diagrams on isoenergy levels whose type is stable with respect to small changes in the physical parameters and the constant of the energy integral.Phase topology of the Kovalevskaya top in an \(\mathrm{SO}(2)\)-symmetric double force fieldhttps://zbmath.org/1487.700542022-07-25T18:03:43.254055Z"Zot'ev, D. B."https://zbmath.org/authors/?q=ai:zotev.dmitrii-bSummary: The phase topology of a completely integrable Hamiltonian system is studied describing the Kovalevskaya top in a double force field under conditions on the parameters that ensure the existence of the symmetry group \(\mathrm{SO}(2)\)-synchronous rotations about the axis of dynamic symmetry and normal to the plane of the force fields. The author computes the bifurcation diagram and the domains of possible motion, and describe geometric features that are characteristic for such a problem. He determines the topological types of the phase space and the isoenergy surfaces, which turn out to be immersed submanifolds with self-intersections, and describe the corresponding singular motions.Bifurcation diagrams on isoenergy levels of the Kovalevskaya top in a double fieldhttps://zbmath.org/1487.700582022-07-25T18:03:43.254055Z"Kharlamov, M. P."https://zbmath.org/authors/?q=ai:kharlamov.mikhail-p"Shvedov, E. G."https://zbmath.org/authors/?q=ai:shvedov.e-gTranslation from the Russian: The authors consider a completely integrable Hamiltonian system with three degrees of freedom describing the motion of a rigid body in a double force field under Kovalevskaya-type conditions \((A=B=2C\), and the force centers lie in the equatorial plane of the ellipsoid of inertia). They consider the plane of independent essential parameters \(h\) (the constant energy integral) and \(\gamma\) (the ratio of the scalar characteristics of the force fields) and the bifurcation diagram \(\Sigma h(\gamma)\) of the map induced on the five-dimensional compact levels of the energy integral by the other two first integrals of our dynamical system. In the plane \(\{h,\gamma\}\) they find the set of points at which the bifurcation diagram \(\Sigma h(\gamma)\) undergoes metamorphoses. The authors give illustrations for the classical Kovalevskaya problem \((\gamma=0)\).Algebraic curves and bifurcation diagrams of two integrable problemshttps://zbmath.org/1487.700592022-07-25T18:03:43.254055Z"Ryabov, P. E."https://zbmath.org/authors/?q=ai:ryabov.p-eTranslation from the Russian: We construct the bifurcation diagrams of two Liouville-integrable Hamiltonian systems on the basis of the isolation of the real part of algebraic surfaces associated with these systems. The first system corresponds to a new case of the integrability of the Kirchhoff equations discovered by V. V. Sokolov. The second system describes the motion of the Kovalevskaya gyrostat in a double force field.A \(K\)-contact Lagrangian formulation for nonconservative field theorieshttps://zbmath.org/1487.700952022-07-25T18:03:43.254055Z"Gaset, Jordi"https://zbmath.org/authors/?q=ai:gaset.jordi"Gràcia, Xavier"https://zbmath.org/authors/?q=ai:gracia.xavier"Muñoz-Lecanda, Miguel C."https://zbmath.org/authors/?q=ai:munoz-lecanda.miguel-c"Rivas, Xavier"https://zbmath.org/authors/?q=ai:rivas.xavier"Román-Roy, Narciso"https://zbmath.org/authors/?q=ai:roman-roy.narcisoSummary: Dynamical systems with dissipative behaviour can be described in terms of contact manifolds and a modified version of Hamilton's equations. Dissipation terms can also be added to field equations, as showed in a recent paper where we introduced the notion of \(k\)-contact structure, and obtained a modified version of the De Donder-Weyl equations of covariant Hamiltonian field theory. In this paper we continue this study by presenting a \(k\)-contact Lagrangian formulation for nonconservative field theories. The Lagrangian density is defined on the product of the space of \(k\)-velocities times a \(k\)-dimensional Euclidean space with coordinates \(s^\alpha\), which are responsible for the dissipation. We analyze the regularity of such Lagrangians; only in the regular case we obtain a \(k\)-contact Hamiltonian system. We study several types of symmetries for \(k\)-contact Lagrangian systems, and relate them with dissipation laws, which are analogous to conservation laws of conservative systems. Several examples are discussed: we find contact Lagrangians for some kinds of second-order linear partial differential equations, with the damped membrane as a particular example, and we also study a vibrating string with a magnetic-like term.Probing the generalized uncertainty principle through quantum noises in optomechanical systemshttps://zbmath.org/1487.830032022-07-25T18:03:43.254055Z"Sen, Soham"https://zbmath.org/authors/?q=ai:sen.soham"Bhattacharyya, Sukanta"https://zbmath.org/authors/?q=ai:bhattacharyya.sukanta"Gangopadhyay, Sunandan"https://zbmath.org/authors/?q=ai:gangopadhyay.sunandanWeyl charges in asymptotically locally \(\mathrm{AdS}_3\) spacetimes in the framework of NMGhttps://zbmath.org/1487.830362022-07-25T18:03:43.254055Z"Setare, M. R."https://zbmath.org/authors/?q=ai:setare.mohammad-reza"Sajadi, S. N."https://zbmath.org/authors/?q=ai:sajadi.seyed-nasehSummary: In this work, following the paper by \textit{F. Alessio} et al. [``Weyl charges in asymptotically locally AdS3 spacetimes'', Phys. Rev. D 103, No. 4, Article ID 046003, 18 p. (2021; \url{doi:10.1103/PhysRevD.103.046003})], we consider Weyl transformations of the boundary metric in Feffermann-Graham gauge in new massive gravity. We construct the phase space, the analyze of the asymptotic structure and the asymptotic symmetry algebra. We show that the surface charges are finite, integrable but not necessarily conserved. Then, we obtain the charge algebra which involves the central extension.Stability of remnants of Bardeen regular black holes in presence of thermal fluctuationshttps://zbmath.org/1487.830932022-07-25T18:03:43.254055Z"Khan, Yawar H."https://zbmath.org/authors/?q=ai:khan.yawar-h"Upadhyay, Sudhaker"https://zbmath.org/authors/?q=ai:upadhyay.sudhaker"Ganai, Prince A."https://zbmath.org/authors/?q=ai:ganai.prince-ahmad