Recent zbMATH articles in MSC 70Hhttps://zbmath.org/atom/cc/70H2023-01-20T17:58:23.823708ZWerkzeugInvariant analysis, optimal system, power series solutions and conservation laws of Kersten-Krasil'shchik coupled KdV-mKdV equationshttps://zbmath.org/1500.350112023-01-20T17:58:23.823708Z"Vinita"https://zbmath.org/authors/?q=ai:vinita.1"Saha Ray, S."https://zbmath.org/authors/?q=ai:saha-ray.santanuSummary: The primary goal of this research is to carry out the closed-form analytical solutions of the generalised \((1+1)\)-dimensional Kersten-Krasil'shchik coupled KdV-mKdV equations, which are frequently used to represent shallow water waves, elastic media, electric circuits, electrodynamics, etc. By following the Lie symmetry analysis for this coupled system, optimal system of subalgebras of the secured symmetries has been constructed. Furthermore, the given system is reduced to the system of ordinary differential equations by utilizing an optimal system of subalgebra. Using power series solutions and their convergence analysis, explicit exact solutions of the reduced ordinary differential equations were obtained. Additionally, the conservation laws of the aforementioned coupled system are derived by applying the ``new conservation theorem'' proposed by Ibragimov.The Herglotz principle and vakonomic dynamicshttps://zbmath.org/1500.370432023-01-20T17:58:23.823708Z"de León, Manuel"https://zbmath.org/authors/?q=ai:de-leon.manuel"Lainz, Manuel"https://zbmath.org/authors/?q=ai:lainz.manuel"Muñoz-Lecanda, Miguel C."https://zbmath.org/authors/?q=ai:munoz-lecanda.miguel-cIn the Herglotz variational principle, one considers a Lagrangian that depends not only on the positions and velocities of the system, but also on the infinitesimal action itself. It has been recently acknowledged that the Herglotz principle provides the dynamics for the Lagrangian counterpart of contact Hamiltonian systems.
In this paper the authors study vakonomic dynamics for contact systems with nonlinear constraints. In order to obtain the dynamics, they consider a space of admissible paths, which are the ones tangent to a given submanifold. Then, they find the critical points of the Herglotz action on this space of paths. This dynamics can be also obtained through an extended Lagrangian, including Lagrange multiplier terms. In the final section, they sketch the relationship between the vakonomic dynamics of contact Lagrangian systems and the Herglotz optimal control problem.
For the entire collection see [Zbl 1482.94007].
Reviewer: Antonio De Nicola (Salerno)On unbounded motions in a real analytic bouncing ball problemhttps://zbmath.org/1500.370472023-01-20T17:58:23.823708Z"Marò, Stefano"https://zbmath.org/authors/?q=ai:maro.stefanoThe paper is concerned with the model of a ball elastically bouncing on a racket moving in the vertical direction according to a given periodic real function \(f(t)\), and the ball is reflected according to the law of elastic bouncing when hitting the racket. The only force acting on the ball is the gravity, with acceleration \(g\). Moreover, the mass of the racket is assumed to be large with respect to the mass of the ball so that the impacts do not affect the motion of the racket.
The main focus concerns the existence of unbounded motion, that is, the velocity of the ball tends to infinity. The first result in this direction is due to \textit{L. D. Pustyl'nikov} [Russ. Math. Surv. 50, No. 1, 145--189 (1995; Zbl 0846.70005); translation from Usp. Mat. Nauk 50, No. 1, 143--186 (1995)], who assumed that \(2\dot{f}(t_0) = g\), for some time \(t_0\). The author claims that with the same proof, it is possible to weaken the condition to
\[
\max \dot{f}\geq \frac{g}{4}.\tag{1}
\]
This paper focuses on improving the upper bound given by (1), more precisely, the author gives an explicit example of a trigonometric polynomial \(p(t)\) with
\[
\max \dot{p} < \frac{g}{4}
\]
admitting unbounded bouncing motions. Indeed, \(p(t)\) admits a continuum of unbounded orbits and moreover belongs to a family of trigonometric polynomial \(p_s (t)\), parametrized by \(s\) in a real open interval \(I\). More precisely, the author proves that if the function \(f(t)\) belongs to a class of trigonometric polynomials of degree 2 then there exists a one-dimensional continuum of initial conditions for which the velocity of the ball tends to infinity. This result improves the previous one by \textit{L. D. Pustyl'nikov} and gives a new upper bound to the applicability of KAM theory to this model. Moreover, the author discusses the optimality of condition (1) and construct the family \(p_s\) proving that the corresponding map admits one unbounded orbit. Thus, he shows how to extend the unbounded orbit previously found to a continuum. It is done a detailed discussion of the result on the persistence of the stable manifold under nonautonomous perturbations.
Reviewer: Martha Alvarez-Ramírez (Ciudad de México)Geometric integration by parts and Lepage equivalentshttps://zbmath.org/1500.530852023-01-20T17:58:23.823708Z"Palese, Marcella"https://zbmath.org/authors/?q=ai:palese.marcella"Rossi, Olga"https://zbmath.org/authors/?q=ai:rossi.olga"Zanello, Fabrizio"https://zbmath.org/authors/?q=ai:zanello.fabrizio.1The authors present a comparison between two approaches to the geometric formulation of calculus of variations: from one point of view the approach based on the variational complex and its representation in terms of differential forms, from the other the description in terms of variational morphisms. In both cases there are decomposition formulas which allows for a geometric description of integration by parts and the authors show how to relate the two decompositions. As a derived result, the authors introduce a recursive formula for the derivation of the Krupka-Betounes equivalents of a Lagrangian form for first-order field theories and generalize it to second-order ones. Let us outline with more details the contents of the work.
The paper is divided into 5 sections, the first one being a short introduction. The second one is a brief summary containing the definitions of the main objects which will be used in the central part of the manuscript. In Section 3 and 4 the authors present their main results, which follow from the comparison between the two approaches previously mentioned. The first and fundamental observation in Section 3 consists in the fact that any 1-contact \((n+1)\)-form can be seen as a variational morphism of codegree 0 and vice versa. Then, the authors show that the decomposition of variational morphisms in terms of a volume and a boundary terms due to Fatibene and Francaviglia, is the counterpart of the decomposition of the form using the interior Euler operator and the residual operator.
In Section 4 the previous comparison is extended to the more complicated case of 1-contact \((n-s+1)\)-form. Using the methods employed in Section 3, the authors provide a decomposition formula for these forms and introduce a local interior Euler operator and residual operator for lower degree forms. These forms can be identified with variational morphisms of codegree s and due to this identification, it is possible to compare the decomposition of variational morphisms in terms of volume and boundary terms with the one derived from the local interior Euler and residual operators: in general, they are different and further analysis is required in order to understand uniqueness and global properties of these alternative integration by parts formulas (it is important to remark that the boundary term in the decomposition of a variational morphism is not uniquely determined so that all the results presented in the paper are coherent with previous ones). Eventually, Section 5 is devoted to the presentation of a recursive formula which allows to derive the Krupka-Betounes equivalent of a Lagrangian form for first-order field theories and to extend it to second-order field theories.
The paper is well organized since the subdivision in sections and subsections is properly adapted to the contents of the work. There are introductory remarks at the beginning of each section which facilitate the reading and allow to rapidly spot the principal results. The authors have tried to make the paper self-contained adding a section containing basic notions and definitions. Nevertheless, a full understanding of the motivation and the results presented can be achieved only after reading of the main references of the paper, which are correctly mentioned whenever required. In particular, [\textit{L. Fatibene} and \textit{M. Francaviglia}, Natural and gauge natural formalism for classical field theories. A geometric perspective including spinors and gauge theories. Dordrecht: Kluwer Academic Publishers (2003; Zbl 1138.81303); \textit{D. Krupka}, Introduction to global variational geometry. Amsterdam: Atlantis Press (2015; Zbl 1310.49001); \textit{M. Palese} et al., SIGMA, Symmetry Integrability Geom. Methods Appl. 12, Paper 045, 45 p. (2016; Zbl 1347.70043)] are highly recommended.
The aim of the authors is to provide a local comparison between two different approaches to geometric integration by parts, leaving global considerations to forthcoming analysis. Therefore, the proofs are mainly based on local descriptions of forms and operators. The theoretical discussion is supported by examples which are clearly presented and help the reader to connect the novelties with known results.
Reviewer: Fabio Di Cosmo (Madrid)Global formulations of Lagrangian and Hamiltonian dynamics on embedded manifoldshttps://zbmath.org/1500.700152023-01-20T17:58:23.823708Z"Lee, Taeyoung"https://zbmath.org/authors/?q=ai:lee.taeyoung"Leok, Melvon"https://zbmath.org/authors/?q=ai:leok.melvon"McClamroch, N. Harris"https://zbmath.org/authors/?q=ai:mcclamroch.n-harrisSummary: This paper provides global formulations of Lagrangian and Hamiltonian variational dynamics evolving on a manifold embedded in \(\mathbb{R}^n\), which appears often in robotics and multi body dynamics. Euler-Lagrange equations and Hamilton's equations are developed in a coordinate-free fashion on a manifold, without relying on local parameterizations that may lead to singularities and cumbersome equations of motion. The proposed intrinsic formulations of Lagrangian and Hamiltonian dynamics are expressed compactly, and they are useful in analysis and computation of the global dynamics. These are illustrated by dynamic systems on the unit-spheres and the special orthogonal group.
For the entire collection see [Zbl 1491.68010].Poisson brackets and truncations in nonlinear reduced fluid models for plasmashttps://zbmath.org/1500.761122023-01-20T17:58:23.823708Z"Tassi, E."https://zbmath.org/authors/?q=ai:tassi.enrico|tassi.emanueleSummary: The Hamiltonian structure for an infinite class of nonlinear reduced fluid models, derived from a Hamiltonian drift-kinetic system, is explicitly provided in terms of the \(N + 1\) fluid moments evolving in each model of the class, with \(N\) an arbitrary positive integer. This improves previous results, in which the existence of the Hamiltonian structure was shown, but the complete explicit expression for the Poisson bracket of each model of the class was not provided. We also show that, whereas the Hamiltonian functional of the fluid models can be derived from that of the drift-kinetic system, by projecting the perturbation of the distribution function onto its truncated series in terms of Hermite polynomials, this is not the case for the Poisson bracket. Indeed, the antisymmetric bilinear form obtained by means of the aforementioned projection, although, interestingly, ``very similar'' to the Poisson bracket of the fluid models, turns out to differ from it. The difference is found to reside in the coefficients \(\mathbb{W}_{(N)l}^{mn}\) of the bilinear form, when the indices are such that \(l + m + n\) is even and \(l \geq N + 1\), \(m \geq N + 1\), \(n \geq N + 1\). We show with a counterexample, related to the case \(N = 2\), that such bilinear form, in general, does not satisfy the Jacobi identity. We provide a physical interpretation of the set of variables \(G_0, G_1, \dots, G_N\), in terms of which the Poisson bracket of the fluid models exhibits a direct-sum structure, and point out an analogy between the present fluid reduction problem and the problem of the truncated quantum harmonic oscillator.Uniform continuity bounds for characteristics of multipartite quantum systemshttps://zbmath.org/1500.810152023-01-20T17:58:23.823708Z"Shirokov, M. E."https://zbmath.org/authors/?q=ai:shirokov.maksim-evgenevichSummary: We consider universal methods for obtaining (uniform) continuity bounds for characteristics of multipartite quantum systems. We pay special attention to infinite-dimensional multipartite quantum systems under the energy constraints. By these methods, we obtain continuity bounds for several important characteristics of a multipartite quantum state: the quantum (conditional) mutual information, the squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information. The continuity bounds for the multipartite quantum mutual information are asymptotically tight for large dimension/energy. The obtained results are used to prove the asymptotic continuity of the \(n\)-partite squashed entanglement, the c-squashed entanglement, and the conditional entanglement of mutual information under the energy constraints.
{\copyright 2021 American Institute of Physics}Complex time method for quantum dynamics when an exceptional point is encircled in the parameter spacehttps://zbmath.org/1500.810332023-01-20T17:58:23.823708Z"Kaprálová-Žďánská, Petra Ruth"https://zbmath.org/authors/?q=ai:kapralova-zdanska.petra-ruthSummary: We revisit the complex time method for the application to quantum dynamics as an exceptional point is encircled in the parameter space of the Hamiltonian. The basic idea of the complex time method is using complex contour integration to perform the first-order adiabatic perturbation integral. In this way, the quantum dynamical problem is transformed to a study of singularities in the complex time plane -- transition points -- which represent complex degeneracies of the adiabatic Hamiltonian as the time-dependent parameters defining the encircling contour are analytically continued to complex plane. As an underlying illustration of the approach we discuss a switch between Rabi oscillations and rapid adiabatic passage which occurs upon the encircling of an exceptional point in a special time-symmetric case.BFV quantization and BRST symmetries of the gauge invariant fourth-order Pais-Uhlenbeck oscillatorhttps://zbmath.org/1500.810522023-01-20T17:58:23.823708Z"Mandal, Bhabani Prasad"https://zbmath.org/authors/?q=ai:mandal.bhabani-prasad"Pandey, Vipul Kumar"https://zbmath.org/authors/?q=ai:pandey.vipul-kumar"Thibes, Ronaldo"https://zbmath.org/authors/?q=ai:thibes.ronaldoSummary: We perform the BFV-BRST quantization of the fourth-order Pais-Uhlenbeck oscillator (PUO) for the first time. We show that although the PUO is not naturally constrained in the sense of Dirac-Bergmann, it is possible to profit from the introduction of suitable constraints in phase space in order to obtain a proper BRST invariant quantum system. Starting from its second-class constrained system description, we use the BFFT formalism to obtain first-class constraints as gauge symmetry generators. After the Abelianization of the constraints, we obtain the conserved BRST charge, the corresponding BRST transformations and proceed further to the BFV functional quantization of the model. We further construct appropriate finite field dependent BRST transformation to establish the interconnections between different BRST invariant effective theories of PUO in different gauges. Our approach sheds light on the open problem of the quantization of general higher derivative quantum field theories.Searching for the charged-current non-standard neutrino interactions at the \(e^- p\) collidershttps://zbmath.org/1500.810762023-01-20T17:58:23.823708Z"Yue, Chong-Xing"https://zbmath.org/authors/?q=ai:yue.chongxing"Cheng, Xue-Jia"https://zbmath.org/authors/?q=ai:cheng.xue-jia"Wang, Yue-Qi"https://zbmath.org/authors/?q=ai:wang.yue-qi"Li, Yan-Yu"https://zbmath.org/authors/?q=ai:li.yan-yuSummary: Considering the theoretical constraints on the charged-current (CC) non-standard neutrino interaction (NSI) parameters in a simplified \(W^\prime\) model, we study the sensitivities of the Large Hadron electron Collider (LHeC) and Future Circular Collider-hadron electron (FCC-he) to the CC NSI parameters through the subprocess \(e^- q \to W^\prime \to v_e q^\prime\). Our results show that the LHeC with \(\mathcal{L} = 100\;\mathrm{fb}^{-1}\) is a little more sensitive to the CC NSI parameters than that of the high-luminosity (HL) LHC, and the absolute values of the lower limits on the CC NSI parameters at the FCC-he are smaller than those of the HL-LHC about two orders of magnitude.Purely radiative Higgs mass in scale invariant modelshttps://zbmath.org/1500.810772023-01-20T17:58:23.823708Z"Ahriche, Amine"https://zbmath.org/authors/?q=ai:ahriche.amineSummary: In this work, we investigate the possibility of having scale invariant (SI) standard model (SM) extensions, where the light CP-even scalar matches the SM-like Higgs instead of being a light dilaton. After deriving the required conditions for this scenario, we show that the radiative corrections that give rise to the Higgs mass can trigger the scalar mixing to the experimentally allowed values. In addition, we discuss the constraints on the parameters space that makes the CP-even scalars properties in a good agreement with all the recent ATLAS and CMS measurements. We illustrate this scenario by considering the SI-scotogenic model as an example, while imposing all the theoretical and experimental constraints. We show that the model is viable and leads to possible modifications of the di-Higgs signatures at current/future with respect to the SM.Exact models of pure radiation in \(R^2\) gravity for spatially homogeneous wave-like Shapovalov spacetimes type IIhttps://zbmath.org/1500.830032023-01-20T17:58:23.823708Z"Osetrin, Konstantin"https://zbmath.org/authors/?q=ai:osetrin.konstantin-e"Osetrin, Evgeny"https://zbmath.org/authors/?q=ai:osetrin.evgeny"Filippov, Altair"https://zbmath.org/authors/?q=ai:filippov.altair-eSummary: Presented are exactly integrable models with pure radiation in \(R^2\) gravity with a cosmological constant related to wave-like Shapovalov spacetimes type II. Spatially homogeneous models of Shapovalov spacetimes were considered. The obtained solutions belong to spaces of type III according to the Bianchi classification and of type N according to the Petrov classification. For the models under consideration, exact solutions for the equations of motion of test particles are obtained in the Hamilton-Jacobi formalism. For the obtained exact models, solutions of the geodesic deviation equations are obtained.
{\copyright 2021 American Institute of Physics}