Recent zbMATH articles in MSC 70Hhttps://zbmath.org/atom/cc/70H2021-06-15T18:09:00+00:00WerkzeugA KAM theorem for finitely differentiable Hamiltonian systems.https://zbmath.org/1460.370602021-06-15T18:09:00+00:00"Koudjinan, C. E."https://zbmath.org/authors/?q=ai:koudjinan.c-eThe following KAM theorem is proved for a Hamiltonian of the form \(H(y,x)=K(y)+P(y,x)\), with \(K,P\in C^l(D\times\mathbb{T}^d)\) and \(D\) being a bounded domain in \(\mathbb{R}^d\). Denote by \(\varepsilon\) the \(C^l\)-norm of the perturbation \(P\). If \(K\) is nondegenerate and \(l>2\nu>2d\geq 4\) then all the KAM tori of the integrable system \(K\) with frequency \((\alpha,\tau)\)-Diophantine, with \(\alpha\sim\varepsilon^{1/2-\nu/l}\) and \(\tau=\nu-1\), do survive being slightly deformed. Moreover, the corresponding family of KAM tori fills the phase space up to a set of Lebesgue measure of \(O(\varepsilon^{1/2-\nu/l})\).
\textit{J. Pöschel} [Commun. Pure Appl. Math. 35, 653--696 (1982; Zbl 0542.58015)] proved that a Cantor family of positive measure of KAM tori survives from a small perturbation \(P\) of class \(C^k\), with \(k>3d-1\), of an analytic integrable Hamiltonian. \textit{D. A. Salamon} [Math. Phys. Electron. J. 10, Paper No. 3, 37 p. (2004; Zbl 1136.37348)]
showed that for the persistence of a single torus it is enough that both the integrable and perturbed part are of class \(C^l\) with \(l>2d\). Moreover, \textit{A. Bounemoura} [J. Éc. Polytech., Math. 7, 1113--1132 (2020; Zbl 07268823)]
proved that if \(K\in C^{l+2}\) and \(P\in C^l\) with \(l>2d\), then a family of KAM tori persists filling the phase space up to a set of measure \(O(\sqrt{\varepsilon})\). The present work is a generalization of all these results.
The proof relies on a KAM scheme à la Arnold. First a quantitative KAM step is proved for the real-analytic case. Then one approximates the \(C^l\) functions \(K,P\) by a sequence of real-analytic functions getting a sequence of real analytic Hamiltonian to which the previous KAM step is applied. This approximation is quantitative and performed in a refined manner, not just truncating the Fourier expansion.
Reviewer: Stefano Marò (Pisa)New integrable two-centre problem on sphere in Dirac magnetic field.https://zbmath.org/1460.370552021-06-15T18:09:00+00:00"Veselov, A. P."https://zbmath.org/authors/?q=ai:veselov.alexander-p"Ye, Y."https://zbmath.org/authors/?q=ai:ye.yisong|ye.yutang|ye.yaohua|ye.yanfei|ye.yanan|ye.yeo|ye.yuanqing|ye.yongan|ye.yifan|ye.yuqi|ye.yuzhang|ye.yuanyuan|ye.yongsheng|ye.yuanting|ye.yizheng|ye.yinfang|ye.yuli|ye.yonggang|ye.yangming|ye.yulin|ye.yinzhong|ye.youpei|ye.yibin|ye.yinyu|ye.yuxin|ye.yin|ye.yicheng|ye.yongxin|ye.yining|ye.yusong|ye.yanfang|ye.yimao|ye.yingwang|ye.yangbo|ye.yanqian|ye.youda|ye.yunming|ye.yongjian|ye.yihan|ye.yuling|ye.yalan|ye.yafen|ye.yangtian|ye.yunwen|ye.yangdong|ye.yun|ye.yujian|ye.yuexiang|ye.yimin|ye.yugong|ye.yaming|ye.yu|ye.yanyan|ye.yuhuang|ye.yaoyao|ye.yichao|ye.yamei|ye.yuanling|ye.yan|ye.ye|ye.yang|ye.yasheng|ye.yaojun|ye.yongxing|ye.yinna|ye.yongchun|ye.yiwei|ye.yingxian|ye.yupeng|ye.yuan|ye.yiming|ye.yali|ye.yicai|ye.yingyu|ye.yuquan|ye.yingying|ye.yong|ye.yongqiang|ye.yanbing|ye.yiyong|ye.yunhua|ye.yuting|ye.ying|ye.yi|ye.yunguang|ye.yuanyuThe authors consider the Euler two-center problem on the sphere \(S^2\) in the presence of a Dirac monopole with an arbitrary charge. The original Euler two-center problem in two dimensions used a Hamiltonian of the form \(H(q_1,q_2,p_1,p_2) = \frac{1}{2}(p_1^2 + p_2^2) - \frac{\mu}{r_1} - \frac{\mu}{r_2}\) where \(r_1 = \sqrt{(q_1 + c)^2 + q_2^2}\) and \(r_2 = \sqrt{(q_1 - c)^2 + q_2^2}\) are the distances between the two centers fixed at the points \((\pm c,0)\).
To get to an explicit formula for the Hamiltonian, coordinates on the dual space \(e(3)^*\) of the Lie algebra of the group E\((3)\) of rigid motions of Euclidean space are introduced. The variables are \(M_i, q_i\) for \(i = 1,2,3\). The corresponding Poisson brackets are \(\{M_i, M_j\} = \epsilon_{ijk} M_k\), \(\{M_i, q_j\} = \epsilon_{ijk} q_k\), and \(\{q_i, q_j\} = 0\). The aim in this paper is to introduce a new family of integrable systems on \(e(3)^*\) with the Hamiltonian \(H = \frac{1}{2} |M|^2 - \mu \frac{|q|}{\sqrt{R(q)}}\) where \(R(q) = Aq_2^2 + Bq_1^2 + (A+B)q_3^2 - 2\sqrt{AB}|q|q_3\) that depends on real parameters \(\mu, A\), and \(B\) with \(A > B > 0\). These systems can be interpreted as the motion on the unit sphere in the external field of a Dirac magnetic monopole with very specific algebraic potentials that have two singularities of Coulomb type.
The authors' main result is that the new systems are integrable in both the classical and quantum cases for all values of the parameters and all values of the magnetic charge (or for all values of the Casimir function \(C_2 = (M,q)\) ). The additional integral \(F = AM_1^2 + BM_2^2 + \frac{2\sqrt{AB}}{|q|} (M,q)M_3 - 2\mu\sqrt{AB} \frac{q_3}{\sqrt{R(q)}}\) is quadratic in \(M\) but has coefficients that depend algebraically on \(q\).
Reviewer: William J. Satzer Jr. (St. Paul)Reeb's theorem and periodic orbits for a rotating Hénon-Heiles potential.https://zbmath.org/1460.370612021-06-15T18:09:00+00:00"Lanchares, V."https://zbmath.org/authors/?q=ai:lanchares.victor"Pascual, A. I."https://zbmath.org/authors/?q=ai:pascual.ana-isabel"Iñarrea, M."https://zbmath.org/authors/?q=ai:inarrea.manuel"Salas, J. P."https://zbmath.org/authors/?q=ai:salas.jose-pablo"Palacián, J. F."https://zbmath.org/authors/?q=ai:palacian.jesus-f"Yanguas, P."https://zbmath.org/authors/?q=ai:yanguas.patriciaSummary: We apply Reeb's theorem to prove the existence of periodic orbits in the rotating Hénon-Heiles system. To this end, a sort of detuned normal form is calculated that yields a reduced system with at most four non degenerate equilibrium points. Linear stability and bifurcations of equilibrium solutions mimic those for periodic solutions of the original system. We also determine heteroclinic connections that can account for transport phenomena.Dark \(f(\mathcal{R},\varphi,\chi)\) universe with Noether symmetry.https://zbmath.org/1460.830712021-06-15T18:09:00+00:00"Shamir, M. F."https://zbmath.org/authors/?q=ai:shamir.m-farasat"Malik, A."https://zbmath.org/authors/?q=ai:malik.adnan"Ahmad, M."https://zbmath.org/authors/?q=ai:ahmad.mushtaqSummary: Using the Noether symmetry approach, we investigate \(f( \mathcal{R} , \varphi ,\chi)\) theories of gravity, where \(\mathcal{R}\) is the scalar curvature, \( \varphi\) is the scalar field, and \(\chi\) is the kinetic term of \(\varphi \). Based on the Lagrangian for \(f( \mathcal{R} , \varphi ,\chi)\) gravity, we obtain the determining equations. We consider \(f( \mathcal{R} , \varphi ,\chi)\) models of a flat Friedmann-Robertson-Walker universe. Using the obtained solutions, we find conserved quantities. In the framework of this scenario, the continuity equation is extremely important for analyzing the energy density and pressure. Using the first integral of motion, we present a graphical analysis of the energy density, pressure component, and parameter of the equation of state. The negativity of the pressure observed in the considered cases in fact suggests that this theory can describe a Noether universe with dark matter.Asymptotics of eigenfunctions of the bouncing ball type of the operator \(\nabla D(x)\nabla\) in a domain bounded by semirigid walls.https://zbmath.org/1460.810322021-06-15T18:09:00+00:00"Klevin, A. I."https://zbmath.org/authors/?q=ai:klevin.a-iSummary: We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla\) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated with the simplest periodic trajectories of the corresponding Hamiltonian system with reflections on the domain boundary.Whiskered tori for presymplectic dynamical systems.https://zbmath.org/1460.370592021-06-15T18:09:00+00:00"de la Llave, Rafael"https://zbmath.org/authors/?q=ai:de-la-llave.rafael"Xu, Lu"https://zbmath.org/authors/?q=ai:xu.luThe main result of the paper is a persistence theorem for the whiskered tori in presymplectic systems. The authors formulate an invariance equation for an embedding and a provide a splitting of the tangent space at the range of the embedding in such a way that zeros of the resulting functional equation define whiskered tori with the corresponding stable and unstable splittings. The proof of the persistence result involves adjusting certain parameters, a common procedure in KAM theory. The method for the proof does not use transformation theory, thus allowing easier numerical implementations.
Reviewer: Mircea Crâşmăreanu (Iaşi)A summary on symmetries and conserved quantities of autonomous Hamiltonian systems.https://zbmath.org/1460.370522021-06-15T18:09:00+00:00"Román-Roy, Narciso"https://zbmath.org/authors/?q=ai:roman-roy.narcisoThe existence of symmetries of Hamiltonian and Lagrangian systems is related to the existence of conserved quantities. As it is well known, the standard procedure to obtain conserved quantities consists in introducing the so-called Noether symmetries, and then use the Noether theorem. However, these kinds of symmetries do not exhaust the set of symmetries. As it is known, there are symmetries which are not of Noether type, but they also generate conserved quantities and they are sometimes called hidden symmetries. In this paper the author establishes a complete scheme of classification of all the different kinds of symmetries of Hamiltonian systems, explaining how to obtain the associated conserved quantities in each case. The author follows the same lines of argument as in the analysis made in [\textit{W. Sarlet} and \textit{F. Cantrijn}, J. Phys. A, Math. Gen. 14, 479--492 (1981; Zbl 0464.58010)] for nonautonomous Lagrangian systems, where the authors obtained conserved quantities for different kinds of symmetries that do not leave the Poincaré-Cartan form invariant.
Reviewer: Nicolai K. Smolentsev (Kemerovo)Stäckel equivalence of non-degenerate superintegrable systems, and invariant quadrics.https://zbmath.org/1460.140772021-06-15T18:09:00+00:00"Vollmer, Andreas"https://zbmath.org/authors/?q=ai:vollmer.andreasThis paper seeks a method that determines the Stäckel class of any given \(2\)-dimensional non-degenerate second-order superintegrable system, i.e., for manifolds of arbitrary (including non-constant) curvature. A non-degenerate second-order maximally conformally superintegrable system in dimension \(2\) naturally gives rise to a quadric with position dependent coefficients. It is shown how the system's Stäckel class can be obtained from this associated quadric. The Stäckel class of a second-order maximally conformally superintegrable system is its equivalence class under Stäckel transformations, i.e., under coupling-constant metamorphosis. This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with some preliminaries. The aim of Section 3 is to construct a certain variety that is invariant under Stäckel transform. It is encoded in a quadric, for a given (non-degenerate) \(2D\) second-order superintegrable system. Here the author relates this variety to the set of all flat realisations of a given Stäckel class, i.e., those non-degenerate \(2D\) second-order superintegrable systems inside the given Stäckel class that are realised on a flat geometry. Section 4 deals with further results and Section 5 with discussion and generalisations.
Reviewer: Ahmed Lesfari (El Jadida)