Recent zbMATH articles in MSC 70H06https://zbmath.org/atom/cc/70H062021-06-15T18:09:00+00:00WerkzeugNew 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)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)