Bifurcations of Magnetic Geodesic Flows on Surfaces of Revolution

IF 1.7 3区物理与天体物理 Q2 PHYSICS, MATHEMATICAL

Russian Journal of Mathematical Physics Pub Date : 2025-05-05 DOI:10.1134/S1061920825600084

I.F. Kobtsev, E.A. Kudryavtseva

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Abstract

We study magnetic geodesic flows invariant under rotations on the 2-sphere. The dynamical system is given by a generic pair of functions \((f,\Lambda)\) in one variable. The topology of the Liouville fibration of the given integrable system near its singular orbits and singular fibers is described. The types of these singularities are computed. The topology of the Liouville fibration on regular 3-dimensional isoenergy manifolds is described by computing the Fomenko–Zieschang invariant. All possible bifurcation diagrams of the momentum mappings of such integrable systems are described. It is shown that the bifurcation diagram consists of two curves in the \((h,k)\)-plane. One of these curves is a line segment \(h=0\), and the other lies in the half-plane \(h\ge0\) and can be obtained from the curve \((a:-1:k) = (f:\Lambda:1)^*\) projectively dual to the curve \((f:\Lambda:1)\) by the transformation \((a:-1:k)\mapsto(a^2/2,k)=(h,k)\).

DOI 10.1134/S1061920825600084

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旋转表面上磁测地线流的分岔

研究了磁场测地线在2球旋转下的不变性。动力系统由一对单变量的泛型函数\((f,\Lambda)\)给出。描述了给定可积系统在其奇异轨道和奇异纤维附近的Liouville纤维的拓扑结构。计算了这些奇异点的类型。通过计算Fomenko-Zieschang不变量，描述了正则三维等能流形上Liouville振动的拓扑结构。描述了这类可积系统动量映射的所有可能的分岔图。结果表明，该分岔图由\((h,k)\) -平面上的两条曲线组成。其中一条曲线为线段\(h=0\)，另一条曲线位于半平面\(h\ge0\)，可以通过变换\((a:-1:k)\mapsto(a^2/2,k)=(h,k)\)从曲线\((a:-1:k) = (f:\Lambda:1)^*\)投影对偶到曲线\((f:\Lambda:1)\)得到。Doi 10.1134/ s1061920825600084

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来源期刊

Russian Journal of Mathematical Physics 物理-物理：数学物理

CiteScore

3.10

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： Russian Journal of Mathematical Physics is a peer-reviewed periodical that deals with the full range of topics subsumed by that discipline, which lies at the foundation of much of contemporary science. Thus, in addition to mathematical physics per se, the journal coverage includes, but is not limited to, functional analysis, linear and nonlinear partial differential equations, algebras, quantization, quantum field theory, modern differential and algebraic geometry and topology, representations of Lie groups, calculus of variations, asymptotic methods, random process theory, dynamical systems, and control theory.