测地线流动的局部高次多项式积分与广义矢状图法

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2025-08-20 DOI:10.1016/j.geomphys.2025.105629

Sergei Agapov

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引用次数: 0

摘要

我们研究了具有可积测地线流的2曲面上的黎曼度量，使得附加的第一个积分是动量的高次多项式。这个问题归结为寻找一类准线性偏微分方程系统的解，而这些系统是半哈密顿的。构造了大量具有3、4、5次多项式一阶积分的局部显式和隐式可积例子。我们的构造基本上是基于应用广义hodograph方法。

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Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method

We study Riemannian metrics on 2-surfaces with integrable geodesic flows such that an additional first integral is high-degree polynomial in momenta. This problem reduces to searching for solutions to certain quasi-linear systems of PDEs which turn out to be semi-Hamiltonian. We construct plenty of local explicit and implicit integrable examples with polynomial first integrals of degrees 3, 4, 5. Our construction is essentially based on applying the generalized hodograph method.

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来源期刊

Journal of Geometry and Physics 物理-物理：数学物理

CiteScore

2.90

自引率

6.70%

发文量

205

审稿时长

64 days

期刊介绍： The Journal of Geometry and Physics is an International Journal in Mathematical Physics. The Journal stimulates the interaction between geometry and physics by publishing primary research, feature and review articles which are of common interest to practitioners in both fields. The Journal of Geometry and Physics now also accepts Letters, allowing for rapid dissemination of outstanding results in the field of geometry and physics. Letters should not exceed a maximum of five printed journal pages (or contain a maximum of 5000 words) and should contain novel, cutting edge results that are of broad interest to the mathematical physics community. Only Letters which are expected to make a significant addition to the literature in the field will be considered. The Journal covers the following areas of research: Methods of: • Algebraic and Differential Topology • Algebraic Geometry • Real and Complex Differential Geometry • Riemannian Manifolds • Symplectic Geometry • Global Analysis, Analysis on Manifolds • Geometric Theory of Differential Equations • Geometric Control Theory • Lie Groups and Lie Algebras • Supermanifolds and Supergroups • Discrete Geometry • Spinors and Twistors Applications to: • Strings and Superstrings • Noncommutative Topology and Geometry • Quantum Groups • Geometric Methods in Statistics and Probability • Geometry Approaches to Thermodynamics • Classical and Quantum Dynamical Systems • Classical and Quantum Integrable Systems • Classical and Quantum Mechanics • Classical and Quantum Field Theory • General Relativity • Quantum Information • Quantum Gravity