Polynomial and non-polynomial first integrals of projective structures and geodesic flows

IF 1.2 3区数学 Q1 MATHEMATICS

Journal of Geometry and Physics Pub Date : 2025-07-18 DOI:10.1016/j.geomphys.2025.105596

Maria V. Demina , Anna R. Ishchenko

引用次数: 0

Abstract

We develop a method based on the Darboux theory of integrability that is able to produce first integrals of geodesic equations on 2-surfaces. We present local explicit examples of two-dimensional metrics with polynomial in momenta first integrals of arbitrary degrees. We also find metrics admitting transcendental first integrals. In particular, we express some first integrals via the hypergeometric function. Our metrics are parameterized by an arbitrary function of one variable.

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投影结构和测地线流动的多项式和非多项式第一积分

本文提出了一种基于达布可积性理论的求2曲面上测地线方程第一积分的方法。给出了具有多项式的任意度动量第一积分的二维度量的局部显式例子。我们也发现了允许超越第一积分的度量。特别地，我们用超几何函数来表示一些第一积分。我们的指标是由一个变量的任意函数参数化的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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