Hilbert’s Fourth Problem in Two Dimensions I
نویسنده
چکیده
Hilbert’s fourth problems asks to construct and study the geometries in which the straight line segment is the shortest connection between two points. In this paper the reader shall find an elementary introduction to the problem and its solutions in dimension two by Busemann, Pogorelov, and Ambartzumian. The relationship between integral geometry and inverse problems in variational calculus is emphasized.
منابع مشابه
Remarks on Magnetic Flows and Magnetic Billiards, Finsler Metrics and a Magnetic Analog of Hilbert’s Fourth Problem
We interpret magnetic billiards as Finsler ones and describe an analog of the string construction for magnetic billiards. Finsler billiards for which the law “angle of incidence equals angle of reflection” are described. We characterize the Finsler metrics in the plane whose geodesics are circles of a fixed radius. This is a magnetic analog of Hilbert’s fourth problem asking to describe the Fin...
متن کاملOn Kuroda’s proof of Hilbert’s fourteenth problem in dimensions three and four
We generalize [3, Lemma 2.2] and [4, Proposition 2.3] and deduce a positive result on Hilbert’s fourteenth problem. Further, we give a relatively transparent and elementary proof of [3, Theorem 1.1].
متن کاملBifurcation of Limit Cycles in a Fourth-Order Near-Hamiltonian System
This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows thatH(4) ≥ 20, whereH(n) is the Hilbert number for the second part of Hilbert’s 16th problem. It is well known that ...
متن کاملHilbert's 16th Problem and bifurcations of Planar Polynomial Vector Fields
The original Hilbert’s 16th problem can be split into four parts consisting of Problems A–D. In this paper, the progress of study on Hilbert’s 16th problem is presented, and the relationship between Hilbert’s 16th problem and bifurcations of planar vector fields is discussed. The material is presented in eight sections. Section 1: Introduction: what is Hilbert’s 16th problem? Section 2: The fir...
متن کاملar X iv : d g - ga / 9 61 10 10 v 1 2 5 N ov 1 99 6 PROJECTIVELY FLAT FINSLER 2 - SPHERES OF CONSTANT
After recalling the structure equations of Finsler structures on surfaces, I define a notion of ‘generalized Finsler structure’ as a way of micro-localizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of ‘generalized path geometry’ analogous to that of ‘generalized Finsler structu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002