Higher Elastica: Geodesics in the Jet Space

Jet空间中不存在周期测地线

No Periodic Geodesics in Jet Space

本研究针对单实变量实函数k阶jet空间J^k的次黎曼流形结构下哈密顿测地流的周期解存在性问题，通过构造T^*J^k中的作用-角坐标，证明了J^k中不存在周期测地线。

No Periodic normal Geodesics in $J^k(\mathbb{R},\mathbb{R}^n)$

The space of $k$-jets of $n$ real function of one real variable $x$ admits the structure of a Carnot group, which then has an associated Hamiltonian geodesic flow. As in any Hamiltonian flow, a natural question is the existence of periodic solutions. Does the space of $k$-jets have periodic geodesics? This study will demonstrate the integrability of subRiemannian geodesic flow, characterize and classify the subRiemannian geodesics in the space of $k$-jets, and show that they are never periodic.

Generic Properties of Geodesic Flows on Analytic Hypersurfaces of Euclidean Space

Consider the geodesic flow on a real-analytic closed hypersurface $M$ of $\mathbb{R}^n$, equipped with the standard Euclidean metric. The flow is entirely determined by the manifold and the Riemannian metric. Typically, geodesic flows are perturbed by varying the metric. In the present paper, however, only the Euclidean metric is used, and instead the manifold $M$ is perturbed. In this context, analogues of the following theorems are proved: the bumpy metric theorem; a theorem of Klingenberg and Takens regarding generic properties of $k$-jets of Poincaré maps along geodesics; and the Kupka-Sma…

$J^2(\u211d^2,\u211d)$中的混沌次黎曼测地流

Chaotic subRiemannian geodesic flow in $J^2(\mathbb{R}^2,\mathbb{R})$

本文研究$J^2(\u211d^2,\u211d)$空间的次黎曼测地流，通过阿贝尔子群的哈密顿作用构造约化哈密顿量$H_\u03bc$，证明其对部分参数不具备亚纯可积性，进而推导出原测地流的亚纯不可积结论。