论文标题
Sturm理论与几何学和经典力学中的应用
Sturm theory with applications in geometry and classical mechanics
论文作者
论文摘要
经典的sturm sturm非振荡和比较定理以及二阶差分方程解的零上的结构定理具有自然的符号形式，因为它们描述了方程相位平面中线的旋转。在这些定理的较高维度符号版本中，线路被拉格朗日子空间代替，而与给定线路的交叉点被以杰出的lagrangian子空间替换为非转换瞬间。因此，符合性结构定理描述了马斯洛夫指数的某些特性。 从关于光学哈密顿的符合性结实理论的著名论文开始，我们将其结果概括为哈密顿将军。最终，我们将这些结果应用于检测有关半摩尼亚歧管上共轭和焦点分布的一些几何信息，并研究了在天体机械中产生的单数拉格朗日系统解决方案空间的几何特性。
Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol'd on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.
