Conservative dynamical systems involving strong forces
Conservative dynamical systems involving strong forces
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DOI：
10.1090/s0002-9947-1975-0377983-1
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发表时间：
1975-04
影响因子：
1.3
通讯作者：
W. Gordon
中科院分区：
文献类型：
--
作者：
W. Gordon
我们把在集S：V(X)a处有奇点的势V所对应的保守动力系统看作x-：+S。证明了只要势满足一定的强力(SF)条件，各种“作用量”积分就满足Palais和Smear的条件C。因此，例如，我们在SF系统中建立了缠绕在S周围且具有任意给定拓扑(同伦)型且具有任意给定周期的周期轨迹的存在性，以及绕着S形成任意紧环的周期轨迹的存在性。对于缠绕在S周围并连接两个给定点的轨迹的存在性也得到了类似的结果。SF条件与与势V相关的某些雅可比度量的完备性(在黎曼意义下)密切相关，这一事实允许在SF系统的分析中使用黎曼几何的标准结果。SF条件排除了引力情况，在引力情况下作用积分不满足Palais-Smear条件。与引力势相关的雅可比度规是不完整的。对于SF系统，存在连接两个给定点并围绕S形成任意紧环的轨迹，而在引力两体问题中则不是这样。另一方面，对于SF系统，任何光滑的X周期轨迹族(A固定)都是远离S的有界的，而对于引力系统也不是这样。因此，SF条件的定义是“有良好动机的”，并导致SF系统的行为与引力(和其他弱力)系统之间的某些差异的揭示。
We consider conservative dynamical systems associated with potentials V which have singularities at a set S: V(x) a as x -:+ S. It is shown that various "action" integrals satisfy Condition C of Palais and Smale provided that the potential satisfy a certain strong force (SF) condition. Hence, e.g., we establish the existence in SF systems of periodic trajectories which wind around S and have arbitrary given topological (homotopy) type and which have arbitrary given period, and also periodic trajectories which make arbitrarily tight loops around S. Similar results are also obtained concerning the existence of trajectories which wind around S and join two given points. The SF condition is shown to be closely related to the completeness (in the riemannian sense) of certain Jacobi metrics associated with the potential V, and this fact permits the use of the standard results of riemannian geometry in the analysis of SF systems. The SF condition excludes the gravitational case, and the action integrals do not satisfy the Palais-Smale condition in the gravitational case. The Jacobi metrics associated with gravitational potentials are not complete. For SF systems there exist trajectories which join two given points and make arbitrarily tight loops around S, and this is not the case in the gravitational two body problem. On the other hand, for SF systems any smooth family of X-periodic trajectories (A fixed) is bounded away from S, and this also is not the case for gravitational systems. Thus the definition of the SF condition is "well motivated", and leads to the disclosure of certain differences between the behavior of SF systems and gravitational (and other weak force) systems.