We consider a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q,t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t → ±∞ and vanishes at a unique point t₀ ∈ ℝ. Let X₊, X₋ denote the sets of isolated critical points of V(x) at which U(x,t) as a function of x attains its maximum for any fixed t > t₀ and t < t₀, respectively. Under nondegeneracy conditions on points of X_± we apply the Newton – Kantorovich type method to study the existence of transversal doubly asymptotic trajectories connecting X₋ and X₊. Conditions on the Riemannian manifold and the potential which guarantee the existence of such orbits are presented. Such connecting trajectories are obtained by continuation of geodesies defined in a vicinity of the point t₀ to the whole real line.

中文翻译：

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非平稳力场中拉格朗日系统的横向连接轨道：牛顿-坎托罗维奇方法

我们考虑一个自然的拉格朗日系统，该系统定义在一个完整的黎曼流形上，该流形受势为U（q，t）= f（t）V（q）的非平稳力场的作用。据推测，该因子˚F（吨）倾向于∞作为吨→±∞和消失在一个独特的点吨₀ ∈ℝ。让X ₊，X _-表示的分离临界点的集合V（X），在该ù（X，吨）作为x的函数分别对于任何固定的t > t ₀和t < t ₀达到最大值。在对点非退化条件X _±我们应用牛顿-学习连接横向双渐近轨迹的存在Kantorovich型方法X _-和X ₊。给出了黎曼流形上的条件和能保证此类轨道存在的条件。通过将在点t ₀附近限定的测地线延续到整个实线来获得这种连接轨迹。