We settle the question of how to compute the entry and leaving arcs for turnpikes in autonomous variational problems, in the one-dimensional case using the phase space of the vector field associated with the Euler equation, and the initial/final and/or the transversality condition. The results hinge on the realization that extremals are the contours of a well-known function and that the transversality condition is (generically) a curve. An approximation algorithm is presented, and an example is included for completeness.

中文翻译：

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收费公路的入口和出口弧：变分计算中的精确计算

我们解决了如何在自主变分问题中计算收费公路的入口和出口弧的问题，在一维情况下，使用与欧拉方程相关的矢量场的相空间，以及初始/最终和/或横截性健康）状况。结果取决于这样的认识：极值是众所周知的函数的轮廓，并且横向条件（通常）是一条曲线。给出了一种近似算法，并且为了完整性还提供了一个示例。