We give simple proofs of some simple statements concerning the Lambert problem. We first restate and reprove the known existence and uniqueness results for the Keplerian arc. We also prove in some cases that the elapsed time is a convex function of natural parameters. Our statements and proofs do not distinguish between the three types of Keplerian conic section, elliptic, parabolic and hyperbolic. We also prove non-uniqueness results and non-convexity results. We do not develop any algorithm of resolution, limiting ourselves to such obviously useful a priori questions: How many solutions should we expect? Can we be sure that the Newton method will converge?

中文翻译：

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关于Lambert问题的一些简单结果

我们给出有关兰伯特问题的一些简单陈述的简单证明。我们首先重申并证明开普勒弧线的已知存在性和唯一性结果。在某些情况下，我们还证明了经过时间是自然参数的凸函数。我们的陈述和证明没有区分开普勒圆锥曲线的三种类型，椭圆形，抛物线形和双曲线形。我们还证明了非唯一性结果和非凸性结果。我们没有开发任何分辨率算法，只能将自己限制在如此明显有用的先验问题上：我们应该期望多少个解决方案？我们可以确定牛顿法会收敛吗？