最速降线问题的充分性

Proof of Sufficient Conditions for the Solution of the Brachistochrone Problem

考虑最速降线解的充分性证明问题,给出质点沿曲线轨道下滑的时间公式,在曲线方程的两种形式下,分别给出时间公式的两种形式。这导致最速降线问题的两种表述形式,对等时曲线问题的也给出了表述公式。最速降线问题转化为求一个泛函的最小值问题。首先考虑最速降线问题的必要条件,证明了泛函临界点的存在性,求出了临界点曲线的参数方程,最速降线问题的必要条件是曲线为摆线的一部分。通过泛函的二阶方向导数的正定性,对泛函的临界点是泛函的最小值给出了直接简单的证明方法。利用泛函二阶方向导数的正定性,对泛函的临界点的唯一性也给出了证明。

To consider the sufficient proof problem of the fastest descending line,the time formula of the particle sliding along the curve track is given. This leads to two forms of expression for the problem of the fastest descending line. The problem of the fastest descending line is transformed into the minimum problem of solving a functional. The existence of the functional critical point is proved, and the parametric equation of the critical point curve is obtained. Through the positive definite of second order derivative of functional, a direct and simple proof method is given for that the critical point of functional is the minimum value of functional. The uniqueness of the critical points of functional is also proved by using the positive definite of second order derivatives of functional.