最速降线问题的充分性  

Proof of Sufficient Conditions for the Solution of the Brachistochrone Problem

作　　者：邢家省[1,2] 杨义川[1,2] 王拥军[1,2] XING Jiasheng;YANG Yichuan;WANG Yongjun(School of Mathematics,Beihang University,Beijing 100191,China;LMIB of the Ministry of Education,Beihang University,Beijing 100191,China)

机构地区：[1]北京航空航天大学数学与系统科学学院,北京100191 [2]数学、信息与行为教育部重点实验室,北京100191

出　　处：《四川理工学院学报（自然科学版）》2019年第4期76-80,共5页Journal of Sichuan University of Science & Engineering(Natural Science Edition)

基　　金：国家自然科学基金资助项目(11771004);北京航空航天大学校级重大教改项目:北航培育项目(2019.01-2021.12)

摘　　要：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.

关 键 词：最速降线问题 泛函的临界点 二阶方向导数 正定性 

分 类 号：O175.29[理学—数学]