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应用数学学报  1984, Vol. 7 Issue (1): 36-47    DOI:
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欧拉-拉格朗日方程的形式讨论
柳长茂
西南交通大学
ON THE FORMS OF THE EULER-LAGRANGE EQUATIONS
Liu Chang-mao
Southwestern Tiaotong University
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摘要 对已给方程(组),变分学逆问题有两种提法:一种是先判断方程(组)的解,是否可能为某泛函的逗留值,可能时则找出相应泛函,如[1, 16, 5]中某些后补变分原理.实质上是将方程(组)看成等价类来探讨.另一种是求泛函,使泛函的欧拉一拉格朗日方程为已给方程,自然,这里也有个判断可能性问题[4-7].本文即在这两种提法下来讨论问题,即将欧拉一拉格朗日方程左端视为算子,这两法提法自然不同,单就变分原理存在性而论,前一种提法较后一种为广,它容许方程变形,但在讨论所谓后补变分原理时[5],后一种提法更方便,还可更加扩大讨论变分原理的范围.
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柳长茂
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Abstract: The purpose of this paper is to discuss the forms of the Euler-Lagrange equations.The polynomial-index law has been obtained:For higher derivatives of these Pquations, their order and (algbraic) degrse have to satisfy a sim ple inequality. Especially, in the two dimensional space, a second order partial differential equation, which is a Euler-Lagrange equation, must be a Monge-Ampere equation.
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收稿日期: 1981-08-06;
引用本文:   
柳长茂. 欧拉-拉格朗日方程的形式讨论[J]. 应用数学学报, 1984, 7(1): 36-47.
Liu Chang-mao. ON THE FORMS OF THE EULER-LAGRANGE EQUATIONS[J]. Acta Mathematicae Applicatae Sinica, 1984, 7(1): 36-47.
 
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