A revisit to higher variations of a functional

  • Guobin Gong Liverpool University
  • Xi'an Jiaotong Liverpool University
Keywords: A, revisit, higher, variations, functional


Two definitions of higher variations of a functional can be found in the literature of variational principles or calculus of variations, which differ by only a positive coefficient number. At first glance, such a discrepancy between the two definitions seems to be purely due to a definition-style preference, as when they degenerate to the first variation it leads to the same result. The use of higher (especially second) variations of a functional is for checking the sufficient condition for the functional to be a minimum (or maximum), and both definitions also lead to the same conclusion regarding this aspect. However, a close theoretical study in this paper shows that only one of the two definitions is appropriate and the other is advised to be discarded. A theoretical method is developed to derive the expressions for higher variations of a functional, which is used for the above claim.

How to Cite
Guobin Gong, & Xi’an Jiaotong. (2019). A revisit to higher variations of a functional. Journal of Advance Research in Mathematics And Statistics (ISSN: 2208-2409), 6(2), 01-15. Retrieved from https://jiaats.com/index.php/MS/article/view/523