A revisit to higher variations of a 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.