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.
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