  In the stability analysis of an equilibrium, given by a stationary point of a
functional F[n] (free energy functional, e.g.), the second derivative of F[n]
plays the essential role. If the system in equilibrium is subject to the
conservation constraint of some extensive property (e.g. volume, material, or
energy conservation), the Euler equation determining the stationary point
corresponding to the equilibrium alters according to the method of Lagrange
multipliers. Here, the question as to how the effects of constraints can be
taken into account in a stability analysis based on second functional
derivatives is examined. It is shown that the concept of constrained second
derivatives incorporates all the effects due to constraints; therefore
constrained second derivatives provide the proper tool for the stability
analysis of equilibria under constraints. For a physically important type of
constraints, it is demonstrated how the presented theory works. Further, the
rigorous derivation of a recently obtained stability condition for a special
case of equilibrium of ultrathin-film binary mixtures is given, presenting a
guide for similar analyses. [For details on constrained derivatives, see also
math-ph/0603027, physics/0603129, physics/0701145.]
