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In the stability analysis of an equilibrium, given by a stationary point of afunctional 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 theconservation constraint of some extensive property (e.g. volume, material, orenergy conservation), the Euler equation determining the stationary pointcorresponding to the equilibrium alters according to the method of Lagrangemultipliers. Here, the question as to how the effects of constraints can betaken into account in a stability analysis based on second functionalderivatives is examined. It is shown that the concept of constrained secondderivatives incorporates all the effects due to constraints; thereforeconstrained second derivatives provide the proper tool for the stabilityanalysis of equilibria under constraints. For a physically important type ofconstraints, it is demonstrated how the presented theory works. Further, therigorous derivation of a recently obtained stability condition for a specialcase of equilibrium of ultrathin-film binary mixtures is given, presenting aguide for similar analyses. [For details on constrained derivatives, see alsomath-ph/0603027, physics/0603129, physics/0701145.]

Stability of equilibrium under constraints: Role of second-order constrained derivatives