On the variational process in parameter optimization

The simplest constrained optimization problem is used to investigate the concept of variations of variations. It is shown that variations of dependent variations are nonzero whereas variations of independent variations are zero. The existence of these variations adds to the second variation a term which vanishes because of first-variation conditions but which contributes directly to the third and higher variations. These variations are needed in particular if the second variation vanishes and in general if the whole process is to be done correctly. Finally, the same results are derived using the Taylor series expansion.