An Example of a Variational Approach to Mechanics Using “Different” Rules of Algebra

In many instances duality does not help in deriving complementary variational principles due to “wrong” signs of abstract second derivatives. The Lagrangian formulation “works”, the Hamilton’s canonical equations are displayed, everything seems to be in the right form. Upon a closer examination only one variational principle can be formulated at best. Even if the signs of the second FrSchet derivatives can be determined in the vicinity of a critical point of the cost functional, both are of the same sign. Suppose that the Lagrangian for the problem L,(x,Ax) vanishes at x = x.