This paper presents alternative proofs of the well-known strict negative monotonicity and convexity properties of the fundamental frequency-load pair curve of the pure eigenproblem of free vibrations/buckling of linear conservative systems subject to a single loading parameter, by studying the first and second order variations of the system's Hamiltonian action (over the fundamental or hghest eigenperiod), as one moves along the fundamental eigenpair curve (FEC). Specifically, by setting the action's first and second frequency derivatives equal to zero it is shown that, all along the FEC, both the first and second derivatives of the load with respect to the frequency keep a strict negative sign; these action derivatives are special cases of the general first and secotad variations of a variable endpoints variational problem. This energetic method opens the way for the development of an asymptotic scheme (by taking third, fourth and higher action derivatives) for the approximate study of the FEC shape in the neighborhood of a given point (reminiscent perhaps of Koiter's asymptotic elastostatic post-buckling theory). At this point, the method does not seem to be restricted by linearity neither in the equations of motion nor in their eigenload(s) dependence.