Hamilton's principle is applied to analyze the problem of the stability of equilibrium of a discrete, holonomic, and scleronomic mechanical system under conservative and non-conservative (position and/or velocity dependent) forces. At the stability limit, the vanishing of the second order terms (in the deviations from equilibrium) of the total action change functional leads to the condition that the matrix of a certain quadratic form be singular; this yields the eigenvalue (frequency-load) curve. The flutter loads follow by setting the frequency derivative of the determinant of this matrix equal to zero; the energetic interpretation of this latter is also given. When the non-conservative forces go to zero it is shown that one recovers the well-known discrete conservative system stability criterion. An application follows, and finally in an Appendix various relevant time-integral equalities are summarized and interpreted.