This paper is concerned with the dynamics of a discrete or discretized system \(\Sigma_{\mathrm{free}}\). After having started with different possible nonlinear settings, convenient assumptions and approximations lead to a dynamic evolution of \(\Sigma_{\mathrm{free}}\) governed by the following equation of motion: \(M\ddot{X}+K(p)X=0\) where \(K(p)\) is a generally nonsymmetric matrix corresponding to the nonconservativity of \(\Sigma_{\mathrm{free}}\), whereas \(M\) is a symmetric positive definite matrix. In fact, the authors focus on families of kinematic constraints that could convert \(\Sigma_{\mathrm{free}}\) into a conservative system. More precisely, this work addresses both the problems of the existence of a minimal family (according to the number of constraints) of such constraints and that of building the set of such families. The minimal number of constraints required to convert the nonconservative system \(\Sigma_{\mathrm{free}}\) into a conservative system is then a measure of the nonconservativity of \(\Sigma_{\mathrm{free}}\) and will be called the geometric degree of nonconservativity of \(\Sigma_{\mathrm{free}}\).