Mechanical systems, equivalent in Lyapunov's sense to systems not containing non-conservative positional forces

Developing results obtained previously (Refs. Koshlyakov VN. Structural transformations of the equations of perturbed motion of a certain class of dynamical systems. Ukr Mat Zh 1997; 49 (4): 535–539; Koshlyakov VN. Structural transformations of dynamical systems with gyroscopic forces. Prikl Mat Mekh 1997; 61 (5): 774–780; Koshlyakov VN, Makarov VL. The theory of gyroscopic systems with non-conservative forces. Prikl Mat Mekh 2001; 65 (4): 698–704; Koshlyakov VN, Makarov VL. The stability of non-conservative systems with degenerate matrices of dissipative forces. Prikl Mat Mekh 2004; 68 (6): 906–913), the general problem of eliminating non-conservative positional structures from the second-order differential equation with constant matrix coefficients, obtained when modelling many mechanical systems, is considered. It is assumed that the matrices of the dissipative and non-conservative positional structures may, in particular, be degenerate. Under fairly general assumptions, theorems containing the necessary and sufficient conditions for a Lyapunov transformation to exist are proved. This converts the initial matrix equation to an equivalent autonomous form (in Lyapunov's sense) with a symmetrical matrix of the positional forces. An illustrative example is considered.