On passive stabilization in critical cases
It is well known that a (practically) lossless mechanical system can be stabilized by coupling it to a system which contains a friction device (standard practice for terrestrial vehicles). Here the equations of motion of such a system are derived from a Hamiltonian with a damping term. The resulting system of fourth-order ODEs exhibits a Lyapunov critical case with two purely imaginary eigenvalues. Non-relevant terms are removed by a polynomial transformation. The details of this procedure can be found in chapter 4 of \textit{I. G. Malkin}'s book [Theorie der Stabilität der Bewegung. (Russian), Moskau: Verlag `Nauka'. Hauptredaktion für physikalisch-mathematische Literatur (1966; Zbl 0136.08502)] which extends an earlier publication by \textit{Frommer} [Math. Ann. 99, (1928)] which, in turn, is based on the works of I. Bendixson (1900) and H. Poincaré (1881).
scientific article; zbMATH DE number 1441390