Synge's concept of stability applied to non-linear normal modes

Synge's concept [J.L. Synge, On the geometry of dynamics, Philos. Trans. R. Soc. London, Ser. A 226 (1926) 33–106] of stability is introduced and shown to be equivalent to the orbital stability in holonomic conservative systems of two-degrees-of-freedom. This furnishes an analytical tool to study the orbital stability in strongly non-linear systems. This concept is shown to be applicable to the stability analysis of non-linear normal modes, for which Liapunov's first method generally fails. Integrally related numbers are found such that, if the ratio of linear natural frequencies is close to one of the numbers, then a normal mode may lose stability at a small amplitude. These numbers depend on the symmetry or asymmetry of system with respect to the origin of the configuration space. Some examples are given to demonstrate the stability analysis of the normal modes and to verify the integrally related numbers.