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where ax , 61 > 0; a2, b2 , e > 0. For example, such a system may arise in nonlinear mechanics in the following way. One considers the idealized spring-mass system pictured in Fig. 1, where the outside springs are identical, both masses are equal to one, and the system is constrained to move horizontally. The displacements from equilibrium are denoted by and x2, with yt = xt . If the springs exert nonlinear restoring forces Ft{A) = a,A + eb,A3 corresponding to a displacement A, then the equations of motion form a Hamiltonian system with the above integral. In what follows, it may be useful to keep this model in mind. Associated with the system defined by (1) are two normal modes of vibration. The inphase mode is the periodic solution defined by xt = x2 ,yi=y2 and the out-of-phase mode is defined by = —x2, = — y2 . We are interested in the iso-energetic stability of these modes; that is, we fix a total energy h for the system and ask whether the given mode is orbitally stable in the energy surface H = h. This question has been treated by several authors ([3, 8, 9]) under various assumptions, usually for weak nonlinearity (small e) or small energy. For example, in [3] it was shown that if the at , bt satisfy certain conditions, then the in-phase mode is iso-energetically stable for fixed £ and all sufficiently small h or for fixed h and all sufficiently small e. The methods used cannot give information about large values of eh. The purpose of this paper is to deal with stability for arbitrary values of these parameters. For the in-phase mode our results are summarized in the following Theorem. Fix , a2 and b2 . There exist numbers ki and k2 depending on at , a2 and b2, with 0 < ki < k2 < °°, such that if eh lies in the set D = [0, /cj U [k2 , 00) then the inphase mode is elliptic or parabolic (in the surface H = h) while if eh lies in the set H = (ki , k2) it is hyperbolic (and hence unstable). Moreover, if a2 ± 0, then there is a set E of measure 0 in D such that the in-phase mode is iso-energetically stable for all values of eh in D-E.

An example of elliptic stability with large parameters: Lamé’s equation and the Arnold-Moser-Rüssmann criterion