A note on isoenergetic stability

In this note we consider certain two-degree-of-freedom Hamiltonian systems which may be regarded as perturbations of integrable systems governed by a real parameter ε. We wish to study the stability, at fixed energy, of certain periodic solutions. Two constants are defined, computable in terms of the original Hamiltonian function and the energy. The main theorem then states that if these constants are not zero, the periodic solutions are isoenergetically stable for sufficiently small ε. The proof is an application of the Twist Theorem of Kolmogorov-Arnol'd-Moser. By way of illustration, we apply the theorem to a mechanical system consisting of coupled non-linear oscillators. The periodic solutions are the “normal modes” and ε governs the non-linearity of the system. One obtains stability criteria for arbitrary energies and small ε, or, alternatively, for arbitrary ε and small energies.