The variational and virial-like theory of oscillations and stability of non-conservative and/or non-linear mechanical systems

Abstract

This paper presents the general theory and some applications of Hamiltonian action and virial-like methods to the exact and/or approximate study of the periodic solutions of non-conservative and/or non-linear, but holonomic, oscillators. Specifically, the first (theoretical) part covers: (i) a generalization of “Hamilton's law of varying action”, to include variable time-endpoints (i.e., frequency variations) and variable system parameters, such as elasticity and/or inertia, and (ii) a general formulation of the “virial” theorem and its use in determining the stability/instability of given oscillatory motions. Applications of the above (of the Rayleigh-Ritz type) to the following systems are then presented: (i) linear circulatory; (ii) linear conservative, loading parameter dependent, (iii) Duffing's (cubic) oscillator; (iv) van der Pol's oscillator, and related general non-linear and non-conservative case. Comparison of the method with other existing ones, and related open problems (such as limit-cycle stability), are finally discussed.