Some results and current problems in the theory of periodic motions and nonlinear mechanics (Review)

Recent results obtained in the theory of periodic motions and nonlinear mechanics are reviewed, and some problems yet to be solved are stated. In particular, attention is given to the direct (second) Liapunov method, stability of periodic motions, applications of the mathematical theory of parametric resonance to acoustic and electromagnetic waveguides, and oscillations in essentially nonlinear systems. Work in the field of rigid body dynamics is also reviewed, including some particular cases of Kovalevskaia's gyroscope, oscillations of a rigid body with a fixed point, and the dimensionless form of Euler-Poisson equations.