Decomposition of motion in systems with fast phase

Abstract

Among the problems considered in the previous chapter special attention was paid to slowly changing motions parameters, for example, slowly increasing amplitudes in Sec. 4.2, and slow Magnus drifts in Sec. 4.3. The Magnus drifts were obtained by averaging the right-hand side of Eqs. (4.29) with respect to the time variable which is a part of high-frequency oscillating components. During the process of averaging, slow variables (ϕ1, (ϕ2 were assumed to be constant and equal to their initial values. This averaging procedure is correct only on bounded intervals of the dimensionless nutational time t = T/T₁ ∽ 1 where the Poincaré theorem is valid and (ϕ1, (ϕ2 changes by values of the order ε according to (4.26). An attempt to use Magnus’s formula for large intervals of time t ∽ 1/ε contradicts the theorem. The foregoing procedure of averaging the right-hand sides of (4.29) loses its meaning because during the time interval t ∽ 1/ε the variables ϕ1, ϕ2 change by finite value.