Abstract: | A procedure, closely related to the averaging method, is given for the construction of asymptotic approximations for the solutions of the Lagrangian system where ε > 0 and δ > 0 are small parameters, aij = aji and βij = βji are real constants. The solutions can be represented as follows: The asymptotic approximations (in a certain well-defined sense) of a, b, ξ and ψ are given. It is shown that a rapid variation of the phase, ψ(t), will take place if b → 0 for the case that δ ≈ ε or δ ? ε. In addition for the case δ ≈ ε the phenomenon of bifurcation will be discussed. |