*(English)*Zbl 1146.70011

Summary: Small oscillations of an undamped holonomic mechanical system with varying parameters are described by the equations

A nontrivial solution ${q}_{1}^{0},\cdots ,{q}_{n}^{0}$ is called small if

It is known that in the scalar case $(n=1$, ${a}_{11}\left(t\right)\equiv 1$, ${c}_{11}\left(t\right)=:c\left(t\right))$ there exists a small solution if $c$ is increasing and tends to infinity as $t\to \infty $.

Here we give sufficient conditions for the existence of a small solution of the general system $(*)$ in the case when coefficients ${a}_{ik}$, ${c}_{ik}$ are step functions. The method of proof is based on a transformation reducing the ODE $(*)$ to a discrete dynamical system. The results are illustrated by examples of coupled harmonic oscillator and double pendulum.