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On small oscillations of mechanical systems with time-dependent kinetic and potential energy. (English) Zbl 1146.70011

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

$\sum _{k=1}^{n}\left({a}_{ik}\left(t\right){\stackrel{¨}{q}}_{k}+{c}_{ik}\left(t\right){q}_{k}\right)=0\phantom{\rule{1.em}{0ex}}\left(i=1,2,\cdots ,n\right)·\phantom{\rule{2.em}{0ex}}\left(*\right)$

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

$\underset{t\to \infty }{lim}{q}_{k}\left(t\right)=0\phantom{\rule{1.em}{0ex}}\left(k=1,2,\cdots ,n\right)·$

It is known that in the scalar case $\left(n=1$, ${a}_{11}\left(t\right)\equiv 1$, ${c}_{11}\left(t\right)=:c\left(t\right)\right)$ 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 $\left(*\right)$ in the case when coefficients ${a}_{ik}$, ${c}_{ik}$ are step functions. The method of proof is based on a transformation reducing the ODE $\left(*\right)$ to a discrete dynamical system. The results are illustrated by examples of coupled harmonic oscillator and double pendulum.

##### MSC:
 70J30 Free linear oscillatory motions 70F20 Holonomic systems (particle dynamics)