Small oscillations of an undamped holonomic mechanical system with
varying parameters are described by the equations
$\sum$nk=1$(a_{ik}(t)\ddot q_k+c_{ik}(t)q_k)=0,
(i=1,2,\ldots,n).$(*)
A nontrivial solution $q_1^0,\ldots ,q_n^0$ is called small
if
$\lim _{t\to \infty}q_k(t)=0 (k=1,2,\ldots n).
It is known that in the scalar case ($n=1$, $a_{11}(t)\equiv 1$,
$c_{11}(t)=:c(t)$) there exists a small solution if $c$ is increasing and
it tends to infinity as $t\to \infty$.
Sufficient conditions for the existence of a small solution of the
general system (*) are given in the case when coefficients $a_{ik}$,
$c_{ik}$ are step functions. The method of proofs is based upon a
transformation reducing the ODE (*) to a discrete dynamical system.
The results are illustrated by the examples of the coupled harmonic
oscillator and the double pendulum.
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