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On small oscillations of mechanical systems with time-dependent kinetic and potential energy

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  • 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.

    Mathematics Subject Classification: Primary:70J25, 34D05; Secondary: 39A11.


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