Small oscillations of an undamped holonom mechanical system with varying parameters are described by equations \[\sum^n_{k=1}\left (a_{ik}(t)\ddot q_k+c_{ik}(t)q_k\right )=0,\qquad (i=1,2,\ldots,n).\qquad \rm {(*)}\] A nontrivial solution $q_1^0,\ldots ,q_n^0$ is called \emph {small} if \[\lim _{t\to \infty}q_k(t)=0\qquad (k=1,2,\ldots n).\] 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 zero 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 results are illustrated by the examples of the coupled harmonic oscillator and the double pendulum.
On small oscillations of dynamical systems with time-dependent kinetic and potential energy
GUGLIELMI, NICOLA;
2008-01-01
Abstract
Small oscillations of an undamped holonom mechanical system with varying parameters are described by equations \[\sum^n_{k=1}\left (a_{ik}(t)\ddot q_k+c_{ik}(t)q_k\right )=0,\qquad (i=1,2,\ldots,n).\qquad \rm {(*)}\] A nontrivial solution $q_1^0,\ldots ,q_n^0$ is called \emph {small} if \[\lim _{t\to \infty}q_k(t)=0\qquad (k=1,2,\ldots n).\] 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 zero 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 results are illustrated by the examples of the coupled harmonic oscillator and the double pendulum.Pubblicazioni consigliate
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