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 N;
2008

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.
Small solution, coupled oscillators, double pendulum
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.12571/2806
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus ND
  • ???jsp.display-item.citation.isi??? ND
social impact