October  2008, 20(4): 911-926. doi: 10.3934/dcds.2008.20.911

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


Dipartimento di Matematica Pura e Applicata, Università de L'Aquila, I-67100 L'Aquila


Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary

Received  January 2007 Revised  September 2007 Published  January 2008

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.

Citation: Nicola Guglielmi, László Hatvani. On small oscillations of mechanical systems with time-dependent kinetic and potential energy. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 911-926. doi: 10.3934/dcds.2008.20.911

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