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On small oscillations of mechanical systems with time-dependent kinetic and potential energy. (English) Zbl 1146.70011

Summary: Small oscillations of an undamped holonomic mechanical system with varying parameters are described by the equations

k=1 n (a ik (t)q ¨ k +c ik (t)q k )=0(i=1,2,,n)·(*)

A nontrivial solution q 1 0 ,,q n 0 is called small if

lim t q k (t)=0(k=1,2,,n)·

It is known that in the scalar case (n=1, a 11 (t)1, c 11 (t)=:c(t)) there exists a small solution if c is increasing and tends to infinity as t.

Here we give sufficient conditions for the existence of a small solution of the general system (*) in the case when coefficients a ik , c ik are step functions. The method of proof is based on a transformation reducing the ODE (*) to a discrete dynamical system. The results are illustrated by examples of coupled harmonic oscillator and double pendulum.

70J30Free linear oscillatory motions
70F20Holonomic systems (particle dynamics)