Quasi-normal and normal oscillations in conservative systems: PMM vol. 40, n≗ 2, 1976, pp. 260–269

Abstract

Particular kinds of periodic solutions in unison — quasi-normal oscillations similar to those of an oscillator — are separated in conservative multidimensional systems. A new definition of normal oscillations, more precise than known ones is proposed. It is applicable to a wider class of nonlinear systems. A method of approximate determination of quasi-normal oscillations for a particular kind of nonlinear systems is described and some examples are presented. In [1, 2] the supposition was made that singular analogs of characteristic solutions, often called normal oscillations, can exist in the class of nonlinear conservative systems of the form x…i = ∂U ∂x i , U (0) = 0, U (−x) = U (x) and x = x 1, x 2, …, x n . It was assumed that normal oscillations are determined by the following characteristic properties: oscillation frequencies of all coordinates are equal, all coordinates attain their maximum deflection and vanish simultaneously, and the displacement of coordinates at any instant of time is a single-valued function of one of these. From the physical point of view the above definition of normal oscillations has the following shortcomings: the characteristic properties of normal oscillations are noninvariant under the change of the coordinate system, are interdependent, comprise a narrow class of nonlinear systems, and do not permit the formulation of the problem of determining normal oscillations. In this paper the concept of normal oscillations of nonlinear systems is extended, a new definition of strictly normal oscillation is presented, and the algorithm for determining quasi-normal oscillations is formulated for the class of strongly nonlinear systems which is of practical interest.