Isochronous Oscillations in a Conservative Newton System
by
Vladimir Ivanov
Sobolev Institute of Mathematics, Novosibirsk, Russia

Consider the simplest Newton equation x'' = f (x) or the corresponding system on the phase plane:

x' = y,     y' = f (x),

where f (x) stands for a real continuous of x defined in some neighborhood of the origin. The potential energy of the system we call the integral

U (x) = - x ó õ 0  f (s) d s.

If zero is a strict minimum point of this function then the origin serves as an equilibrium of the system and the phase trajectories near to this equilibrium are closed and encircle the origin. Moreover, if their periods coincide than we call the system isochronous. The classical example of an isochronous system is the harmonic oscillator with linear force function f (x) = - k x and potential energy U (x) = k x²/2 . This talk discusses the following results:

Theorem 1. Oscillations in a conservative Newton system are isochronous with given frequency if and only if the graphs of potential energy of the system under study and the harmonic oscillator with the same frequency have the same width at each not very large level .

Theorem 2. If the potential energy of a conservative Newton system is an entire analytic function then the system is isochronous if and only if it is a harmonic oscillator.

Theorem 3. A conservative Newton system with meromorphic potential, other than a harmonic oscillator, is isochronous if and only if its potential energy is the square of the Zhukovsk function up to linear transformation.

Date received: February 25, 2000