A Nonlinear Oscillator With Velocity Dependent Frequency

We study the mathematical properties of a harmonic type oscillator in which the square of the angular frequency is an even, quadratic function of the velocity. After calculating the first integral of this system, phase-space and symmetry arguments are used to prove that all solutions are periodic. From the first-integral, a closed form, integral representation is obtained for the period of the oscillations; but, this formula does not allow for the explicit determination of the period as a function of the initial amplitude. (The initial conditions are taken to be x(0) = A and dx(0)/dt = 0.) However, the application of the mean-value theorem to the integral, along with information on the small amplitude behaviors of the frequency gives a theoretical, closed form expression for the dependence of the frequency on A. We compare this theoretical expression to ``data'' gotten from the numerical integration of the differential equations.