Classical Dynamics of Particles and Systems · Problem 4-4

Gravitation — Marion & Thornton

Physics — Mechanics · rayleigh-van-der-pol

Problem

Lord Rayleigh used the equation x¨−(a−bx˙2)x˙+ω02x=0\ddot{x} - (a - b\dot{x}^2)\dot{x} + \omega_0^2 x = 0 in his discussion of nonlinear effects in acoustic phenomena. Show that differentiating this equation with respect to time and making the substitution y=y03b/ax˙y = y_0\sqrt{3b/a}\dot{x} results in van der Pol's equation: y¨−ay02(y02−y2)y˙+ω02y=0\ddot{y} - \frac{a}{y_0^2}(y_0^2 - y^2)\dot{y} + \omega_0^2 y = 0

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