Classical Dynamics of Particles and Systems · Problem 4-7

Gravitation — Marion & Thornton

Physics — Mechanics · plane-pendulum-nonlinear

Problem

Consider the free motion of a plane pendulum whose amplitude is not small. Show that the horizontal component of the motion may be represented by the approximate expression (components through the third order are included) x¨+ω02(1+x02l2)x−εx3=0\ddot{x} + \omega_0^2\left(1 + \frac{x_0^2}{l^2}\right)x - \varepsilon x^3 = 0 where ω02=g/l\omega_0^2 = g/l and ε=3g/2l3\varepsilon = 3g/2l^3, with ll equal to the length of the suspension.

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