A generalization of the undamped pendulum equation

Chapter 9, Problem 6

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A generalization of the undamped pendulum equation isd2u/dt2 + g(u) = 0, (i)where g(0) = 0, g(u) > 0 for 0 < u < k, and g(u) < 0 for k < u < 0; that is, ug(u) > 0for u = 0, k < u < k. Notice that g(u) = sin u has this property on (/2, /2).(a) Letting x = u, y = du/dt, write Eq. (i) as a system of two equations, and show thatx = 0, y = 0 is a critical point.(b) Show thatV(x, y) = 12 y2 +x0g(s) ds, k < x < k (ii)is positive definite, and use this result to show that the critical point (0, 0) is stable.Note that the Liapunov function V given by Eq. (ii) corresponds to the energy functionV(x, y) = 12 y2 + (1 cos x) for the case g(u) = sin u.