(a) Show that if the amplitude of oscillation is small the energy of a simple pendulum is given by 𝐸 = 𝐽𝑣.
(b) Consider a simple pendulum formed by a point weight on a string which passes vertically through a hole in a horizontal board. Suppose the length of the pendulum is decreased by slowly pulling the string up through the hole. The rate of decrease of length is to be extremely slow compared with the frequency of swing, so that one can still speak of a period of oscillation for any given length. Compute the change in the energy of the pendulum, as the length is shortened, from the work done in pulling against the tension of the string. Show as a result that the action variable 𝐽 = 𝐸/𝑣 is constant throughout the process. A change in the external parameters of the system at a rate slow compared to the intrinsic frequencies is known as an adiabatic variation and the action variable of the pendulum is therefore an adiabatic invariant. It is possible to prove in general the adiabatic invariance of the action variables for any system in which degeneracy does not occur. The action variables are thus stable constants of the system under the influence of slowly varying external conditions. Now, the quantum state of a system is also an adiabatic invariant; a slow change of the external parameters does not induce transitions from one state to another. We have here another indication of the suitability of the action variables for describing the quantization of the system states.
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