It is well known that action can be easily expressed as a function of energy of the pendulum in terms of elliptic integrals. So, the question is if it is possible to obtain the inverse functions for liberation and rotation cases?
Sure. The relation between the energy and action of a pendulum is dH/dI=\omega (a), where H is the energy, I the action, \omega the frequency, and a the amplitude (see, e.g., https://www.researchgate.net/publication/260783019_Energy_Methods_in_Dynamics). So, principally one can integrate this equation to obtain energy as function of action. The explicit dependence of the energy on the action in terms of the special function for the pendulum is not known to me. For this one needs to express the action explicitly in terms of the amplitude. Such expression may include special functions other than elliptic integrals. To plot energy as function of the action we can use, for instance, the command ParametricPlot in Mathematica and the amplitude a as a parameter.
Book Energy Methods in Dynamics
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