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1. Consider the Hamiltonian where A and B are positive constants. (i) Find the critical points and critical values of the Hamiltonian (allow for a crit- ical point at infinity). Determine the range of energies for which the motion is bounded. Find the turning points for bounded orbits. Sketch a contour-plot of the Hamiltonian, including at least one bounded orbit, one unbounded orbit, and the orbit separating bounded from unbounded orbits. (ii) Show that the action variable is J = V A² + B² - V A² - 2E for 2E < A². For C < 1 we - = - - (iii) Calculate the frequency of the motion and express it as a function of energy. Explain why it is possible to find a motion with arbitrarily small frequency in this potential. (iv) Expand the potential at the equilibrium point to quadratic order. The resulting truncated Hamiltonian is the harmonic oscillator approximation. Verify that the frequency obtained in the previous part for E = Emin coincides with that of the harmonic oscillator approximation.

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