Transcribed TextTranscribed Text

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.

Solution PreviewSolution Preview

These solutions may offer step-by-step problem-solving explanations or good writing examples that include modern styles of formatting and construction of bibliographies out of text citations and references. Students may use these solutions for personal skill-building and practice. Unethical use is strictly forbidden.

    By purchasing this solution you'll be able to access the following files:

    for this solution

    or FREE if you
    register a new account!

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Mathematics - Other Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats