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    The action of a harmonic oscillator

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    Consider the harmonic oscillator, for which the general solution is

    x(t) = A cos(wt) + B sin(wt)

    1. Express the energy in terms of A and B and show it is time independent.
    2. Choose A and B such that x(0)=x1 and x(T)=x2.
    3. Write down the energy in terms of x1, x2 and T.
    4. Calculate the action S for the trajectory connecting x1 and x2
    5. verify the dS/dT = -E.

    See the attached file.

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    Solution Summary

    8 pages of full drivations of the Lagrangian and the action of a harmonic oscillator.