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