For a harmonic oscillator in one dimension the Lagrangian is $$L = {1\over 2}\,m\, \dot{x}^2 -{1\over 2}\,m\,\omega^2 x^2 $$ Write the solution of the equations of motion subject to the boundary conditions at times $t_1$ and $t_2$ given by $$ x(t_1) = x_1 \qquad x(t_2) = x_2 $$ Use the solution to compute the action integral. Show that the action integral is given by $$ S(x_2, t_2; x_1,t_1) = \frac{m\omega}{2\sin \omega(t_2-t_1) }\big[ (x_1^2 + x_2^2) \cos\omega(t_2-t_1) -2 x_1x_2\big] $$
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4727:Diamond Point
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