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[QUE/CM-03001] Action for harmonic oscillator

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 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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