I have been calculating the classical action of the harmonic oscillator, the problem I have is that I am only able to solve it if I set the integration limits of the action integral to be $t=T$ and $t=0$. I have looked online and this seems to be the solution. What I don't understand is why we can set $t=0$ as one of the limits, for a general treatment do we not have to set the limits as an arbitrary $t_b$ and $t_a$?
If there is no external force with explicit time dependence, then the harmonic oscillator contains no explicit time dependence. Then the system has time translation symmetry, i.e. the result can only depend on the difference $T =t_b-t_a$, not on $t_a$ and $t_b$ individually.