Argue for formula about one-dimensional harmonic oscillator

Solution 1:

I think you've miscalculated; you don't need $S(\gamma_0)=0$ because$$\begin{align}S(\gamma)-S(\gamma_0)&=\int_0^T\left[\frac12m\left((\dot{\gamma}_0+\dot{\nu})^2-\dot{\gamma}_0^2\right)-\frac12k\left((\gamma_0+\nu)^2-\gamma_0^2\right)\right]dx\\&=\int_0^T\left[\frac12m\left(2\dot{\gamma}_0\dot{\nu}+\dot{\nu}^2\right)-\frac12k\left(2\gamma_0\nu+\nu^2\right)\right]dx,\end{align}$$which gives the desired result iff$$\int_0^T\left[m\dot{\gamma}_0\dot{\nu}-k\gamma_0\nu\right]dx=0.$$But that integral is$$\int_0^Tm\left[\dot{\gamma}_0\dot{\nu}+\ddot{\gamma}_0\nu\right]dx=[m\dot{\gamma}_0\nu]_{t_1}^{t_2},$$which vanishes as desired because $\nu(t_1)=\nu(t_2)$.