I was reading about symmetry of action when I came before the symmetry variation in Particles and Quantum Fields by H. Kleinert; there he wrote:
Symmetry variations must not be confused with ordinary variations $\delta q(t)$ used to derive the Euler-Lagrange equations. While the ordinary variations $\delta q(t)$ vanish at initial and final times, $\delta q(t_b) = \delta q(t_a) = 0,$ the symmetry variations $\delta_s q(t)$ are usually non-zero at the ends.
So, isn't $\delta_s q$ a virtual variation? For, if it would be, it should have vanished in the fixed initial and final time, $t_a$ and $t_b$, isn't it?
Could anyone explain me why the symmetry variation is different from the ordinary variation?