When covering Huygen's principle: the wavelets always give a backwards-moving wave in addition to the forwards-moving wave. In other words, Huygen's principle i

When covering Huygen's principle: the wavelets always give a backwards-moving wave in addition to the forwards-moving wave. In other words, Huygen's principle is time symmetric. So in real life, why don't you get the backwards wave? When covering the principle of least action: often, applying it will give infinitely many solutions, or none at all. An example with no solutions would be the harmonic oscillator with x(0) = 0, x(t) = 0, and t not a multiple of half the period. An example with infinitely many solutions would be the principle of least time applied between the two foci of an ellipse with reflecting walls. So what happens in these cases? Does Lagrangian mechanics just not work?