What does this equation mean?

A beam of light crossing from air into glass bends at the interface by exactly the angle that gets it through in the least possible time. Pierre de Fermat stated this as a principle in the 1660s, and the consequence collapses into one line of algebra: for a ray crossing between two media of refractive index $n_1$ and $n_2$ , $n_1 \sin\theta_1 = n_2 \sin\theta_2$. Snell’s law, in other words, is not an independent fact about glass. It is what falls out of a single global stipulation — least time — applied to one path among infinitely many. Pierre-Louis Maupertuis generalized the same move to all of mechanics in the 1740s, proposing that nature always acts by the smallest possible “action,” and Euler, Lagrange, and Hamilton spent the following century turning that guess into the working machinery of classical physics: a moving body’s entire trajectory falls out of minimizing one integrated quantity over the whole path, not from adding up forces moment to moment. General relativity is this idea pushed as far as it goes. Einstein’s field equations recast gravity as the curvature of spacetime itself, and a body in free fall — no rocket, no resistance — simply follows the straightest available route through that curved geometry, a geodesic, obeying the same one-line logic as light and no other: $\ddot{x}^\mu$ + $\Gamma^\mu_{\alpha\beta}$$\dot{x}^\alpha$$\dot{x}^\beta$ = 0 , the geodesic equation, in which curvature has replaced force as the reason anything bends [ 11 ] . Newton needed a force to explain why a thrown ball leaves a straight line; Einstein needed nothing but geometry, because the ball was never leaving the straightest line available to it [ 11 ] .