Snell's Law and X, T Independence in the Classical Action

Note May 25, 2022 The argument of Et1 = Et2 so t1=t2 which is then linked to different distances d1=v1 t and d2=v2 t which are then compared to L sin(theta1) and L sin(theta2) applies to the component of the angled L path which does not undergo a momentum change i.e the "sin(theta)" component. Snell’s law of refraction may be derived in a number of ways. In (1) these approaches include: Fermat’s principle, Huygen’s principle, electromagnetism and conservation of energy and momentum. In a previous note (1) we argued that a free particle classical action (relativistic or nonrelativistic i.e. -mo t sqrt(1-vv) (c=1) or t .5mvv) may be written in terms of v=x/t and x and t varied independently to yield: dA/dx partial= p and dA/dt partial=-E. From this A= -Et+px with x and t independent. If one writes eigenvalue equations, then exp(iA) is a solution. We argue that a photon moving from one medium into another does not lose energy. Thus exp(-iEt) represents a kind of cycling in each medium which we take to be the same. In other words if we consider t medium 1 = t medium 2 and also consider either vertical or horizontal motion by itself then d medium 1 = v (medium 1) t and d (medium 2) = v(medium 2) t. In other words a constant Et action in each medium leads to different horizontal and vertical lengths. These may be shown to yield Snell’s law, we argue. If one uses the angled light then constant -Et leads to different lengths L1 and L2 and does readily yield information about refraction