Differing Interpretations of the Lagrangian in Classical and Quantum Mechanics

Classical mechanics, which is a deterministic theory, was developed largely with the work of Newton. Later, the Lagrangian approach of obtaining stationary paths of an action was introduced to obtain the Newtonian results. Thus a deterministic theory was considered in the interpretation of varying paths in the action with the stationary path being the classical mechanical result. In (1) we have argued that both relativistic and nonrelativistic quantum mechanics follow from varying x and t independently in v= x/t = constant in the classical action for a free particle with constant momentum. In this case, it seems one considers a fluctuating distance or a stochastic approach, yet in this note we argue that for dAction/dx =p for the quantum case is very similar to dL/dv = p for the classical. It is the interpretation which seems to differ. We argue that an energy flow through space i.e. a p in quantum mechanics seems to follow from a certain kind of spatial probability fluctuation i.e. a periodic one. On average the free particle case and also the bound state case yield classical results so the Lagrangian approach of classical mechanics is linked to quantum rms values.