Lagrangian/Action Formalism As a Flow Scheme

In a previous note (1), we argued that one may obtain either the Schrodinger or Klein-Gordon equation from the classical free particle Action = Integral dt L = T L(v) by writing v=X/T and using:  p= dAction/dx partial  and -E = dAction/dt partial. This may be used together with an energy-momentum relationship  E=pp/2m (nonrelativistic) or EE=pp + mm (relativistic) to create an eigenfunction equation by using exp(i Action) as a solution. This is equivalent to exp(-iEt+ipx). 

   In (2), we argued that if one generalizes momentum p to Ev where E(v) is energy and uses nonrelativsitic Lagrangian equations:  p=dL/dv  and H=E= pv-L, one may find the relativistic Lagrangian for a free particle which in turn yields p= d/dx (TL) partial and -E= d/dt (TL) partial i.e. the form of energy and momentum for a free particle in special relativity.

    In this brief note, we argue that the Action/Lagrangian formalism for a free particle seems to follow directly from a flow(flux)  type equation namely:  d/dx Action partial = p = Ev (= mo v nonrelativistic limit). If energy is flowing out of a region given by the usual Gaussian flux equation d/dx function (i.e. grad dot vector) represented by d/dx Action partial, then one is losing energy E(v) in this region. Thus d/dt Action partial = -E. As a result, one should analyze the problem in special relativity i.e. use p=E(v) v (but no explicit form for E(v)) and then take the nonrelativistic limit to get the nonrelativistic approximation. 

   As a result, assuming a flow problem with a function Action(x,t) leads to the Lagrangian formalism. From this one may obtain the forms of special relativity, classical equations of motion and quantum mechanics for a free particle (relativistic or nonrelativistic).