Special Relativity from Lagrangian Formalism Part II?

 In part I of this note we argued that setting p=E(v,mo)v where p is momentum and E energy of a free particle and using d/dx (partial) A(x,t) = p and d/dt (partial) A(x,t) = -E one may obtain special relativistic results for a free particle with a rest mass of mo. In particular  A(x,t) is set equal to L(v)t with v=x/t yielding p=dL/dv and E= pv-L from which one may create a differential equation using p=Ev. From L, one has p=dL/dv and finally E=p/v and finds that E=mo/sqrt(1-vv) (c=1) and p=Ev. This leads to the well-known EE=pp+momo. If one sets mo=0, one has E=pc, but for mo, E=mo/sqrt(1-vv)=0 so we wish to examine the photon result in more detail. The photon solution is linked to a special solution of A=-Et+px which does not hold for a particle with rest mass, namely dA=0= -Edt + pdx. This yields the classical wave equation for a photon, which also follows from Maxwell’s equations. 

    In addition one may consider rest mass to be created by bouncing photon. In such a case, the bouncing particle need not have a rest mass, yet its localized energy acts as rest mass and this system may be given “momentum” i.e. Ev. Given that the particle, however, has no rest mass, it is all momentum and this must be linked to energy in a manner consistent to EE=pp+momo and also to E=pc. Thus one finds a Doppler shift for the photon.

  We examine some of these ideas in this note.