Full Lagrangian Theory from Constant Speed Lorentz Invariant -Et+px ?

A free particle’s motion may be described by the simple equation x=vt. One may ask: What more does one need to present than that? We suggest that due to special relativity, x=vt is not sufficient, especially in cases for which one has v(x)=constant to first order, but with acceleration occurring at the same time. Furthermore, even in the free particle case, x=vt is not sufficient quantum mechanically. The reason for this is that is that a key piece of physics is missing from x=vt, namely Newton’s notion of rest mass as inertia. Inertia means that some force is required to move a rest mass to the speed v. One may argue that once it is moving with a constant v and not interacting, then x=vt is the full description. According to special relativity, however, this is not the case due to the Lorentz invariant -EE + cc p dot p = -momo cccc ((1)), which is used to derive the formulas for E and p (energy and momentum in the first place). ((1)) holds even if the particle is not interacting. There is also another Lorentz invariant -Et+px = A(t,v), and we suggest that this also carries essential physical information as it is independent of x=vt -> x=E/p t (particle with rest mass). Given -Et+px=A(t) for a free particle, it is understood that x=vt. Nevertheless, -Et+px=A(t) contains independent information and we suggest trying to extract x=vt information from -Et+px=A(t). This may seem unnecessary given that one knows x=vt, a priori. Any expression one obtains from -Et+px = A(t) cannot contradict x=vt with v constant and so is essentially a statement of v=constant. In particular, we will show/argue that the expression obtained from -Et+px=A(t) is d/dt dL/dv=0 where L = -E+pv= -sqrt(1-vv/cc), i.e. t factored out of A(t,v) and then only v used as a variable. d/dt dL/dv=0 reduces to dv/dt=0 or v=constant and so one may ask: What possible relevance can d/dt dL/dv =0 have? We argue that its relevance emerges when one considers motion due to a conservative force F = -grad V(x). In such a case, each dx region is considered to have a constant v(x) and so the idea of a constant speed must still be relevant, but one must still respect special relativity. The simplest thing to do would be to write dv(x)/dt = function(x) linked to F(x), because when F=0, dv/dt=constant. This is in fact what one does for a constant rest mass in Newtonian (nonrelativistic physics). To be more general, however, one must include special relativistic information, even for a constant speed, and that full information is contained in d/dt dL/dv partial. Instead of setting this equal to 0, one must set it equal to an appropriate function of x, but one sees that there is a difference been a formalism using dv/dt = Function of x linked to force and d/dt dL/dv partial = Different function of x linked to force. The latter respects special relativity and L is the same function of v obtained from the special relativistic treatment of a free particle. This is why L= -mo sqrt(1-vv/cc) - V(x) in special relativity. In short, we argue that full Lagrangian theory emerges from special relativistic considerations of a free particle which contain more information than x=vt.