Classical Free Particle Lagrangian, Special Relativity and Quantum Mechanics

 It is known that classical waves such as waves on a string, sound and light (from Maxwell’s em equations) take the argument -Et+px where E is associated with a frequency and p with 1/wavelength. We argue that these classical nonrelativistic treatments already bring in features of special relativity (as does Schrodinger equation quantum mechanics for a free particle) as -Et+px is a Lorentz invariant. In fact we argue that Lagrangian formalism: E=pv-L and p=dL/dv is compatible with special relativity. In a previous note (1) we argued that d/dx partial A(x,t) = p and d/dt partial A(x,t) = -E for a free particle with A=tL(v). We also noted that A= -Et+px is a solution, but that one may also write E(v)= E(x/t) and p(x/t) and take the derivatives d/dx partial A and d/dt partial A and obtain the same results. In this note we show how this follows from regular Lagrange theory and note that one already associates d/dx partial with p and d/dt partial with -E i.e. the quantum theory for a free particle is present.