After some discussions with Horst the derivative of a scalar with respect to a vector is defined as the well known vector gradient (Google "derivative with respect to a vector") so that is how the Euler Lagrange equation (4) of the note should be defined. I exemplify this definition by deriving Eq. (16) of UFT377. I also give an example of a lagrangian differentiated with respect to bold r dot from Appendix 21 of Atkins, "Molecular Quantum Mechanics", third edition and also show how this equation can be interpreted. So in vector algebra
del phi : = partial phi / partial bold r
That is what I meant in UFT377. The important result is that the relativistic Newton equation gives retrograde precession. Discussions such as this are very important in order to check fundamentals. The definition (1) of this note can be extended to n dimensions, notably four dimensions, and the Lagrange method is important throughout modern physics.