This note reviews the lagrangian analysis and shows that it is rigorously self consistent. The formal Euler Lagrange equation (2) is valid in any coordinate system. In the Cartesian system it is equivalent to Eq. (6), and in the next note it will be expressed in plane polars. These are the fundamentals, and it is always important to keep on checking the fundamentals. These fundamentals can now be augmented by vacuum interaction in many interesting ways.