First integrals and osculating elements

First integrals are functions of position and velocity components, which are conserved on orbits. Functions of first integrals are also first integrals and do not count when we talk about independent first integrals. We can, however, use any set of independent first integrals. How many independent first integrals does the Kepler motion have?

Answer: 5. These are: semi-major axis, eccentricity, inclination, longitude of ascending node and argument of pericenter. The vectors ${\bf A}$ and ${\bf L}$ can be expressed in terms of these first integrals. In particular, the components of ${\bf A}$ and ${\bf L}$ and the orbital energy are not independent. Since ${\bf A}$ and ${\bf L}$ are perpendicular, ${\bf A}\cdot {\bf L}= 0$.

Exercise: Derive an equation which relates the lengths of ${\bf A}$ and ${\bf L}$ to energy. It is a matter of convenience what set of first integrals to use.

When ${\bf F}\neq 0$, the first integrals of Kepler motion are not conserved anymore, i.e. they are not first integrals of the perturbed motion. Orbital elements change in time. The instantaneous orbital elements in this situation are called the osculating elements. You can visualize them as having Keplerian ellipses shadowing the perturbed motion. At any moment, there is a Keplerian ellipse which touches the actual trajectory, having the same position and velocity vector as the actual trajectory.