I was told that in order to solve the Kepler problem (6 degrees of freedom in total) you have to proceed, step by step, to reduce those degrees of freedom using the integrals of motion. You do so until you have no one left. Now you can do so in "clever" ways (you see that the motion can happen only on a certain plane, then that the two bodies can be seen as a single body with a certain mass, and so on…) or in a more general and theoretical way (Hamilton-Jacobi). What matters is that you should find six integrals of motion.
Now energy (1 scalar) plus angular momentum (3 components) plus linear momentum (3 components) make a total of 7 integrals of motion. I was told that one of the components of angular momentum is really dependent on the other components, so it cannot be considered as an independent integral of motion. My question is: what's the relationship between the components of angular momentum that makes them dependent?
Best Answer
A single particle moving under any conservative central force has at least four constants of motion, the total energy $E$ and the three Cartesian components of the angular momentum vector $L$. The particle's orbit is confined to a plane defined by the particle's initial momentum $p$ (or, equivalently, its velocity $v$) and the vector $r$ between the particle and the center of force. The planar nature impose thus $\boldsymbol r \cdot \boldsymbol L=0$. In the special case of the Kepler problem, the force satisfies a inverse-square law: $F=-k/r^2$.
As you said, this particular form allows many simplification in the description of the Kepler motion. One of this is the possibility to introduce another vector instead of the linear momentum, that is also conserved, called the Runge-Lenz vector $\boldsymbol A$.
This is a conserved quantity, and so we can say that the seven scalar quantities $E$, $\boldsymbol A$ (which substitutes $p$)and $\boldsymbol L$ are conserved.
One of the main reason to introduce the RL vector is just to answer to your question. In fact, if we solved the motion by means of any technique, we would find that the constants of the motion are all algebraic functions of $r$ and $p$ that describe the orbit traced by the body as a whole (orientation in space, eccentricity, etc.). None of these seven conserved quantities relate to where the particle is located in the orbit at the initial time. Since one constant of the motion must relate to this information, say in the form of $T$, the time of the perihelion passage, there can be only five independent constants of the motion describing the size, shape, and orientation of the orbit. We can therefore conclude that not all of the quantities making up $\boldsymbol L$, $\boldsymbol A$, and $E$ can be independent; there must in fact be two relations connecting these quantities.
These are $\boldsymbol r · \boldsymbol L = 0$ that becomes $\boldsymbol L\cdot\boldsymbol A=0$ and $A^2 = m^2 k^2 + 2 m E L^2$, giving five independent constants of motion. This is consistent with the six initial conditions (the particle's initial position and velocity vectors, each with three components) that specify the orbit of the particle, since (as we noticed before) the initial time is not determined by a constant of motion. Since the magnitude of $\boldsymbol A$ (and the eccentricity e of the orbit) can be determined from the total angular momentum $\boldsymbol L$ and the energy $E$, only the direction of $\boldsymbol A$ is conserved independently; moreover, since $\boldsymbol A$ must be perpendicular to $\boldsymbol L$, it contributes only one additional conserved quantity.
Resuming, the angular momentum vector and the energy alone contain only four independent constants of the motion: The Laplace-Runge-Lenz vector thus adds one more.
Notice that, it is natural to ask why there should not exist for any general central force law some conserved quantity that together with $\boldsymbol L$ and $E$ serves to define the orbit in a manner similar to the Laplace-Runge-Lenz vector for the special case of the Kepler problem. The answer seems to be that such conserved quantities can in fact be constructed, but that they are in general rather peculiar functions of the motion.