I've looked at this problem, and the preceding pages in Spiegel's book, and it would seem to me that any particle travelling in this orbit would have an infinite velocity at O, at least if the velocity was non-zero anywhere else in the orbit since the potential is infinitely negative when r=0 so the kinetic energy would have to be infinite. Wouldn't it be the same situation as two massive particles moving towards each other until they are zero distance apart?
If that's right than your angular momentum at O becomes a zero times infinity situation...
Here's a Q&A from Quora that addresses a particle moving through the origin in central force motion.
Conservation of angular momentum really is a new phenomenon, one that does not follow from the Newtonian mechanics you already know; therefore it deserves its own place as a law. Specifically, you have proven that
If a system experiences no torque, then its angular momentum is conserved.
However, this statement by itself is useless. Maybe all systems always experience torque; maybe a system can exert a torque on itself. What we really want to say, i.e. the actual law of conservation of angular momentum, is more like
An isolated system's angular momentum is conserved.
To see how these are not equivalent, suppose we have a system of two isolated particles, one above the other. Newton's third law does not forbid the particles from pushing left and right on each other. But then the system will begin spontaneously rotating! It changes its own angular momentum by exerting a torque on itself.
To force conservation of angular momentum, we need to use the strong form of Newton's third law,
Forces between particles come in action/reaction pairs, and these forces are directed along the line of separation between the two particles.
This is a fundamentally new assumption, so angular momentum really is its own thing. On a deeper level, conservation of linear and angular momentum follow from translation and rotational symmetry of space, and it's possible to have spaces which only are translationally symmetric, or only are rotationally symmetric. The two are independent.
Best Answer
Well, yes and no. You need to use conservation of angular momentum but you also need to use the radial equation, which is specific for the case of gravity. A different radial force law would still conserve angular momentum but it wouldn't have elliptical orbits. Not to mention that conservation of angular momentum can be deduced from Newton's laws and the law of gravitation. So it doesn't seem very useful to say that the first law is a consequence of conservation of angular momentum, since you need a lot more than that to prove it. The same goes for the third law.
The second law, however, can be deduced just from conservation of angular momentum, so it holds for any central force, not just gravity.