How to learn celestial mechanics? I'm a PhD student in math and am really excited about celestial mechanics.
I was wondering if anyone could give me a roadmap for learning this subject. The amount of information about it on the internet is overwhelming, and I honestly don't know where I should start.
Any help would be greatly appreciated.
 A: For something so specialized, I'd suggest you read books instead of dive into the  big melting pot of confusing information called the internet. 
Nevertheless, there are a few good sites. This is certainly one of them.  
If you have access to journals, Celestial Mechanics and Dynamical Astronomy is a good one, as is Journal of Guidance, Control, and Dynamics (although generally more intended for spacecraft orbits etc., it still covers a vast amount of celestial mechanics).
Richard H. Battin has written the classic "An Introduction to the Mathematics and Methods of Astrodynamics". It is rather mathematically inclined, which you might appreciate.  
The big names are Poisson, Lagrange, Laplace, Poincare. Google-scholar for them with "celestial mechanics" and you'll get tons of papers, books, and other material. 
A: Apart from reading books about celestial mechanics, may I suggest an incredible computer game, called Kerbal Space Program? 
It is incredibly fun to use your knowledge of physics to build stuff in space -- and in process you develop your intuitions and skills around the subject enormously. As XKCD puts it (and, currently working at NASA, I  can confirm):    

A: This book, written by two (prominent) mathematicians, may be of interest to you. I would also follow Rody Oldenhuis' suggestion to look at the research literature and find examples you can work out by yourself and check against references. The mathematics is not very difficult, but the exciting part is in the computations.
A: This textbook is used in Universities (and by amateur astronomers such as myself with an interest in computing). Hope this is useful. 
Fundamentals of Celestial Mechanics, 2nd Revised, 1992, J. Danby
A: Space Mission Design and Operations - edx. This is one of the best course for the one who and has interests in space science and the pre-requisites is an engineer with high-school level math & physics. This course motivated me to go back to school studying a PhD space engineering in nearly future. 
