How do you politely ask someone to be on your committee?
I suggest an in-person meeting with each of them, which you can initiate with a brief but explanatory email: "I would like to meet with you for an hour in your office to discuss the possibility of you joining my dissertation committee.  Based on your work and experience in X, Y, and Z, I think you might be a valuable committee member. ".

When you meet in person, you can treat it as a two-way interview.  They should be interviewing you to determine if this is a good match and if they believe in your research topic and approach.  Likewise, you should be interviewing them to determine that they will perform well on your particular committee.  Do they have enough time?  Do they get along with other committee members? How do they evaluate dissertations and the process of writing dissertations?  What sort of feedback or support are they capable of giving you outside of the official meetings? What do they dislike or seek to avoid in the process of evaluating dissertations or being on the dissertation committee?  And so on.  You should also ask questions regarding their future plans, including sabbaticals, possible changes in job or employer (esp. for non-tenured professors), or possible retirement.

These questions that you ask a prospective committee member can help you avoid many committee problems later on.