Derivation of the Lagrangian method using discretized time axis

I'm watching this video lecture by Leonard Susskind of Stanford: http://www.youtube.com/watch?v=3apIZCpmdls

After some preliminaries, at 34 minutes he jumps into a discretization of the time axis and proceeds to mess around with it and the Lagrangian, completely losing me in the process. I haven't come across this discrete method before. What is this called and are there any resources on the net (or textbooks) that use this approach rather than a continuous approach? Is it necessary to do this discretely rather than continuously?

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Joebevo, is this really a homework question? (i.e. did it arise in the context of a homework or self-study exercise?) – David Z Mar 11 '12 at 6:05
@David Zaslavsky: self-study (but I thought it was of educational value, hence the homework tag). – Joebevo Mar 11 '12 at 8:12
@Joebevo: I didn't mean to ask which it was, homework or self-study, since the two are equivalent for our purposes. The thing is, this doesn't read like a homework question at all. Just because the question has educational value doesn't mean it deserves the homework tag; if that were the case, every question here should have it. If it's related to some specific exercise, whether you are doing that exercise for your own benefit (self-study) or because someone told you to (homework), then you should include that exercise in the question. Otherwise, the homework tag can probably be left off. – David Z Mar 11 '12 at 8:26
P.S. It's a good question either way, so it's not like you're doing anything wrong here, whether it is appropriate to include the homework tag or not. I'm just trying to make sure the question is categorized correctly. – David Z Mar 11 '12 at 8:27
I have updated the link, which now points to the correct video. – Joebevo Mar 11 '12 at 8:47

1) Susskind does not mention time discretization in the first video. Time $t$ is there a continuous real parameter throughout the video, and the derivation is very similar to e.g. Chapter 2 in Herbert Goldstein, Classical Mechanics.