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I'm watching this video lecture by Leonard Susskind of Stanford:

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

Both the first linked video to the first version of the question(v1), and the second link video in later versions of the question, is about deriving Euler-Lagrange equations from the principle of stationary action.

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.

2) Susskind does indeed use time discretization in the new video. Many places in physics we can approximate a continuous model with a discrete lattice model, where derivatives get replaced with finite differences. In the end, we let the lattice spacing go to zero, and recover the continuous model. In the last chapter of his book, Goldstein speaks about transitions between discrete and continous models. This may help you to get the big picture. It is useful in physics to understand both the discrete and continuous approach. When done correctly, they should be equivalent, but for certain applications one approach is often more convenient than the other. I'll try to make a future update if I find an exact reference to Lenny's discretization argument.

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The standard textbook for discrete to continuous stuff is anything in computational physics. Numerical Recipes in C is a widely used reference, and it has many such things. Feynman does discretizations in Feynman and Hibbs, also in the Feynman lectures, and it is commonly picked up by writing simulation codes.

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