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First, I believe there is a trivial error. The second equation should have another $\Delta t$ multiplying everything on the right. It is divided out later when the equation I set equal to 0.

  1. Given that $L$ is a function of $ x_7, x_8, x_9$ how can he justify evaluating it at single points?

  2. Also, why are points 8 and 9 used rather than 7 and 8?

For context, this is the last equation on page 112 and the first equation on page 113 of Susskind's The Theoretical Minimum.

rob
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MadScientist
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2 Answers2

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Yes, I agree it looks like the second equation is missing an overall factor of $\Delta t$ on the right.

The action is a function of all the $x$s for the whole trajectory, but the Lagrangian is not. It is only a function of a position and a velocity. So it makes perfect sense to evaluate it using the position and velocity for a single time.

Points 8 and 9 are used because (I guess) the action is defined as a sum of terms involving $x_n$ and $x_{n-1}$, so the terms that involve $x_8$ are the 8th and 9th terms. If he had defined the action using $x_n$ and $x_{n+1}$ then he would have had to take the 7th and 8th terms instead. It's just a choice of convention.

Nathan Reed
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  • I don't understand how he can have an equation with two variables and then treat it as though it is composed of one in the next step. Also, the $x_7$ and $x_8$ terms are included in the series also. See the second line of the first equation. – MadScientist Dec 11 '14 at 08:12
  • @MadScientist No, the $n$th term depends on $x_n$ and $x_{n-1}$. So the $n = 8$ term depends on $x_8$ and $x_7$, while the $n = 9$ term depends on $x_9$ and $x_8$. Those are the only terms that depend on $x_8$. So to get the total dependence on $x_8$ he only needs to evaluate things at $n = 8$ and $n = 9$. – Nathan Reed Dec 11 '14 at 08:15
  • I'm not seeing why he can't choose points 7 and 8. – MadScientist Dec 11 '14 at 08:18
  • @MadScientist The $n = 7$ term only depends on $x_7$ and $x_6$, so it's irrelevant to $x_8$. And if you leave out the $n = 9$ term you don't capture all the dependence of the action on $x_8$. – Nathan Reed Dec 11 '14 at 08:19
  • Thanks. I see it now. Everything makes sense. I was confused. – MadScientist Dec 11 '14 at 08:25
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I still think it might be wrong. the 8th term is dependant on x8 and x9 and the 7th term is dependant on x7 and x8. Therefore L should be evaluated at n=8 and n= 7.

user191106
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