- #1

gionole

- 281

- 24

Landau first starts to mention the variational calculus technique and how euler lagrange equation is given. He uses ##q(t)## such as for the path, but his proof is the general one we all know. So he derives EL equation and writes it down. Then starts and builds his logic why ##L = K - U##.

His first argument is that equations of motions from different reference frames(inertial and non-inertial) have the different EOM form and in that case, he decides that we can always find inertial frame and stick with it. Ok, clear till now. Then starts to derive ##L = K - U## for the freely moving particle. He says that in inertial frame, the freely moving particle would only change speed if external force acts on it and he says that in this frame, space and time are homogeneous. i.e in this frame, freely moving particle's equation of motion would be the same(wouldn't change) at all times and at all points in this inertial frame. So because of this, he says that Lagrangian can not depend on ##q## and ##t##.

Question 1: Well, Everything was clear, till when he said that Lagrangian can not depend on ##q## and ##t##. He bases this logic on the homogenous time and space argument, but how homogeneous time and space implies anything about Lagrangian ? they imply that equations of motions exactly stay the same, but not sure why it implies anything about ##L## not containing the ##q## and ##t##.

**It seems to me that because Landau already has derived the EL before this moment, in his head, he imagines what if ##L## contained ##q## ? then for sure equation of motion would end up to be ##m\ddot x \neq 0##(due to EL) which is wrong due to Newton. Then what if ##L## contained ##t##? then EOM wouldn't be the same as of newton's. So if you think as well, that Landau imagines ##L## to be containing ##q## and why it would be wrong and thats why he says, ##L## couldn't contain ##q##, then why in the first place does he mention homogenity, inertial frames, .e.t.c ?**

Question 2: I like the logic of how he gets to the point(excluding Question 1's thoughts) that L must be ##cv^2## where ##c## is something he doesn't know yet. Then he out of nowhere says that ##c## must be ##\frac{1}{2}m## and that's it. It seems to me that he might have used the Euler lagrange on ##cv^2## and tries to find ##c## such as he arrives to Newton's equations and he would only arrive at newton's equations if ##c == \frac{1}{2}m##.

**What was the whole point of using: inertial, non inertial frames, "adding total time derivative of Lagrangian doesn't change equations of motion" if he in the end still relies on Euler Lagrange and specifically newton's already derived equations for free particle. He could have just never mention the whole explanation and just wrote something like this: "We're looking for the Lagrangian for the free particle, we know newton arrives with a constant speed equation for such particle, so what should our L be such that using Euler lagrange on it would get us ##m\ddot =0## and it's super easy to come up with ##L = \frac{1}{2}mv^2##".**

Question 3: Not only that, I think he does the same to arrive at ##K - U## for the system of particles and still relies on Euler lagrange equation.

If Landau uses Euler Lagrange + Newton's already derived EOM's, then what's the need at all to be mentioning such huge explanations about inertial frames, homogenity, lagrangian total time derivatives, e.t.c. It's pointless.

Here is the book link. (from page 4)