Recent content by Rick16

I just thought about something. I am new to Landau and I am not familiar with his style. When he writes the displacement as ##dl_1^2 =a^2 d\theta^2 + a^2 sin^2 \theta d\phi^2##, perhaps this is not meant to be an intermediate step towards the final solution, but just a helpful statement?

This problem has been addressed before under https://www.physicsforums.com/threads/trouble-understanding-coordinates-for-the-lagrangian.1006528/ I also copied the following problem statement with Landau's very sketchy solution from this old post, because I don't have the English edition of the...

In any case, this is my first reading of the text, and I am satisfied that I can show that ##\vec v## is constant when I assume that the Lagrangian is known. I don't need to show it for an arbitrary function ##L(v^2)## at this point. I may come back to it later, but now I must move on. Thank you...

@Vincf already had a counter example in #5.

The fact that dL/dv is constant was never the question. The question was why v is constant.

This would be my next problem: I don't understand the paragraph before equation 4.1. But like I already wrote, my head is already spinning, mostly because of problem 3, and it looks like I won't be able to understand anything more today.

I also tried out several different functions for ##L(v^2)##, but I found that the velocity was constant for all of them. You apparently found one for which this is not the case. When Landau writes that the Lagrangian can only be a function of the square of the velocity, he actually does not...

Yes, I know, but PeroK's suggestion is one way to see why the velocity would have to be constant (provided I already know the form of the Lagrangian). Landau must have had something else in mind.

I just wanted to ask why on earth the Lagrangian would be ##L=\sqrt {v_x^2 +v_y^2 +v_z^2}##, when I discovered that you had changed that. I spent quite a while this morning trying to work through the problems at the end of chapter 1, and my head was already spinning. This Lagrangian really...

Thank you, that is an interesting trick with the ratios. It is not likely that I would have thought of it. I suppose you know how to do it? Because I don't.

This question does not seem to evoke a lot of interest. Could somebody just tell me if it is legitimate to write ##\frac{d}{d\vec v}(\vec v \cdot \vec v)=2\vec v##? I.e. I wonder if I can take the derivative directly with respect to the vector as a whole or if I must take the derivative...

I have begun reading Landau's Mechanics. In chapter 1, §3 he writes ##\frac {\partial L} {\partial \vec v}=const.##, where ##L## is a function of ##v^2##: ##L=L(v^2)##. He then writes that from this it follows that ##\vec v=const.## I want to try to show formally that v is constant, but I am...

Thank you. This is it. It is not even new to me. I read about it probably more than once, but I had completely forgotten about it. This shows what happens when you just read physics texts without doing problems.

One more comment to show where exactly my problem lies: Potential energy is defined at a specific position. A System does not have a specific position. How then can I define/understand what the potential energy of a system would be?

Here is the solution: The kinetic and potential energies of the system are $$U_k=2\cdot\frac{1}{2}mv^2=mv^2,~~~U_p=-\frac{Gm^2}{2r}.$$ To show how these are related, apply Newton's second law to the motion of one of the particles: $$F=ma\Rightarrow \frac{Gm^2}{(2r)^2}=m\frac{v^2}{r}.$$ Multiply...