Question on: Classical disintegration of particles, Landau-Lifshitz series on Physics

MManuel Abad

Hi there.

I'm a second-year physics student. I'm from Mexico (so please, forgive my english), and I'm on winter holidays. I've been studying Mechanics with the first volume of the "Course of Theoretical Physics", by Landau and Lifshitz, that awesome though hard book series. My study plan is to do all exercises and to explain every mathematical detail Landau skips in his exposition. Two days ago, everything was alright, but then I found, in chapter IV "Collisions between particles", something I cannot fully comprehend.

In the first section (from the last paragraph in page 43 in the 3rd english edition), Landau considers the disintegration of many identical particles, isotropically oriented in space and wants to know the distribution of the resulting particles in direction etc.

Please, I can't get the picture of the problem. ¿Can someone explain it to me?
I think it's this:

You have this bunch of particles, all identical, randomly oriented. Each of them disintegrates in two resulting particles of masses m1 and m2, with equal momentums in the C (center of mass) frame of reference, and the direction of m1 making the same angle phi (say) with that of m2 for every original particle. As a result of this, there's a uniform distribution of the directions of the resulting particles. But then there's the problem:

So far in the text, the angle LaTeX Code: \\theta 0 was the angle the velocity (seen from the C frame) of one of the particles made with the direction of the velocity V of the original particle as seen from the LAB frame of reference (see fig. 14). Now, what does LaTeX Code: \\theta 0 mean?? When you consider that system of particles disintegrating in the LAB system, are they moving with velocity V? In what direction? Is V the velocity of the center of mass of the system of particles as seen from the LAB reference frame?? Then LaTeX Code: \\theta 0 is the angle, say, particle 1 makes with the velocity of the center of mass? But then LaTeX Code: \\theta 0 varies with the resulting particle of mass m1 you choose, cuz they're resulting particles from different original particles differently oriented. If this is so, then how do they measure the solid angle dLaTeX Code: \\omega ??? And another question: how do they obtain eq. 16.8?? I mean, I know they differentiate v2 and all that (paragraph between eq 16.7 and 16.8), but d(cos(LaTeX Code: \\theta 0)) is -sin(LaTeX Code: \\theta 0)dLaTeX Code: \\theta 0!!! WITH A MINUS SIGN!!! When you substitute in eq. 16.7 and obtain eq 16.8, what happened to the sign???

Please, I need your help!!!
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