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I am reading Landau's Classical Mechanics book and I'm having trouble understanding the concept of "internal energy". In Landau's words: the energy of a mechanical system which is at rest as a whole is usually called its internal energy.

He then considers a particle which spontaneously disintegrates (there are no external forces, so this system is closed) into two particles, which will be labeled 1 and 2. If we're in the frame K in which the initial particle is at rest, then by conservation of momentum we have that the two particles will move away from each other with momentums which are equal in modulus (let's call it $p_0$). Also, he says that conservation of energy implies \begin{equation} E_i = E_{i1} + \frac{p_0^2}{2m_1} + E_{i2} + \frac{p_0^2}{2m_2} \end{equation} where $E_i, E_{i1}, E_{i2}$ are the internal energies of the initial particle, particle 1 and particle 2 respectively. This is where I get lost because Landau has only explained (at least up to this point) how to write the Lagrangian of a closed system, but a single particle is not a closed system, since it could be interacting with the other particle. And if the two particles weren't interacting we would have $E_{i1} = E_{i2} = 0$, so what is the point of the formula above? Also, what is the expression for $E_{ik}, \; k = 1, 2$?

Am I missing something?

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The fact that the 'particle' is able to decay implies that it must be a composite body, and so we have to assume that its constituents are also. For example, consider the splitting of a dense cloud of particles into two smaller groups. So long as we look long enough after the splitting, so that they no longer interact when they are far away, it is still valid to treat all the bodies involved as particles.

In this case, the internal energies of the bodies might be some interaction energy between the molecules making up the cloud. More generally, the internal energy of the body is just any sort of 'self-contained' energy.

L&L love this sort of terse writing where you're expected to fill in the details for yourself. In theory it's possible, but we don't all have minds like theirs!

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  • $\begingroup$ Ok so, if I understood your argument, I have two make two assumptions, namely 1) that the particles 1 and 2 are actually composite bodies and therefore carry about "their own" potential energies $U_1$ and $U_2$ and 2) that I observe the two particles when any possible interaction between them is negligible (assuming thus that this interaction goes to zero when the distance between the two particles goes to infinity). Am I correct? $\endgroup$
    – fresh
    Commented Apr 15, 2020 at 22:38
  • $\begingroup$ Yep, that's exactly right! $\endgroup$
    – xzd209
    Commented Apr 15, 2020 at 22:40
  • $\begingroup$ Ok, last one I promise. I still don't understand how to describe the system soon after the disintegration, that is when the hypothetical interaction between the two particles is not negligible. Am I missing tools that maybe haven't been introduced yet? $\endgroup$
    – fresh
    Commented Apr 15, 2020 at 22:47
  • $\begingroup$ The above formalism isn't able to deal with that sort of thing, as it is impossible to describe without knowing exactly what the bodies are made of or how they interact etc: rather, it is only applicable when you look after the particles have separated. To look again at the splitting gas example in my answer, in order to know what happens directly after the collision, you would need to know exactly how the different parts of the gas are flowing, and how hot each part is, which is simply not something we deal with in such a 'high level' calculation. So yes, you need a different set of tools! $\endgroup$
    – xzd209
    Commented Apr 15, 2020 at 22:53
  • $\begingroup$ I hope that helps, feel free to ask anything else if you're still not sure though $\endgroup$
    – xzd209
    Commented Apr 15, 2020 at 22:54

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