Energy balance from the Boltzman tansport equation I'm trying to understand the derivation of the energy balance equation from the 2nd moment of the Boltzman transport equation, following this lecture notes.
Unfortunately the authors only derived the mass continuity equation as well as the momentum balance but did not explain how to arrive at the energy balance presented by equation 6.37.
I feel like the derivation of the energy equation is by far the hardest, and I have no clue how to arrive there. I'm missing a lot of tricks to get there. Everybody mentions you need to utilize the momentum balance equation as well as the continuity equation, add or subtract something ...yada yada...The problem is, you need to have some intuition to recognize the terms which might cancel out .
I've googled for hours now and it seems like nobody ever made an effort to actually derive it. Everybody is just referencing somebody else, who is then referencing Laundau & Lifshitz. 
Is there any full derivation (a classical one, I dont care about quantum mechanics) available online?
Any suggestions?
EDIT: figured it out using this excellent lecture
 A: I haven't done the derivation for this system before, but I've done it for the Navier-Stokes equations and there's similarities in how to do it. In the notes you linked to, they derive the zeroth order moment (continuity) and that's equation 6.11. Then they go through a series of steps where they are just re-arranging the terms using the product rule from calculus and defining things. And that keeps going. 
Then, in section 6.3.2, they derive the first order moment and call it the momentum equation. Here, they do the same basic tricks, but they start from the zeroth-order moment and multiply it by $m$. Then they just go through a bunch of steps of rearranging things again using the product rule. They also do some common steps where an integral over all space of something gives you an average (6.23) and any quantity can be written as an average component and a fluctuation (6.24). Then they expand out those products (6.25) and remark that the average of the fluctuating quantities are 0. Then they define some more stuff to clean up notation. They also keep applying things like the divergence theorem or product rule to figure out which terms can be simplified or cancel. 
Then they move into 6.3.3 and just say that the energy equation is derived from the momentum equation. The energy of a particle moving is just $m v^2$, and so to go from momentum equation to energy equation, you take the dot product of the velocity $\vec{v}$ and the momentum equation. That's the trick to get from one to the next.
This is the same general process they did to go from continuity -> momentum -- they multiplied by mass and then did a bunch of simplifications using some calculus rules. 
You'll need to do the same thing here. Dot the momentum equation with velocity, and then apply a lot of calculus rules to rearrange and introduce notation to simplify. The best way I would recommend to get there is to really practice deriving the ones they show all the steps for so you become fluent at rearranging things with the product rule, divergence theorem, and so on. Then you'll see when you can apply those principles in deriving the energy equation.
