Indeed, it is at the start of the book. It also relies on the assumption that the Lagrangian is invariant up to a total time derivative under Galilean transform

Indeed, it is at the start of the book. It also relies on the assumption that the Lagrangian is invariant up to a total time derivative under Galilean transformations, which leads to `dL/d(v^2)` being a constant (1/2 m). This is, I believe, the most fundamental way to look at it. Otherwise you could ask why the Lagrangian can't have other terms like v^4.