Every time I teach mechanics (and this is something like the 21st year I have taught it at the undergraduate level) I learn something new. This week we are talking about many-body systems; I had two epiphanies (both trivial, but still): The first is that the description of the object in terms of a center-of-mass vector and then many difference vectors away from the center of mass (one per "atom") is purely a coordinate transform. Indeed, it is just generalized coordinate system that is related to the Newtonian coordinates by a holonomic transformation. Awesome! So when the Lagrangian separates into external and internal terms, this is just a result of the appropriateness of that transformation.
The second is that the definition of the many-body system is completely arbitrary. It should be chosen not on the grounds of being bound or solid or connected but rather on the grounds of whether choosing it that way simplifies the problem solution. Both of these realizations are simple and obvious, but it took a lot of teaching for me to get them fully. I am reminded as I realize these things that the physics concepts we expect first-year undergraduates to manipulate and be comfortable with are in fact pretty damned hard.
