I recently came across a very cool book called Div, Grad, and Curl are Dead by Burke. This is apparently a bit of a cult classic among mathematicians, not to be confused with Div, Grad, Curl, and All That. Burke was killed in a car accident before he could put the book in final, publishable form, but it can be found in various places online, e.g., on scribd.com. At the beginning of section 12, in the context of Newtonian mechanics, he says, In the language of Einstein-style index gymnastics, applied in a nonrelativistic context, this amounts to a statement that energy is a scalar, and displacement is a contravariant (upper-index) vector, so force should naturally be considered as a covariant (lower-index) vector. The first thing I'm unsure about here is whether energy is really a scalar in the context of nonrelativistic mechanics, if "scalar" is taken to have its full Einstein-style interpretation of "invariant under any change of coordinates." Doesn't nonrelativistic energy change when you rescale your coordinates, implying that it's a scalar density rather than a true scalar? (After all, in relativity, rescaling coordinates changes all the components of the energy-momentum tensor, which means it changes the mass-energy.) The second thing that bugs me is that if you were to reason from Newton's second law, it seems like you would "naturally" conclude the opposite, that force is a contravariant vector. If we follow the usual but arbitrary convention of saying that upper indices are used for distances measured on a ruler, then this breaks the otherwise perfect symmetry between vectors and their duals. It then seems clear that things like velocity, which can be obtained by differentiation with respect to a scalar, should also take upper indices (be contravariant). But I'm less convinced that this then breaks the duality symmetry in the case of Newtonian force, or a relativistic quantity like the stress-energy tensor...?