Orodruin said:
robphy said:
While true, the issue I have with these attempts is to motivate WHY these forms arise as such...
what is the interpretation, beyond being a convenient way to find the non-relativistic case (using substitution, rather than a limit)?
Why does there have to be a deeper interpretation than being a didactic trick for people who are not yet familiar with Taylor expansion?
Of course, there doesn't have to be...
but I think we would prefer to have a deeper interpretation [using physical principles and methods developing intuition, not a collection of tricks]
rather than...
here's a mathematically equivalent form [motivated algebraically because we know the limit we expect].
Does this algebraic form teach us anything else?
Maybe.
It would be nice if the result of the "didactic trick" ("didactrick")
naturally fell out of a physics calculation (like the work-energy theorem)...
or had a Minkowski-spacetime-geometric interpretation.
But if such a "didactic trick" is all that one has, then it would be better than not having one.
When developing relativistic kinetic energy, Kleppner (An Introduction to Mechanics) notes
"This expression for kinetic energy bears little resemblance to its classical counterpart."
The works I cited above attempt to develop a resemblance algebraically...
but I don't think those attempts tell us anything more.
(I am writing an article that develops the resemblance using Minkowski-spacetime-geometry,
with analogues in Euclidean geometry and Galilean-spacetime-geometry.)