Actually I agree with you completely. This is one of the reasons why I was insisting on taking Math classes when I was working on my Physics PhD. (Which I haven't gotten yet, so I'll probably be ticking off a whole new set of professors in the future.) Especially in fields like statistical mechanics. They were doing approximations that even _I_ thought were suspect. (Fermion distribution functions especially.) Of course, in stat mech if you don't know how to solve an equation, the rule of thumb is to approximate it. (Yes, TPH, even more so than in Classical Mechanics! )
-Dan
I know this is getting of topic...
But that is the ugly side of physics.
I never, never, ever been able to understand those derivations in physics. For example, my professor was showing why a hanging cable is a caternary. He set up a differencial equation and shown that hyperbolic functions solve it, good that I understand. But how did he get that equation! He just divided the section into a very small piece. And said, well it is almost like a line... some more crazy statement... and wonderful we have an equation. However, if you mimick his approach say with a parabola instead of a line, because you can say that curve is almost a parabola... the same crazy manipulations... and we have an equation... which is different!! So my problem is, how do you understand these derivations? There is no way. Which is why I consider these laws discovered by luck rather then by derivation.
It seems the rule in physics is: that it is important that it works rather than how it was derived.
I know what you mean about the approximations. Actually, in Intro Physics there is the concept of a (2-D) radius of curvature, which essentially models a curve as a series of arcs of a circle as opposed to a series of line segments. (I don't know how the concept is taken into 3-D, but there is a definition somewhere. I ran across it in my graduate Electrodynamics class.) One would hope that the caternary would at least also solve your "parabolic approximation" equation being that the parabola approximation would be a second order level approximation.
But in general, yes, if you use a different approximation you tend to get a different equation. The clearest example I can think of to this is a lecture I attended at Purdue U. where one of the Nuclear Field Theorists presented a talk that mentioned that he and his team had discovered a (theoretical) stable energy state lying at a lower level than the previously determined ground state for nuclear particles. For some 20 - 30 years Physicists had used an approximate solution for the ground state of nuclear matter and now they found out it wasn't really the ground state. (I don't think it was an earth-shattering development since I haven't heard anything about it since. But it's the principle of the matter that counts.) Some approximations seem to work better than others and as far as I know it's a crap shoot to decide which will be the best one to model nature on.
-Dan