physics test cases

[gusl]

What's your favorite equational derivation in physics? I'm testing my equational theorem prover.

If you know any of physics arguments that use any non-standard reasoning (e.g. using diagrams), please please let me know.

Also, if you have any problems which involve semantic reasoning... maybe boring textbook examples will be good enough.

If you're wondering why I'm asking, it's because my thesis is titled "Automating Normal Science".

Ok.. back to Halliday & Resnick.

Comments

[spoonless.livejournal.com]

What's your favorite equational derivation in physics? I'm testing my equational theorem prover.

My favorite is the derivation of the least action principle of classical mechanics from Feynman's path integral formulation of quantum mechanics. But it's way too cumbersome to try and write here.

Something quick and simple that I can think of that would be good to test out is deriving the non-relativistic kinetic energy of a moving mass by Taylor expanding the total relativistic energy:

E = mc²/sqrt(1-v²/c²)

~= mc² + 1/2*mv² + higher order powers of v

where the first term is then interpretted as the "mass energy" and the second term is then interpretted as the "kinetic energy" due to motion. This was very pleasing to me the first time I saw it because I always wondered where the "1/2" comes from in the famous 1/2*mv² which they usually just have people memorize the first time they see it.

[gustavolacerda.livejournal.com]

It's not at all clear why you should consider 1/2*mv2 as the KE, instead of a higher power of v.

Do the higher order powers of v have meaning? I thought that the total energy should be simply "mass energy" + "KE".


I always wondered where the "1/2" comes from in the famous 1/2*mv2 which they usually just have people memorize the first time they see it.
Me too! I suspect this was discovered empirically, though... so it's not a totally artifical situation brought on by the teachers.

[spoonless.livejournal.com]

It's not at all clear why you should consider 1/2*mv2 as the KE, instead of a higher power of v.

Well, the 1/2*mv² term is just the non-relativistic kinetic energy. The full kinetic energy is the entire series (aside from the mc² term). But when the velocity is very small compared to c, then only the lowest power of v matters and so that gives you the "non-relativistic" kinetic energy which is used in Newtonian physics. It's a "leading order approximation" to the (more) exact theory given by relativity.

Me too! I suspect this was discovered empirically, though... so it's not a totally artifical situation brought on by the teachers.

Well, most things in physics were discovered empirically at some point. What's neat is that theorists have been smart enough to come up with a limited number of axioms (quantum field theory plus general relativity) from which you can derive all the rest of the equations of physics. The only semi exception to this might be statistical mechanics which... as I mentioned in another thread, sort of has its own axioms... which follow mostly from pure logic & statistics. The only missing link is proving that quantum field theory and general relativity give rise to more complicated sturctures that obey the statistical laws we'd expect... I'm not sure if this has been fully proven yet, there may be a few holes in it yet.

[spoonless.livejournal.com]

Actually, I just realized why you're probably confused... and it's my fault. I should have written "powers of v/c" rather than "powers of v". The way I wrote it you would expect the terms to get larger and larger, but in fact they get smaller and smaller.

I'm used to doing everything in units where c=1 in which case what I wrote would work fine... but in SI units you'd have to factor out a c before doing the expansion. So I should have written:

E ~= mc² [ 1 + 1/2*(v/c)² + higher powers of (v/c) ]

which when multiplied out gives you exactly the terms I wrote, except that the expansion is in terms of v/c not just v. So for v/c much less than 1, every term will be negligible compared to the last... which makes only the first term matter when you actually go to measure the energy. Only when v gets close to c do the other terms become significant, although they're always there to some extent.