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

[smandal.livejournal.com]

If you want to see some serious derivations (which you would comprehend), get a copy of Jackson Classical Electrodynamics or Sakurai's Modern Quantum Mechanics.

You'll find that much of it is semantic. "Theoretical physics is science locally isomorphic to mathematics."

[gustavolacerda.livejournal.com]

in what sense is it "semantic"?

[smandal.livejournal.com]

Physical reasoning is often inserted in the middle of derivations, to inform the math.

[gustavolacerda.livejournal.com]

Can you give me an example of "physical reasoning"?

It sounds like this "inform" the math always comes in the form of adding assumptions... am I correct?

[smandal.livejournal.com]

How do physics derivations work? The most basic involves physical laws and boundary/symmetry conditions, which are easy to express algebraically. (They might have to be transcribed from a diagram.)

The next level of sophistication is thinking about what you can throw away, via Taylor expansions and other approximations. This is harder to express algebraically, but still possible. It would require significant skill to for see the possible approximations, or computer power to see which approximations would work.

Finally, there is coming up with a whole new model. This would be entirely semantic/diagrammatic. For example, as with novel objects, like composite fermions or path integrals or Feynman diagrams.

[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.

[bram.livejournal.com]

Simple harmonic motion from a restoring force (trivial)?

Blackbody spectrum from E=h nu?

Lorentz transformations from speed of light constant (and other assorted reasonable assumptions)

[gustavolacerda.livejournal.com]

Simple harmonic motion from a restoring force (trivial)?
Seems like a good one!


Blackbody spectrum from E=h nu?

This equation describes the energy of a photon, right? Is this related to the infamous T^4 result?


My problem is precisely to make these "reasonable assumptions" explicit. Hopefully this won't be too hard to dig out of the derivations.

Symmetry Arguments

[http://users.livejournal.com/_greg/]

My success when I was studying physics in college was due to my being uninterested in memorizing formulas and being good at deriving what I need on the spot. My biggest tool has always been symmetry, including abstract symmetries. Often an argument based on symmetry made a problem trivial (problems given in physics classes usually are trivial, but disguised). Other times the symmetries were the clue to the approach and to deriving what was needed. I suppose another way of saying it is invariances. I find the work of Emma Noether particularly inspiring.

Re: Symmetry Arguments

[gustavolacerda.livejournal.com]

My success when I was studying physics in college was due to my being uninterested in memorizing formulas and being good at deriving what I need on the spot.

Me too, although you could say I only "minored" in physics. It sounds like we have a similar "psychoepistemology".

I've always been one to do everything from first principles, especially abstract principles of mathematics, rather than "physical laws"... which is the reason I like information theory and proofs that use Kolmogorov Complexity.

Re: Symmetry Arguments

[gustavolacerda.livejournal.com]

So for example, I'm the kind of guy who always asks whether the hypothesized result is invariant under Lorentz transforms (speed-relativity is an abstract philosophical principle).

[tdj.livejournal.com]

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

I suppose a Feynman diagram is too obvious?

[gustavolacerda.livejournal.com]

Do you know a specific argument / derivation that uses Feynman diagrams?

[spoonless.livejournal.com]

There are tons of examples, but one of the most common is the Ward-Takahashi identity for QED. Like any perturbative statement in quantum field theory, it can be written entirely with Feynman diagrams (aka the "Feynman calculus"), but if you really want to prove it explicitly it's better to write it out in terms of path integrals.

[veddersbetter00.livejournal.com]

my memory of such things is really, really fuzzy, but i believe there is a way to derive moment of inertia from density formulas? i found it to be rather simple and fun, but that was lifetimes ago and this suggestion might be not at all along the lines of what you were thinking of.

[(Anonymous)]

I once saw a derivation of F = ma starting from Schroedinger's equation. I think it's outlined in Eisberg and Resnick.