What is nature's formulary?

March 18, 2009 2:45 PM

Physics-filter: in the context of a diatribe against memorization, a physics professor of mine once mentioned that she fell in love with the discipline because all of modern physics can be derived from a few foundational equations. So, what are they?

This was an intro physics course in its first semester, and so, as I recall, my professor offered only a couple of examples: F=MA for all of classical mechanics, which seems reasonable to me, and Einstein's famous mass-energy equivalence formula.

posted by perissodactyl to science & nature (16 comments total)

Maxwell's Equations?

But good luck with deriving all of modern physics from them by hand during an exam.

From the wikipedia page: "And, of course, some situations demand that Maxwell's equations and the Lorentz force be combined with other forces that are not electromagnetic. An obvious example is gravity. A more subtle example, which applies where electrical forces are weakened due to charge balance in a solid or a molecule, is the Casimir force from quantum electrodynamics."
posted by GuyZero at 2:53 PM on March 18

I have a t-shirt, similar to this one, only with about a dozen more equations, that claim to cover everything that goes between

And god said ... [equations go here]
and then there was light

They are referred to as "Maxwell's Equations".
posted by nomisxid at 2:54 PM on March 18

perissodactyl, take a look at Sander Bais's beautifully designed little book The Equations: Icons of Knowledge. Also, here is a brief, brief summary by Martin Rees of the argument in his book Just Six Numbers, also well worth reading.
posted by cgc373 at 3:00 PM on March 18

Oh, to clarify: I don't think she was suggesting that deriving all of physics was practical for exam-taking or for any other purpose. But if you were a physics equivalent of a Ramanujan, say, working in isolation and with unlimited time in which to work, what's the smallest set of equations you would you need to recreate all of physics?
posted by perissodactyl at 3:03 PM on March 18

That's a great question. What is the smallest set of equations you would need to recreate all of physics?

The answer to that is the Theory of Everything (T.O.E.). We do not have it yet.

Maxwell's equations are what you'd need to recreate bog-standard E&M. You'd need a bunch more if you want to do classical physics (the domain of Newton). More if you would like to do quantum mechanics. Then you've got equations for special and general relativities. And it goes staggering on: quantum chromodynamics, M-theory, etc. Each one of these is a bit like the parable of the blind men describing the elephant.

For what it is worth, it wasn't uncommon in physics exams to go in with just a handful of equations memorized which were germane to that particular topic, then derive particular equations on the fly. It's a bit like how you would do it in trigonometry.
posted by adipocere at 3:18 PM on March 18

Special relativity can be derived from trying to reconcile Maxwell's equations in all frames of reference - that was Einstein's big breakthrough although it seems obvious in retrospect. And according (again) to wikipedia, if you use 5 dimensions, there's a relation between general relativity and Maxwell.

As usual, gravity is the big missing force.

And "all of physics" can't be re-derived from equations alone. The issue of whether electromagnetic waves propagate in a preferred frame of reference (the aether) can't be resolved by staring into a copper bowl. You have to go out and do an experiment or two at some point. Michelson & Morley were a pretty big deal.
posted by GuyZero at 3:31 PM on March 18

Here is a wikipedia article with a few more examples. I also recommend the article on Physical laws as background reading.
posted by metastability at 3:36 PM on March 18

As has been said, our theory of everything is not quite on board yet.

To get a really good understanding of what we know very well, check out Feynman's short and relatively easy to read book QED.
posted by ErWenn at 4:14 PM on March 18

I think Schrodinger's equation is kind of an important one too. For quantum mechanics.
posted by number9dream at 4:15 PM on March 18

Maxwell's equations.
Newton's 2nd law.
Shrödinger equation
Dirac equation.
Einstein's equation of General Relativity.

This should be enough to derive most stuff of interest. With the addition of Lagrangian for Gauge theories - with gauge groups SU(3) [quantum chromodynamics] and SU(2)x(U1) [electroweak theory] supplemented by the Higg's mechanism, you should have, in theory, everything you need to derive all known stuff - except for a few holes in cosmology (e.g. dark matter) and leaving you with one inconsistency (how to make General Relativity consistent with Quantum mechanics).
posted by aroberge at 4:50 PM on March 18

To give a sense of what she may mean: I can never remember Schrodinger's equation, so I derive it when I need it. It goes something like this.

I know I need de Broglie's equations, but I usually can't exactly remember those either. So I remember that a free wave goes something like
Psi = exp(i kx - i w t)
Then since momentum p = hBar k, I can calculate p Psi = -i d/dx Psi
energy E = hBar w, so E Psi = i d/dt Psi
Now energy is can be broken down into kinetic and potential energy
K + U = E
K = p^2 / (2m)
So Schrodinger's equation is
-(hBar^2 / (2m)) d^2/dx^2 Psi + U Psi = i hBar d/dt Psi

I haven't written down any of those equations in years (I'm a neuroscientist now) and can't remember shit, but bam, there they are, from pretty simple principles. You just need to know the form a wave takes, and that energy and momentum are quantized in units of hBar. You can do similar stuff to derive the equations of special relativity. You just need a few basic principles (the transform is linear, the speed of light is constant in all reference frames, and if Bob sees Alice as moving at velocity v, Alice sees Bob as moving with velocity -v).

If you actually use this stuff regularly, you WILL memorize it without trying. But you don't need to. And it's amazing that the principles of physics yield up the results of physics so easily. It gives it the feeling of being "deep". Once you understand it, it's all easy. Except the spinning top, which is always a sonofabitch.
posted by Humanzee at 6:44 PM on March 18 [2 favorites has favorites]

My favorite is V=IR
posted by salvia at 7:09 PM on March 18

While taking both electrical engineering and calculus I noticed that methods to work with complex numbers presented in the electrical engineering class were easily implemented in the calculus class and saved a whole lot of time. These methods were never mentioned in the calc class at all, for some odd reason.
posted by odinsdream at 7:41 PM on March 18

We're forgetting how big "physics" is. I suspect someone will prove me wrong, but Maxwell's Equations are all electromagnetic, Newton's are all classical mechanics and I guess you can derive rotational dynamics from that, Einstein did a lot of work in relativity but as GuyZero said above (I'm not sure on this one but I'll trust it), general relativity can be derived from Maxwell's Equations.

But! There's also a lot of stuff about waves, light and of course thermodynamics that aren't covered so far in this thread. There are also fields like physical chemistry where it's kind of a hybrid science that would require you also know a thing or twenty about chemistry. Mathematical physics tries to prove abstract mathematics through physics; I guess those two subjects are more in the "applied physics" field though.

Like any science, there are new discoveries in physics all the time and there are also some pretty abstract and controversial fields. Off the top of my head, I can remember hearing someone who tried connect pure physics with human conversation, but my Google skills are failing me right now.

Anyways, physics is huger than most people think. Your teacher was right about always being able to derive things from equations, but it would be easier to narrow your field.

(On a side note, I had never heard of this Ramanujan guy before! Wow!)
posted by battlebison at 10:05 PM on March 18

I think the question is more along the lines of high school or 1st year university physics can be accomplished with a surprisingly small set of equations that you can then build on.
This was certainly my experience in school, where s=ut+1/2at² and F=MA got me through motion, V=IR was enough for electrical along with that one where you make a fist with the thumb stuck out to determine the direction of the electromagnetic field (I really used to know this 15 years ago!) etc.
posted by bystander at 12:50 AM on March 19

Defintely not something you would actually want to work with, but here(pdf) is the Standard Model Lagrangian.

Add gravity, and you'll be cooking with gas.
posted by toftflin at 3:23 AM on March 20