More Dimensional Analysis
In the last post I neglected to mention some additional uses of dimensional analysis:
1) Characteristic scales - let's say you have some system and there is exactly one combination of symbols that has dimensions of T^-1 (frequency). If asked to guess what the natural frequency is, what do you respond with? Obviously that. An example is a Quantum Harmonic Oscillator, that is, a particle bound within a quadratic potential:
E = p^2/2m + 1/2kx^2 = (1/2)(P^2/m + kx^2)
The relevant quantities we have are: mass, spring constant, Planck's constant. Immediately, we remember dealing with springs before, and decide to rewrite w^2=k/m because we know that has the right units. We guess that the ground state frequency might be about w. We also know some quantum mechanics and guess the ground state energy to be E=hf as usual, using w=2pi*f. Are we done? Not quite. There's one more quantity we might want to guess - how "spread" out the particle is. For this we need a length scale. We might try to work it out by doing something like this:
We know that W=Fs , and an energy scale is hw and a force scale can be kL. We then set L=hw/(kL) and get L = sqrt(hw/k) = sqrt(h/mw) = sqrt(hsqrt(m)/sqrt(k))
That's enoguh of that - we can compute three natural scales without any work. The energy scale we calculated turns out to be twice the ground state and equal to the spacing between all the energy levels.
2) De-dimensionalisation - you can go through a process to get rid of all dimensional numbers in your equation. Why do this? Because dimensions and constants clutter things up! Once you have a dimensionless equation, it's very likely that a mathematician will have already sovled it.
3) Error checking - If you're good, you can see at a glance when you've made a mistake. This stops you from having to fix things down the road.
4) Robustness- Let's say that we have some equation that we derived properly using mechanics and maths and all the rest. Let's take as an example, Bernoulli's equation:
Pressure + Kinetic energy/volume + potential energy/volume = consant
(Usually written P + rho v^2/2 + rho gh = C, where rho is density.)
This is, however, derived under certain restrictive assumptions. What's to say it's true at all in other cases? Well, firstly, we expect small deviations from out initial assumptions to only have small-ish impacts on the final answer, so there's that. But the other thing is this:
The form of the equation is that a bunch of energy densities add up to a constant. That looks eirily like conservation of energy. And indeed it is. If the other assumptions aren't met, but energy is still conserved, we might well expect a Bernoulli-like equation to hold (maybe with a couple of other terms etc. We'd expect similar qualititave results like pressure increasing when velocity decreases.
5) Adaptability - We saw this earlier with the QHO. We have seen that classical harmonic oscillator, and sure that gives us a good guess at what the quantum version will do (w^2=k/m) but how do we know that nothing new happens? Because the dimensions still need to be correct, and w=sqrt(k/m) is the ONLY frequency available. Maybe you get a factor of 2 when going from classical to quantum, but the basic behaviour has to be pretty much the same.
6) Loads more - You can easily show that if the period of a pendulum depends on g, the mass and the length of the pendulum, that it cannot depend on the mass at all. This is even independant of the case for large initial swings, because the initial amplitude is an angle and hence dimensionless.
You can guess formulas as solutions to very complex equations, like the Navier-Stokes equations. This is often done by finding dimensionless quantities and writing them as functions of each other. If there is only one dimensionless group, then it must be a constant of the system. (For more on this, read "Streetfighting Mathematics" and the Wiki entry on the "Buckingham Pi Theorem", the latter requires you to know what the null space/kernel of a matrix is.)
The formula for the cross product of a cross product is quite hard to prove. It's AX(BXC)=(A.C)B-(A.B)C
Now, one way prove this is to let the vector B lie along the x axis, the vector C lie in the x,y plane. Then the vector (BXC) must lie along the z axix. You can then algebraically cross this vector (which only has one component) with an arbitrary vector A and start solving simultaneously to prove the formula.
You might be a little bit smarter and realise that the resultant vector must lie wholly in the x,y plane. That's because AX(BXC) is perpendicular to both A and (BXC) and (BXC) lies along the Z axis. That means it's a linear combination of B and C and the correct answer is a 'natural choice'. It becomes even more natural once you realise that e.g. the coefficient of B needs to be (a) a scalar and (b) a quantity with the same dimensions as AB. The dot product of the two is the simplest guess.
In some electromagnetic examples, you can show that a typical "scale" for one term is proportional to the "velocity scale" over the speed of light. For non-relativistic systems that means the term is negligibly small and can be ignored. That's great news because of Maxwell's four equations all but one have only one term on the right hand side, and that one has a factor of 1/c^2 in it. That saves the day.
And, as before, once you have solved a problem by dimensional analysis, not only might the original problem make more sense (hmm... I need a frequency and a time...), but others might start to as well. E.g. here's a simple problem:
"A man is walking home with his dog. His dog runs home 5 times faster than the man walks, and the man starts 5 miles out from his home. If the dog runs back and forth between the man and the house, how far will the dog have travelled?"
Here we have a length scale already in the problem - 5 miles. We do, however, suspect that the man and the dog are not going to cover the same distance. What else do we have? The number 5! And that's dimensionless, so the dog must cover 25 miles and that's the end of the story.
That's it for now, that turned out longer than I expected - as an exercise, give a (non-dimensional) explanation of why 25 miles is right, and come up with a similar question where there is some parameter whose dimensions make it unnecessary (e.g. mass) but which looks like it might matter (as in the pendulum example, or a projectile based question). Then you can be smug and know why it CAN'T POSSIBLY MATTER whilst everyone around you is trying to get it to cancel out of their equations.
