Decoupling harmonic oscillators

Wow!  Here's the short version of what's going on.  My quantum II final was a bit rough because I didn't know how to decouple a pair of coupled harmonic oscillators.  However, I found an excellent article on decoupling in my second favorite journal the American Journal of Physics.  I wanted to capture a few notes here on the process and on one of the math operations left out of the original article, but hashing out the level of background to provide has taken awhile.  I decided to go with the terser representation of the subject.  It may be extremely sketchy since I'm more than a bit scattered studying for finals.  Basically, these are my notes, and I hope they're helpful :)  Any ideas, comments, corrections, or suggestions, as always, are more than welcome! For more detail and an excellent article on how to decouple coupled oscillators see [1].

OK, here goes.

What does 'decoupling harmonic oscillators' mean and what is it used for?
Pretty frequently in physics, we come across problems where you have to decouple a system of coupled harmonics oscillators.  For folks who haven't had analytical mechanics yet, this can be a bit dicy.  I'm one of those folks.  Decoupling means taking the generic Lagrangian that describes the system of two coupled harmonic oscillators that looks like this (picture 1)



and modifying/rewriting it so that it looks like this (picture 2)

In doing this, we get rid of the terms with the b and q coefficients that mix the characteristics of the two oscillators and arrive at an equation where the energy of the system can be calculated easily and directly from the two new terms lambda_sub_1 and lambda_sub_2.

Why does everyone say 'Just diagonalize it'?

What they're referring to is that the above two equations can be written in matrix notation as (picture 3)

If you do the matrix multiplies through, you'll wind up back at the first equation above.  If we actually manage to modify the equation so that the b and q coefficients go to zero, then the two square matrices will be diagonal, they'll only have non-zero entries on a diagonal going from upper left to lower right.  Hence, diagonalizing is just another way of saying that we want to rearrange things so that b and q go to zero.

Why is the next thing everyone says 'Rotate the coordinates'
Looking back at the first equation, you might notice that it can be broken into two parts, each of which describe an ellipse when they're set to zero (picture 4).

If the b and q terms aren't zero, then you wind up with diagonal ellipses like those shown in (a) and (b) in picture 6 [1].  Paradoxically, diagonalizing the matrices and setting b and q to zero will force the ellipses to not be diagonal.  Aravind points out a very cool way to visualize the entire coupling process using the diagram to the left (picture 5).

First, we perform a rotation in the eta_dot plane to force the kinetic energy ellipse to no longer be diagonal.  Cool, the b term of our matrix that describes the kinetic energy of the system is now zero!  

As you can see however, the ellipse in the eta plane is still tilted off diagonal.  We need to rotate it as well.  There's a bit of a problem though.  When we rotate one of the ellipses we rotate them both.  To get around this, we rotate the potential energy ellipse and scale both ellipses at the same time in the same way so that the kinetic energy ellipse becomes a circle.  You can rotate a circle to your heart's content in any way you please and it's still a circle, and, (Cool!), it's still parallel to the coordinate axes in the way we want so that the b entries in the kinetic energy matrix are still zero.  When we're done, we've rotated the potential energy ellipse as well so that the q terms are now zero!

The step that Aravind kind of glossed over was the details on how to calculate the angle of rotation.  My version of the calculation follows.

The matrix equation that describes rotating the eta axes is shown in picture 6.  The squiggles on the right that are hard to distinguish are primes that indicate the new set of axes.

The matrix in the middle is called the rotation matrix and the angle theta is what we need to calculate.  We know that we want to rotate the axes so that the b components will become zero in the square matrix shown in picture 3.  We also know that the rotation matrix has to be applied to the square matrix from picture 3 in the following way, (picture 7), to bring it into the new coordinate system.  The square matrix with the a, b, and c components is denoted by T and the rotation matrix is denoted by F.  

We know that in the new version of T, we want the matrix entries where the b coefficients go to be zero.  So, to calulate the angle, we perform the matrix multiply, then set the resulting equation that winds up in that position to zero and solve.  Here goes (picture 8)

The first multiply of T times F is shown above and it gives the result below, (picture 9), that is multiplied by the transpose of F, (the term on the left).

The resulting equation for the coefficient in the lower left-hand corner of the T' matrix, (the position where the coefficient resides that we want to set to zero), and its solution for zero are shown below, (picture 10).  Gotta get back to studying now.

References:

1.  Geometrical interpretation of the simultaneous diagonalization of two

quadratic forms
http://dx.doi.org/10.1119%2F1.16069
Aravind P.K. (1989). Geometrical interpretation of the simultaneous diagonalization of two quadratic forms, American Journal of Physics, 57 (4) 309. DOI: 10.1119/1.16069