Sunday, July 11, 2010

DANGER, Will Robinson! It DOES NOT Commute!

As I begin a detailed study of the Mathematics of Physics (known Physics) I shall begin with William Rowan Hamilton and the study of Quaternions, and Group Theory beginning with Galois Theory.

Each is dip and rich and old and ancient, but way important.

Consider the following equation, which is beautifully simple yet very key. It is the quaternion equation Hamilton came up with in an inspirational flash, and scratched on a bridge in Ireland:

i^2 = j^2 = k^2 = ijk = -1

Sweet. Vectors do NOT commute, not when multiplied together, meaning if the order is taken into account, you will get different results based on which vector you multiply first. Vectors do commute when added, but not when multiplied.

From that simple fact, all good things follow.


Phil Warnell said...

Hi Steven,

A good observation with Hamilton indicating there is more to be assigned to the importance of ordering within nature than first suspected. Similarly Joseph Louis Lagrange was to show that Pierre de Fermat’s intuitions in respect to nature’s economy of action could also be expressed as mathematically consistent. In truth if one takes their two discoveries together you have much of the mathematical foundations of modern physics which at the same time have exposed many of what still remains as its greatest mysteries.

DANGER Will Robinson, DANGER,DANGER!!!!!



Steven Colyer said...

Oh yeah, DANGER Will Robinson! Lol, I forgot. I orginally had WARNING! but will have to change that now, thanks.

What a schlocky show Lost in Space was, huh? At least Zorro was the Dad, and Timmy's mom from Lassie was the Mom! And Penny! I had a thing for Penny at that age, remember when? :-)

Dr. Smith and the Robot were the best though. I lol that CBS turned down Gene Roddenberry and his "Star Trek" idea, because they already had a sci-fi series! Sure they did, and not much of one at that. What a decade that 60's was, huh? For example, click here for a modern "re-telling" of The Beatles British Invasion. :-)

Well I hear you on Lagrage/Fermat. In this our modern era, Emily Noether's contribution being the duality between Conservation Laws and Symmetry is no less important, as Plato at Dialogos of Eide points out here.

Phil Warnell said...

Hi Steven,

The Beatles depicted as musical terrorists is certainly consistent with how other political documentaries attempt to rewrite the history of our times:-) As for Noether’s revelation which Einstein himself called one of the most pivotal finding of all time I would remind it to was known to be true to at least to a limited degree all the way back to the times of Plato. What Noether actually did was take this to be generalized to be true for all places where either a symmetry becomes evident or a conservation present. I myself wrote about how this can be appreciated from the aspects of something we are all familiar and that being a circle, which for me has always possessed the most fearsome of symmetries.

”TIGER, tiger, burning bright
In the forests of the night,
What immortal hand or eye
Could frame thy fearful symmetry?”

-William Blake