What mathematical algebra and notation should we use to encode our physical models?
We often have a choice, especially when we are dealing with vector like quantities (quantities with magnitude and direction as described on this page).
One of the most important considerations is that the expression of our laws should be independent of the coordinate system that we are using.
As an example of this consider the relationship between the force on a lever and the torque it produces:
T = r × F
| where: | |||
symbol |
description |
type |
units |
| T | torque | bivector | N m |
| r | distance from the centre of rotation | vector | m |
| F | the force | vector | N |
| × | the cross product vector operator | ||
Provided that we use orthogonal coordinates and provided that we are constant about using the same units for all three quantities this equation is independent of the coordinate system.
For example we could rotate the coordinate system so that x becomes y, y becomes z and z becomes x and the above equation is still valid (bare in mind though that changing between left and right hand coordinates will change the sign of the equation).
So if we used a different mathematical algebra/notation to express the same physical relationship do we still have coordinate independence? Lets try encoding the same law using matrices:
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= |
|
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To some extent we could replace x with y, y with z and z with x and the equation is still valid, but to do this we have to change the standards and conventions we usually use with matrices, we would normally expect x to be the top value, then y then z. Somehow the vector cross product equation seems to give us a level of abstraction above the world of coordinates but matrix equations mix the two together. Whatever system we use we need to use actual coordinates when working out actual forces, but its better to remain independent of coordinates as long as possible.
