Background

The concept of reference system(RS) and coordinate system(CS) is confusing in theoretical mechanics. Conceptually, they are like chalk and cheese. However, since we always connect a reference system with a coordinate system, some confusion may happen when we write our equations.

A very basic equation is \(\frac{d \vec{A}}{dt} = \frac{\tilde{d}\vec{A}}{dt}+\omega\times\vec{A}\) which describes the relationship between what we call absolute derivative and relative derivative. What bothers me a lot is whether it is an equation about RS or CS?

Wrong Comprehension

At first I think that, by definition, vector (or 1-order tensor) is a quantity that has don’t vary with coordinate system, so do its derivative. So I suppose this equation to be used to describe the relationship of the components for a vector in two CS. Namely, I take the derivative of \(\vec{A}\), then get its components in systme A and B, then two set of components will satisfy the equation.

But a problem comes, if a vector doesn’t vary with time in one CS, it certainly remains constant in any another system. So both absolute and relative derivative equals zero, then we get \(0 = \omega\times\vec{A}\), what’s wrong?

Conclusion

The problem above is that the changes of components come from two parts: the change of \(\vec{A}\) and the change of CS itself. However when we are talking about The change of a coordinate system, we are actually talking about RS. So this equation actually connects two RS rather than two CS.

In two different RS, the derivative of \(\vec{A}\) are two vectors, we can choose in which CS we write their relationship.

The usage of this formula is just as those CS changing methods in mechanics.