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Trying2Learn
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- The derivative of a vector in a rotating frame (or even inertial)
I apologize: despite my verbosity, this is, I hope, a simple question.)
Consider the following relationship between a rotating reference frame and an inertial reference frame (both Bold), through a rotation matrix:
(the superscript is to designate the rotating frame e(1) and the I is for the inertial)
e(1)=eIR(1)
Now suppose I wish the time derivative of the rotating frame.
(I do not know how to put the dot overhead, sorry about this)
(The dot shoud be above the first e and above the R)
.e(1)=eI.R(1)
Now, did you see how I did not take the time derivative of the base frame?
Yes, PHYSICALLY, I know it is not rotating, so I do not have to do that.
But suppose I am a stubborn person (I am) who has memorized the "product rule" in calculus -- the first times the derivative of the second + the second times the derivative of the first.
And, now I see the product of a Rotation matrix (functions -- sine, cosine) and this bold base frame: eI
If I apply the rule, strictly, I put the dot over it and assert the result is zero.
But suppose I want to be stubborn and say: "but that base frame is not a function! I only know how to use the product rule for functions"
How do I answer myself?
The ONLY guess I can make is the following:
We began with coordinate functions: x1(1), x2(t) and x3(t)
We take time derivatives as we move a point, P(x1,x2, x3)
∂P(x1,x2, x3)/∂x1,
∂P(x1,x2, x3)/∂x2,
∂P(x1,x2, x3)/∂x3
Then, we decide to create the frame by dropping the point P, notation and looking at only the partial notation
e(1)≡∂/∂x1,
e(2)≡∂/∂x2,
e(3)≡∂/∂x3
We do this like Ted Frankel did on page (3) of this:
http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf
And while we have a frame, its three axes "began their lives as derivatives of coordinate FUNCTIONS, so we CAN use the product rule."
Can anyone advise a better way?
For example, suppose BOTH frames are moving and we want the derivative. We have to use the product rule and get the following sequence.
e(2)=e(1)R(2/1)
(and please forgive me for the placement of the dot--all dots should be directly overhead)
.e(2)= .e(1)R(2/1) + e(1).R(2/1)
Again, how do I know (in my ignorance and stupidity), to apply a product rule (proven for functions) to a base frame? (I have no issue, by the way, with the matrix structure formulation of the time derivative -- that is not an issue).
(And if you can tell me how to get the dot directly overhead, that would be nice, too.)
Consider the following relationship between a rotating reference frame and an inertial reference frame (both Bold), through a rotation matrix:
(the superscript is to designate the rotating frame e(1) and the I is for the inertial)
e(1)=eIR(1)
Now suppose I wish the time derivative of the rotating frame.
(I do not know how to put the dot overhead, sorry about this)
(The dot shoud be above the first e and above the R)
.e(1)=eI.R(1)
Now, did you see how I did not take the time derivative of the base frame?
Yes, PHYSICALLY, I know it is not rotating, so I do not have to do that.
But suppose I am a stubborn person (I am) who has memorized the "product rule" in calculus -- the first times the derivative of the second + the second times the derivative of the first.
And, now I see the product of a Rotation matrix (functions -- sine, cosine) and this bold base frame: eI
If I apply the rule, strictly, I put the dot over it and assert the result is zero.
But suppose I want to be stubborn and say: "but that base frame is not a function! I only know how to use the product rule for functions"
How do I answer myself?
The ONLY guess I can make is the following:
We began with coordinate functions: x1(1), x2(t) and x3(t)
We take time derivatives as we move a point, P(x1,x2, x3)
∂P(x1,x2, x3)/∂x1,
∂P(x1,x2, x3)/∂x2,
∂P(x1,x2, x3)/∂x3
Then, we decide to create the frame by dropping the point P, notation and looking at only the partial notation
e(1)≡∂/∂x1,
e(2)≡∂/∂x2,
e(3)≡∂/∂x3
We do this like Ted Frankel did on page (3) of this:
http://www.math.ucsd.edu/~tfrankel/the_geometry_of_physics.pdf
And while we have a frame, its three axes "began their lives as derivatives of coordinate FUNCTIONS, so we CAN use the product rule."
Can anyone advise a better way?
For example, suppose BOTH frames are moving and we want the derivative. We have to use the product rule and get the following sequence.
e(2)=e(1)R(2/1)
(and please forgive me for the placement of the dot--all dots should be directly overhead)
.e(2)= .e(1)R(2/1) + e(1).R(2/1)
Again, how do I know (in my ignorance and stupidity), to apply a product rule (proven for functions) to a base frame? (I have no issue, by the way, with the matrix structure formulation of the time derivative -- that is not an issue).
(And if you can tell me how to get the dot directly overhead, that would be nice, too.)