Dr/Dt world coordinates problem

[yetar]

Homework Statement

In rigid body, r is the position of a rigid body relative to world coordinates,
dr/dt is the rate of change of r in world coordinates.
I encountered the term Dr/Dt as the rate of change of r in world coordinates relative to the body coordinates system.
And that Dr/Dt = dr/dt-wxr
My question is, how does Dr/Dt is called? (terminology)
Also the operator D/Dt has the same rules as the operator d/dt, how do you proove it? There is some relationship to proove this, but what is this relation ship?

Homework Equations

Dr/Dt = dr/dt-wxr

The Attempt at a Solution

I tried to use Dr/Dt = dr/dt-wxr to proove that both operators has the same rules, but didnt succed.
In matter of fact, I found some contradiction.
For instance: d(AxB)/dt = Ax(dB/dt)+(dA/dt)xB
So D(AxB)/Dt = Ax(DB/Dt)+(DA/Dt)xB should also be true.
However using Dr/Dt = dr/dt-wxr showed that there is inequality.

Thank you.

 

[christianjb]

I prefer the notation

dr^s/dt=dr^b/dt-wxr

where r^s = space fixed r (with subscript s) and r^b = body fixed r

The d/dt operator is the same in both cases.

There's a good, but lengthy article here

http://python.rice.edu/~chem630/Polyatomics.pdf

 

[m2003]

The term Dr/Dt is called the rate of change of position in world coordinates relative to the body coordinates system. This is also known as the absolute velocity of the rigid body. The operator D/Dt is called the total time derivative and it is used to describe the rate of change of a quantity with respect to time, taking into account not only the change in value but also the change in the coordinate system.

To prove that D/Dt and d/dt have the same rules, we can use the chain rule. Let's say we have a function f(x,y,z,t), where x, y, and z are coordinates in the body frame and t is time. We can express this function in terms of world coordinates as f(r,t), where r is the position vector of the body in world coordinates. Now, using the chain rule, we have:

Df/Dt = (df/dr)(dr/dt) + (df/dt)(dt/dt)

But since dt/dt = 1, we can simplify to:

Df/Dt = (df/dr)(dr/dt) + (df/dt)

This is equivalent to the rule for d/dt, where we have:

df/dt = (df/dx)(dx/dt) + (df/dy)(dy/dt) + (df/dz)(dz/dt) + (df/dt)

So, we can see that D/Dt and d/dt have the same rules, with the addition of the term (df/dt) which takes into account the change in the coordinate system. This is why Dr/Dt = dr/dt-wxr, where wxr is the cross product of the angular velocity of the body and the position vector in world coordinates.