Rotating frame: a question of interpretation

[Philip Wood]

I'm reading Lanczos: 'The variational Principles of Mechanics'. I need help resolving a paradox - which is probably trivial…

Lanczos (page 100 Dover edition) introduces a system, S', rotating at angular velocity $\vec \Omega$ about an axis through a fixed point with respect to inertial system S. The radius vectors $\vec R$ and $\vec R'$ in the two systems are, he says, the same: $\vec R = \vec R'$. [I don't have trouble with this: it's fundamental to the idea of a vector that the same vector can be expressed in terms of different basis vectors, in this case, $\vec i, \vec j, \vec k$ and $\vec i', \vec j', \vec k'$.]

Nevertheless, he says, the velocities and accelerations measured in both systems differ from each other because rates of change observed in the two systems are different. If a certain vector $\vec B$ is constant in S' it rotates with the system and thus, if observed in S, undergoes in the time dt an infinitesimal change $d \vec B = (\vec \Omega \times \vec B) dt$. [Again, easily seen - especially for $\vec R$ itself.]

Hence $\frac{d \vec B}{dt} = (\vec \Omega \times \vec B)$ while at the same time, Lanczos says, $\frac{d' \vec B}{dt} = 0$.

Here, Lanczos has introduced the notation $\frac {d'}{dt}$ which refers to the operation of observing the rate of change of a quantity in the moving system S'.

If you've read as far as this, well done and thank you. Now here's my problem. Regarding $\vec R$ and $\vec R'$ as functions of time, we can surely write:
$$\vec R (t) = \vec R' (t).$$
But because $\frac{d \vec B}{dt} = (\vec \Omega \times \vec B)$
$$\vec R (t + dt) = \vec R (t) + (\vec \Omega \times \vec R (t)) dt$$
while because $\frac{d' \vec B}{dt} = 0$
$$\vec R' (t + dt) = \vec R' (t) + 0.$$
It would therefore seem that at time (t + dt), it is no longer the case that $\vec R = \vec R'$. Whereas $\vec R'$ has changed over dt, $\vec R'$ has stayed constant.
Where is my reasoning wrong?

 

[Andrew Mason]

I am not clear on what $\vec{R} \text{ and } \vec{R'}$ are. Are they the vectors from the inertial point to the origins S and S'? In other words, S and S' have a common origin?

AM

 

[UltrafastPED]

In other words, S and S' have a common origin?

I checked my copy of Lanczos; yes, they have a common origin. The discussion is for rotations with no translation.

 

[UltrafastPED]

Where is my reasoning wrong?

Lanczos uses the "superposition principle of infinitesimal processes", while you are not.

Also note that there is a typo on eq. (45.9) of the fourth edition; there should be a "dot" over the Ω in the final term.

 

[Philip Wood]

Thanks for replying, but with respect, UPED, I don't think either of your points solves my problem. L uses the superposition principle when moving on from a vector which is constant in S' to one which is not necessarily constant in S'. My concern is merely with a vector which is constant in S'. [Incidentally I don't much like L's appeal to a superposition principle; there are, imo, nicer treatments in Kibble or in Synge which differentiate $x_i \vec e_i$ as a product.] Equation 45.9 is also beyond the point where my problem lies.

 

[Philip Wood]

No more ideas? Is this because my question is not understood, or is it because it's baffling you, too?

[I apologise for the typo in the last-but-one sentence of post 1. This should read: Whereas $\vec R$ has changed over dt, $\vec R'$ has stayed constant.]

 

[Andrew Mason]

No more ideas? Is this because my question is not understood, or is it because it's baffling you, too?

[I apologise for the typo in the last-but-one sentence of post 1. This should read: Whereas $\vec R$ has changed over dt, $\vec R'$ has stayed constant.]

I am still not clear on $\vec R$ and $\vec R'$. I don't have a copy of Lanczos' book.


Can you post the page in question?

AM

 

[dauto]

When you say R = R' does that mean they have the same numerical values or does that mean they represent the same physical vector? You can't have it both ways. You're trying to have it both ways and that's why you're getting in trouble.

 

[Philip Wood]

AM and Dauto. Thank you for your interest.

AM Please find extracts.

Dauto: I've wondered the same thing. Yet R' = R seems to imply same value.

 

[Andrew Mason]

AM and Dauto. Thank you for your interest.

AM Please find extracts.

Dauto: I've wondered the same thing. Yet R' = R seems to imply same value.

The problem seems to be that we are not sure what $\vec{R}$ and $\vec{R}'$ are.

Lanczos seems to be using 2 dimensional polar co-ordinates. So $\vec{R}(r,θ)$ differs from $\vec{R}'(r,θ')$ only by angle: i.e. the second co-ordinate - angle θ. If that is the case, his statement:

$\vec{R}= \vec{R}'$ cannot be true, generally.

Perhaps he meant to say $|\vec{R}| = |\vec{R}'|$ (i.e. the length (the first co-ordinate, r) is the same for both vectors).

AM

 

[Philip Wood]

AM Thank you. Yet he has R and R' in bold and refers to them (second para in extracts) as radius vectors And he does seem to be pretty scrupulous about nomenclature. Also by starting the third para in the extract with "Nevertheless" he seems to be pointing to the peculiarity of radius vector behaving differently from velocity and acceleration vectors: there wouldn't be anything worth remarking about if radius wasn't to be considered a vector.

 

[Andrew Mason]

AM Thank you. Yet he has R and R' in bold and refers to them (second para in extracts) as radius vectors And he does seem to be pretty scrupulous about nomenclature. Also by starting the third para in the extract with "Nevertheless" he seems to be pointing to the peculiarity of radius vector behaving differently from velocity and acceleration vectors: there wouldn't be anything worth remarking about if radius wasn't to be considered a vector.

Laczos appears to be referring to $\vec{R} \text{ and } \vec{R}'$ at a particular time e.g. $\vec{R}(0) = \vec{R}'(0)$. If $\vec{R}'(t) = \vec{R}'(0)$ for all t, then $\vec{R}'$ is just a fixed vector in S'. So, while $\vec{R}(0) = \vec{R}'(0)$, $\vec{R}(t) \ne \vec{R}'(t)$ (unless $\dot{Ω}t = n2\pi$).

AM

 

[Philip Wood]

AM Yes, this interpretation is free of inconsistentency. I was loathe to accept it because to claim that $\vec R = \vec R'$ (only) at one particular time seemed such a weak claim. Thanks to your post, I'll try to stop worrying about the issue. I have a great gift for getting stuck when trying to learn; it has not faded with advancing years.

Edit: Fickle to the last, I've now gone for a different interpretation. See post 15. Many thanks!

 

[D H]

Now here's my problem. Regarding $\vec R$ and $\vec R'$ as functions of time, we can surely write:
$$\vec R (t) = \vec R' (t).$$

Surely you cannot.

Imagine a merry-go-round with a hole in the middle, big enough so that you can stand on the ground in the middle of the merry-go-round and watch the horses move around you. The horses are moving. From this perspective, the position of some particular horse as a function of time is $R(t) = R\cos(\omega t)\hat x + R\sin(\omega t)\hat y$.

Alternatively, you can jump up on the merry-go-round, put a plank over that hole, and position yourself exactly as before, except now you are rotating with the merry. From this perspective, that horse is stationary: $R'(t) = R\hat x'$. The horse's velocity is zero from this rotating perspective. It's obviously not zero from the perspective of the non-rotating observer.I don't have Lanczos, but the use of "infinitesimal rotations" at the bottom of that page you posted makes it looks like he's about to do some "physics math" hand waving in his derivations. Hand waving is standard fare in this regard, even in many graduate level texts. Even Goldstein waved his hands.

 

[Philip Wood]

DH Thank you. I like your very clear scene-setting with the merry-go-round. I also like your use of unit vectors.

I'm now going to turn your argument on its head and postulate that
$$\textbf {R'} (t) = \textbf{R} (t)$$
in which case, with your notation,
$$R \mathbf {\hat{x'}}=R\ \text {cos} \omega t \ \mathbf {\hat{x}}+R\ \text {sin} \omega t \ \mathbf {\hat{y}}$$
This is the familiar business of expressing the same vector on two different basis sets. It does, though, demand that we regard vectors fixed in the rotating system, such as $\mathbf {\hat{x'}}$, as functions of time.

I've been labouring under the misapprehension that this is inconsistent with Lanczos's equation (in the extract in post 9):
$$\frac {\text{d'} \mathbf{B}}{\text{d} t} = 0$$
He says that $\frac {\text{d'} }{\text{d} t}$ refers to the rate of change of a quantity in the moving system, S'. I now realize that he's not saying that $\frac {\text{d} \mathbf{B'}}{\text{d} t} = 0$. What he means, I think, by $\frac {\text{d'}\mathbf{B}}{\text{d} t}$ is the rate of change of vector B' if we regard the rotating unit vectors $\mathbf {\hat{x'}}$ etc. as constant; in other words a sort of partial differentiation.

This interpretation rings true for me. Many thanks to those who've came to my aid.

 

[D H]

Before I start delving into your last post, Philip, it's important to realize that time derivatives of vector quantities are frame dependent. It's certainly true for translation. Why wouldn't this be the case for rotation?

To illustrate that it's true for translation, suppose you and your best buddy decided to take the long weekend off with a trip to Vegas. The only problem: Who's car? You like your vintage Dodge Charger with its 426 cu. in. engine, he likes his vintage AMC Matador with its smaller but zestier 401 cu. in. engine. He comes up with the perfect solution: "Let's race!"

You're neck and neck as your cross the California-Nevada state line. From your perspective, his velocity as you cross into Nevada is nearly zero. The same is true regarding your velocity from your buddy's perspective. Unfortunately for both of you, you just zoomed past a Nevada highway patrol officer sitting on a lawn chair in the freeway median strip. That officer (along with the radar gun he was holding just before he called you in) had a slightly different perspective on your velocities than did the two of you.

DH Thank you. I like your very clear scene-setting with the merry-go-round. I also like your use of unit vectors.

I'm now going to turn your argument on its head and postulate that
$$\textbf {R'} (t) = \textbf{R} (t)$$
in which case, with your notation,
$$R \mathbf {\hat{x'}}=R\ \text {cos} \omega t \ \mathbf {\hat{x}}+R\ \text {sin} \omega t \ \mathbf {\hat{y}}$$
This is the familiar business of expressing the same vector on two different basis sets. It does, though, demand that we regard vectors fixed in the rotating system, such as $\mathbf {\hat{x'}}$, as functions of time.

The derivatives of those unit vectors is frame-dependent. From the perspective of the non-rotating observer, the $\hat x, \hat y, \hat z$ unit vectors are stationary (time derivatives are zero) while the $\hat x', \hat y', \hat z'$ unit vectors are rotating (time derivatives are non-zero). It's the other way around for the rotating observer.

From the above, it's obvious that $\hat x' = \cos(\omega t)\hat x + \sin(\omega t) \hat y$. Let's finish off the relationship between those unit vectors. The two observers share the same z axis: $\hat z' = \hat z$. This means that to complete a right-handed system, we must have $\hat y' = \hat z' \times \hat x' = -\sin(\omega t) \hat x + \cos(\omega t)\hat y$.

These relationships form the transformation matrix that transforms a vector as represented in the rotating frame to the representation of the same vector in the non-rotating frame:

$$\begin{bmatrix} x \\ y \\z \end{bmatrix} = \begin{bmatrix} \cos(\omega t) & -\sin(\omega t) & 0 \\ \sin(\omega t) & \phantom{-} \cos(\omega t) & 0 \\ 0 & 0 & 1 \end{bmatrix} \; \begin{bmatrix} x' \\ y' \\z' \end{bmatrix}$$
Denoting that transformation matrix as $T_{R\to I}$, the above becomes $\vec R(t) = T_{R\to I}(t) \, \vec R'(t)$. Note this applies to all vectors, not just position vectors. Any vector quantity $\vec q$ can be transformed from its rotating frame representation to its non-rotating frame representation.

Taking the time derivative of above results in $\dot{\vec R}(t) = T_{R\to I}(t) \, \dot{\vec R}'(t) + \dot T_{R\to I}(t) \, \vec R'(t)$. What's the time derivative of this transformation matrix? From the above, that time derivative is
$$\dot T_{R\to I}(t) = \begin{bmatrix} -\omega\sin(\omega t) & -\omega\cos(\omega t) & 0 \\ \phantom{-}\omega\cos(\omega t) & -\omega\sin(\omega t) & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
Pre-multiplying by the transpose of the transformation matrix yields
$$T_{R\to I}(t)^T \, \dot T_{R\to I}(t) = \begin{bmatrix} 0 & -\omega & 0 \\ \omega & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$
Note that the right hand side is the skew symmetric cross product matrix corresponding to the angular velocity vector $\vec \omega = \omega \hat z'$.

My hand wave is the unproven claim (unproven here, that is) that $T_{R\to I}(t)^T \, \dot T_{R\to I}(t) = \text{Sk}(\vec \omega)$ where $\vec \omega$ is the angular velocity vector as represented in the rotating frame is always true for rotations in three dimensional space. This is a consequence that the set of all 3x3 proper real transformation matrices is the Lie group SO(3). The Lie algebra for this group is the set of 3x3 skew symmetric matrices. Proving this claim either requires a class in differential geometry and Lie groups, or five pages of a math.

Another way to write this relation between the transformation matrix, its time derivative, and the angular velocity vector is $\dot T_{R\to I}(t) = T_{R\to I}(t) \text{Sk}(\vec \omega(t))$. With this, the time derivative becomes $\dot{\vec R}(t) = T_{R\to I}(t) \, \dot{\vec R}'(t) + T_{R\to I}(t)\,\text{Sk}(\vec \omega(t)) \, \vec R'(t)$, or
$$\dot{\vec R}(t) = T_{R\to I}(t) \left(\dot{\vec R}'(t) + \vec \omega(t) \times \vec R'(t)\right)$$

 

[Andrew Mason]

Note that the right hand side is the skew symmetric cross product matrix corresponding to the angular velocity vector $\vec \omega = \omega \hat z'$.

My hand wave is the unproven claim (unproven here, that is) that $T_{R\to I}(t)^T \, \dot T_{R\to I}(t) = \text{Sk}(\vec \omega)$ where $\vec \omega$ is the angular velocity vector as represented in the rotating frame is always true for rotations in three dimensional space. This is a consequence that the set of all 3x3 proper real transformation matrices is the Lie group SO(3). The Lie algebra for this group is the set of 3x3 skew symmetric matrices. Proving this claim either requires a class in differential geometry and Lie groups, or five pages of a math.

Laczos must be using polar coordinates.

Using polar coordinates (where $\hat{k}$ is the unit vector in the direction perpendicular to the plane of rotation ), the orthogonal basis vectors in S are $\hat{r} \text{ and } \hat{Ω}$ where $\hat{Ω} = \hat{k} \times \hat{r}$. In S' the basis vectors are: $\hat{r}' \text{ and } \hat{Ω}'$ where $\hat{Ω}' = \hat{k} \times \hat{r}'$. However, since $\hat{r} = \vec{R}/|\vec{R}| = \vec{R}'/|\vec{R}'| = \hat{r}'$, $\hat{Ω}' = \hat{k} \times \hat{r}' = \hat{k} \times \hat{r} = \hat{Ω}$, so the unit basis vectors for S and S' are identical.

The only difference between $\vec{R}$ and $\vec{R}'$ is in the values for Ω and Ω'.

$\vec{R} = r\hat{r}+ Ω\hat{Ω}$ and

$\vec{R}' = r'\hat{r}'+ Ω'\hat{Ω}' = r\hat{r}+ Ω'\hat{Ω}$

Subtracting:

$\vec{R}' - \vec{R} = Ω'\hat{Ω}' - Ω\hat{Ω} = (Ω'- Ω)\hat{Ω}$

Differentiating a rotating vector R in S with respect to time results in:

$\dot{\vec{R}} = \dot{r}\hat{r} + r\dot{Ω}\hat{Ω} = r\dot{Ω}\hat{Ω}$

but in co-rotating frame S', $\dot{Ω} = 0$ so the derivative with respect to time is:

$\dot{\vec{R}'} = \dot{r}'\hat{r} + r'\dot{Ω}'\hat{Ω} = 0\hat{r} + r0\hat{Ω} = 0$

AM

 

[Philip Wood]

DH Thank you for this interesting piece. The case that concerned me was of a vector fixed to the rotating frame, so I think you'll agree that $\left(\begin{array}{cc}x'\\y'\\z'\ \end{array}\right)$ is constant, though the unit vectors $\hat{\mathbf{x'}}$ etc., are functions of time, when expressed in terms of $\hat{\mathbf{x}}$ etc.. This, I believe, is what Lanczos means by $\frac{\text{d'} \mathbf{B}}{\text{d}t} = 0$.

 

[D H]

Laczos must be using polar coordinates.

No, he's not. Almost assuredly. He's using two reference frames, one of which is rotating with respect to another. It doesn't really matter what kind of coordinates you use. This an issue of the time derivative of some vector quantity as observed in one reference frame versus the time derivative as observed in another reference frame.

What he's doing is developing what some call the dynamical transport theorem. You don't need coordinate systems at all to express this theorem:
$$\left(\frac{d\vec q}{dt}\right)_A = \left(\frac{d\vec q}{dt}\right)_B + \vec \omega \times \vec q$$
The subscripts on the derivatives denote the time derivative as observed by two observers, call them observer A and observer B. Observer B is rotating with respect to observer A with an angular velocity of $\vec \omega$.

 

[Andrew Mason]

No, he's not. Almost assuredly. He's using two reference frames, one of which is rotating with respect to another. It doesn't really matter what kind of coordinates you use. This an issue of the time derivative of some vector quantity as observed in one reference frame versus the time derivative as observed in another reference frame.

What he's doing is developing what some call the dynamical transport theorem. You don't need coordinate systems at all to express this theorem:
$$\left(\frac{d\vec q}{dt}\right)_A = \left(\frac{d\vec q}{dt}\right)_B + \vec \omega \times \vec q$$
The subscripts on the derivatives denote the time derivative as observed by two observers, call them observer A and observer B. Observer B is rotating with respect to observer A with an angular velocity of $\vec \omega$.

But by using polar coordinates you can use the same basis vectors for both the rotating and non-rotating reference frames. It is just much easier to do the math.

Without the pages prior to page 100 I am just guessing at what the statement $\vec{R} = \vec{R}'$ means. It just occurred to me that he may mean $\vec{R} = R\hat{r} = R\hat{r}' = \vec{R}'$. The $r\hat{r}$ component then is always the same using polar coordinates. That may be what he is referring to.

AM

 

[UltrafastPED]

Without the pages prior to page 100 I am just guessing at what the statement $\vec{R} = \vec{R}'$ means. It just occurred to me that he may mean $\vec{R} = R\hat{r} = R\hat{r}' = \vec{R}'$. The $r\hat{r}$ component then is always the same using polar coordinates. That may be what he is referring to.

Lanczos means that the geometric vectors are the same; that is the advantage of vector notation - you can say things that are independent of the coordinate systems!

If you find coordinates helpful, then select the coordinates that are easiest to use - polar coordinates would seem the best here.

 

[D H]

But by using polar coordinates you can use the same basis vectors for both the rotating and non-rotating reference frames. It is just much easier to do the math.

Without the pages prior to page 100 I am just guessing at what the statement $\vec{R} = \vec{R}'$ means. It just occurred to me that he may mean $\vec{R} = R\hat{r} = R\hat{r}' = \vec{R}'$. The $r\hat{r}$ component then is always the same using polar coordinates. That may be what he is referring to.

AM

That is not what it means. $\vec R = \vec R'$ means exactly what it says, that $\vec R$ and $\vec R'$ are the same vector. It has nothing to do with polar coordinates. It has to do with two different reference frames giving two different representations of the same vector.

 

[Andrew Mason]

That is not what it means. $\vec R = \vec R'$ means exactly what it says, that $\vec R$ and $\vec R'$ are the same vector. It has nothing to do with polar coordinates. It has to do with two different reference frames giving two different representations of the same vector.

The difficulty is in defining what is mean by "the same vector" in S and S'.

Since the origins of S and S' are same, the radial displacement vector representing the displacement of a point relative to the origin is the same in both frames of reference.

If the observer in S' realizes that he is rotating and not S, then S' can do the math to subtract his rotational movement and conclude that the point is constant over time. But if he doesn't, he concludes that the point (i.e the displacement vector) keeps changing and, with it, the direction of the radial vector from the origin.

AM

 

[Philip Wood]

Thank you again for your help. The penny dropped for me back at post 15. Just a comment on...

What he's doing is developing what some call the dynamical transport theorem. You don't need coordinate systems at all to express this theorem:
$$\left(\frac{d\vec q}{dt}\right)_A = \left(\frac{d\vec q}{dt}\right)_B + \vec \omega \times \vec q$$
The subscripts on the derivatives denote the time derivative as observed by two observers, call them observer A and observer B. Observer B is rotating with respect to observer A with an angular velocity of $\vec \omega$.

I think the danger of this presentation is that it invites my original objection (in post 1) that we have the same vector in two frames but which changes at a different rate in the two frames.

The key, I think, to understanding what's going on is to realize that $\left(\frac{d\vec q}{dt}\right)_A$ is a genuine rate of change of vector $\vec q$. [We need only to consider the change in the set of scalar components, because in frame A the basis vectors are fixed.] But $\left(\frac{d\vec q}{dt}\right)_B$ is not the rate of change of $\vec q$. It is again the rate of change of the component set, holding the basis vectors (those fixed to frame B) constant. This makes a real difference in frame B because the basis vectors in frame B (those fixed to the frame) are actually rotating. [The real rate of change of $\vec q$ calculated in frame B would take account of the rotation of the basis vectors, and would come out to the same value as the rate of change of $\vec q$ in frame A!]

That's why Lanczos, in his wisdom, used a new notation, $\frac{\text {d'} \mathbf q}{\text {d}t}$ for this special sort of partial differentiation in the rotating, B, frame.

So I think tacit use IS being made of co-ordinate systems, though the notation (including that of Lanczos) hides this.

 

[D H]

I think the danger of this presentation is that it invites my original objection (in post 1) that we have the same vector in two frames but which changes at a different rate in the two frames.

Danger? What danger? That the same vector has different derivatives in different frames is entirely the point. It is key to understanding fictitious forces, key to many solutions of the three body problem (which is typically solved in the synodic frame), key to making a robotic arm behave correctly, and key to making one spacecraft safely dock with another.

 

[Philip Wood]

If the (complete) time derivative of a vector is different in frames A and B, how can it stay the same vector in A as in B?

Did you read the rest of my post 24? Here I contend that the so-called time derivative we are taking in the rotating frame treats the base vectors in that frame (and rotating with that frame) as if they were stationary. So it's only a sort of partial derivative.

 

[D H]

What do you mean by "(complete) time derivative of a vector"? You are essentially saying that doing physics in a rotating frame is invalid. It most certainly is not invalid.

Suppose two observers rotating with respect to one another watch the same event. One sees one time derivative, the other sees another. Which observer is right, which is wrong? The answer is that both are "right". What's wrong is that insisting that only one can be right.

 

[Philip Wood]

I'm certainly not saying that doing Physics in a rotating frame is invalid! Nor am I saying that one observer is right and the other wrong.

I'm merely trying to understand how to interpret the equation you write as

$$\left( \frac{d \vec q}{d t}\right)_A = \left( \frac{d \vec q}{d t}\right)_B +\ \vec \omega \times \vec q.$$
I'm arguing that $\left( \frac{d \vec q}{d t}\right)_B$ is only a 'partial' derivative of the vector $\vec q$ because it treats the unit vectors in the rotating (B) frame (rotating with the frame) as constants. The second term on the right is the 'missing' part of the B-frame derivative that takes account of the rotation (non-constancy) of the unit vectors. See footnote.

On the left hand side, there is only one term, $\left( \frac{d \vec q}{d t}\right)_A$ because in the inertial frame the unit vectors are constants. So $\left( \frac{d \vec q}{d t}\right)_A$ is a genuine, complete, derivative of $\vec q$.

Footnote
Mathematically, what I'm saying in the paragraph under the equation is that the 'complete' derivative of $\vec q$ expressed in terms of the rotating unit vectors of the 'B' frame is

$$\frac{d}{dt}\left(q_x \hat {\vec x}+ q_y \hat {\vec y}+ q_z \hat {\vec z}\right)$$
in which $q_x, q_y, q_z$ are (scalar) components of $\vec q$ on the rotating unit vector set $\hat {\vec x}, \hat {\vec y}, \hat {\vec z}$.

Differentiating each term in the last equation as a product, and re-assembling:

$$\frac{d q_x}{dt} \hat {\vec x}+ \frac{d q_y}{dt} \hat {\vec y}+\frac{d q_z}{dt} \hat {\vec z} \ +\ q_x \frac{d \hat {\vec x}}{dt}+q_y \frac{d \hat {\vec y}}{dt}+q_z \frac{d \hat {\vec z}}{dt} \ \ \ = \ \ \ \left( \frac{d \vec q}{d t}\right)_B \ + \ \vec{\omega} \times \vec{q}$$
In the last step I've identified the sum of the first three terms with your $\left( \frac{d \vec q}{d t}\right)_B$, (what Lanczos calls $\frac{d' \mathbf q}{dt}$), and the last three terms with $\vec{\omega} \times \vec{q}$. The latter can easily be justified.

 

[D H]

I'm certainly not saying that doing Physics in a rotating frame is invalid! Nor am I saying that one observer is right and the other wrong.

I'm merely trying to understand how to interpret the equation you write as

$$\left( \frac{d \vec q}{d t}\right)_A = \left( \frac{d \vec q}{d t}\right)_B +\ \vec \omega \times \vec q.$$
I'm arguing that $\left( \frac{d \vec q}{d t}\right)_B$ is only a 'partial' derivative of the vector $\vec q$ because it treats the unit vectors in the rotating (B) frame (rotating with the frame) as constants.

To an observer fixed with respect to that rotating frame, those unit vectors in the rotating frame most certainly are constant. By arguing that they are not you are implicitly saying that doing physics in a rotating frame is invalid. Look at it from the perspective of that rotating observer. From that person's perspective, it's the inertial unit vectors that are rotating.

 

[mattt]

Thank you again for your help. The penny dropped for me back at post 15. Just a comment on...



I think the danger of this presentation is that it invites my original objection (in post 1) that we have the same vector in two frames but which changes at a different rate in the two frames.

The key, I think, to understanding what's going on is to realize that $\left(\frac{d\vec q}{dt}\right)_A$ is a genuine rate of change of vector $\vec q$. [We need only to consider the change in the set of scalar components, because in frame A the basis vectors are fixed.] But $\left(\frac{d\vec q}{dt}\right)_B$ is not the rate of change of $\vec q$. It is again the rate of change of the component set, holding the basis vectors (those fixed to frame B) constant. This makes a real difference in frame B because the basis vectors in frame B (those fixed to the frame) are actually rotating. [The real rate of change of $\vec q$ calculated in frame B would take account of the rotation of the basis vectors, and would come out to the same value as the rate of change of $\vec q$ in frame A!]

That's why Lanczos, in his wisdom, used a new notation, $\frac{\text {d'} \mathbf q}{\text {d}t}$ for this special sort of partial differentiation in the rotating, B, frame.

So I think tacit use IS being made of co-ordinate systems, though the notation (including that of Lanczos) hides this.

D.H. is right. You will understand it all perfectly when you study Differential Geometry.

Different observers are different coordinate charts on the same Differential Manifold (Euclidean Space-time in this case).

One world-curve on the manifod is what it is, some $$\alpha : I \to M$$, but obviously the mathematical expression of the coordinates of the same curve (and its vector velocity) on two different coordinate charts, are different (in general).

The mathematical expression for the transformation from one coordinate chart to another coordinate chart is precisely what D.H. has written for this concrete case (each one observer rotating with respect to the other).

 

[Philip Wood]

To an observer fixed with respect to that rotating frame, those unit vectors in the rotating frame most certainly are constant. […] From that person's perspective, it's the inertial unit vectors that are rotating.

The relative rotation of the sets of unit vectors is undeniable, isn't it? I could agree to say that it's the inertial frame unit vectors which are rotating and the rotating frame unit vectors which are stationary. But would you really be happier with that?

In fact, there are good reasons to choose to call the inertial frame unit vectors stationary...

We can distinguish a rotating frame from a non-rotating frame. For example, on a rotating platform a string running between two balls develops a tension. In general in a rotating frame objects experience certain forces which cannot be attributed to specific objects external to themselves, and which do not, therefore come in Newton's Third Law pairs.

Granted that we can recognise a rotating frame from such observations, it seems to me far more natural, then, that we should regard axes fixed to the rotating frame as rotating, and axes in the inertial frame as stationary. It doesn't matter that the observer in the rotating frame may SEE the inertial frame as rotating and his own frame as stationary.

 

[mattt]

The relative rotation of the sets of unit vectors is undeniable. I could agree to say that it's the inertial frame vectors which are rotating and the rotating frame vectors which are stationary. But would you really be happier with that? If so, why?

That is a different thing. I tried to explain in my previous post that a geometric object (a curve, its vector velocity, etc) on a Differentiable Manifold has different coordinate expressions in different coordinate charts.

In fact, there are good reasons to choose to call the inertial frame vectors stationary...

We can distinguish a rotating frame from a non-rotating frame. For example, on a rotating platform a string running between two balls develops a tension. In general in a rotating frame objects experience certain forces which cannot be attributed to specific objects external to themselves, and which do not, therefore come in Newton's Third Law pairs.

Granted that we can recognise a rotating frame from such observations, it seems to me far more natural, then, that we should regard axes fixed to the rotating frame as rotating, and axes in the inertial frame as stationary. It doesn't matter that the observer in the rotating frame may SEE the inertial frame as rotating and his own frame as stationary.

That is simply the following: the dynamical laws (you'll see this when you study General Relativity, but it is also true in Newtonian Mechanics if you study it under Differential Geometry point of view) are geometric relations among certain geometric objects on a given differentiable manifold (space-time). You can state it in a coordinate-free way.

But, this given dynamical law has (in general) different coordinate expressions in different coordinate charts.

In Newtonian Mechanics with its Euclidean Space-time manifold, there is a subset of coordinate charts (observers) in which the dynamical law (that is a geometric relation between geometric objects, coordinate-free) has a very simple mathematical expression. We call these charts (these observers) "inertial frames".

What you call "rotating observers" are not in this subset; the same dynamical law (geometric relation) has a less simple mathematical expression in these charts.


We are not saying that "inertial frames" and "rotating frames" are the same in Newtonian Mechanics. They are not.

What we are saying is both are just valid charts on the (euclidean space-time) manifold. Any geometric object (curve, vector velocity, vector field, tensor field...) and any geometric relation among geometric objects, is what it is, but will have (in general) different coordinate expression in different coordinate charts.


What D.H. showed you is the following: if you have two different charts (each one correspond to an observer that is rotating with respect to the other observer), then the coordinate expression for the geometric object (dq/dt which is a coordinate-free concept, you will understand this when you study differential geometry) in those two charts are related in exactly that way.

 

[Philip Wood]

Is it possible, would you say, to understand the equation (dq/dt)_A = (dq/dt)_B + ω x q without using differential geometry, or the language of differential geometry?

 

[mattt]

Is it possible, would you say, to understand the equation (dq/dt)_A = (dq/dt)_B + ω x q without using differential geometry, or the language of differential geometry?

Let me start with this (later on I'll try to not use differential geometry) :

Let $$M$$ be a Euclidean Space-time (a four-dimensional Differentiable Manifold with some requisites I won't go into) and let

$$\alpha : I \to M$$ be some differentiable curve (that correspond to some point-particle with mass, for example).


A chart is "a way" to give space-time coordinates to every point of some region of M (again, all this has its perfect definition, I won't go into details here).

So under a chart, it will be $$\alpha(\tau) = (t(\tau), x(\tau), y(\tau), z(\tau))$$

In this particular case of Newtonian Space-time, there is a foliation that allow a reparametrization of the curve in such a way that in any chart it will be:

$$\alpha(t) = (t,x(t),y(t),z(t))$$

(i.e. the "time" is "the same" for all observers; by the way, this is not true in other Space-time manifolds, such as Lorentz Manifolds used in Special and General Relativity as Space-times).


The last three coordinates is what you call "the vector position of the particle with respect to this chart (this observer)"

$$\vec{r}(t) = (x(t), y(t), z(t))$$


x(t), y(t), z(t) don't have to be necessarily escalar components with respect to orthonormal unit vectors. It can be much more general (again, there is a perfect mathematical definition of what is a coordinate chart in a differentiable manifold, but you don't need to know these details now).


In the example of this thread, we have two different charts, each one correspond to an observer that is rotating with respect to the other observer.

Let us put some names:

$$\alpha(t) = (t, \vec{r}(t)) = (t, x(t), y(t), z(t) )$$ for one observer.

$$\alpha(t) = (t, \vec{s}(t)) = (t, a(t), b(t), c(t) )$$ for the other observer.


The curve $$\alpha$$ has a vector velocity concept (a geometric object whose definition is coordinate-free) and this new object will have (in general) different coordinate representation in different coordinate charts.

For example, it happens that:

$$\alpha'(t) = (1, x'(t), y'(t), z'(t))$$ is the mathematical expression of the vector velocity of the curve for on observer (with x'(t) I mean dx(t)/dt) and

$$\alpha'(t) = (1, a'(t), b'(t), c'(t))$$ for the other observer.


Again, the last three components is what you call "vector velocity of the particle (with respect to that observer).


For the example of this thread, $$(x'(t), y'(t), z'(t))$$ is what you call $$\left(\frac{d\vec{q}}{dt}\right)_A$$

and $$(a'(t), b'(t), c'(t))$$ is what you call $$\left(\frac{d\vec{q}}{dt}\right)_B$$


Obviously, in general, the functions $$x'(t)$$ and $$a'(t)$$ are different, i.e. in general, $$x'(t)\not = a'(t)$$, $$y'(t)\not = b'(t)$$, $$z'(t)\not = c'(t)$$


But there is a mathematical relation between those two different coordinate expressions of THE SAME geometric object (just like in any vector space, a given vector has different coordinates in different basis, and there is a mathematical relation between these two set of coordinates; in our example is just the same, this time in the tangent vector space to each point of the manifold).


That relation is precisely what D.H. wrote for this case.


I hope it helps. :-)

 

[Philip Wood]

Hello Matt.
I'm very grateful (this is not ironic) that you've taken the trouble to try to educate me. What your post certainly does is to whet my appetite to study differential geometry. I have to say that regarding the equation (dq/dt)_A = (dq/dt)_B + ω x q, I can't see that your post shows my interpretation to be wrong. But perhaps I've missed something.

I have three textbooks which treat the equation. They are quite old and fairly elementary, but the authors are authoritative: T W Kibble, J L Synge, C Lanczos. All of them treat the equation using the mathematics of vectors; none of them mentions differential geometry. I'm not using this as an argument that DG is irrelevant, but as an indication that the insights of DG can be translated into the ordinary language of vectors.

It seems that both you and DH are trying to point out that I am making some mistake of interpretation. Is it my use of a set of rotating base vectors that you find objectionable? The best post for you to get your teeth into would probably be number 28. Maybe any insights from differential geometry could be expressed in the language of vectors; this could even be a useful exercise for you!