Different coordinate system as opposed to different reference frame

This is a problem I've struggled with before. The issue is that most textbooks are not clear enough in their definitions of the fixed and rotating reference frames. I've tried to be as comprehensive as possible in my answer, I hope it helps!

To describe the motion of a rigid body through space we must make use of two separate reference frames:

  • Firstly, a fixed (i.e. inertial) reference frame with a set of orthonormal basis vectors $\mathbf{e}_i$. These vectors are constant in time to all inertial observers. I will call the inertial reference frame $K$, and this set of vectors $S$.

  • Secondly, the body frame - a non-inertial reference frame which moves with the rigid body. A second set of orthonormal basis vectors $\mathbf{e}'_j$ move with this frame. I will call this frame of reference $K'$, and this set of vectors $S'$.

The diagram below illustrates this picture (the weird blob is the rigid body!).

Two reference frames

Any vector in space can be written as a linear combination of either set of basis vectors: we can write $$ \mathbf{A} = A_i\mathbf{e}_i = A'_j\mathbf{e}'_j. $$ In particular, if the vector $\mathbf{A}$ is fixed to the rigid body, then the components $A_i$ are functions of time, but, since the projection of $\mathbf{A}$ onto each of the $\mathbf{e}'_j$ is constant, the $A'_j$ are constant. This holds in both $K$ and $K'$.

Importantly, an observer from either reference frame can use either set of basis vectors. The difference is in how they view the behaviour of each set of vectors: from the perspective of $K$, the basis vectors $\mathbf{e}_i$ are fixed, but the basis vectors $\mathbf{e}'_j$ are functions of time; however, to an observer in $K'$, i.e. from the perspective of the rigid body, the $\mathbf{e}'_j$ are fixed, and it is instead the vectors $\mathbf{e}_i$ which are time dependent.

Thus, when observed from $K'$, $$ \left(\frac{\text{d}}{\text{d}t}\right)_{K'}\mathbf{e}'_i = \mathbf{0} $$ so the rate of change of a vector $\mathbf{A} = A'_i\mathbf{e}'_i$ appears to be $$ \left(\frac{\text{d}\mathbf{A}}{\text{d}t}\right)_{K'} = \dot{A'}_i\mathbf{e}'_i. $$ In $S$ howver, the $\mathbf{e}'_i$ basis vectors, are functions of time. I show below that, as observed from $K$, $$ \left(\frac{\text{d}}{\text{d}t}\right)_{K}\mathbf{e}'_i = \boldsymbol{\omega}\times\mathbf{e}'_i\tag{1} $$ for some time-dependent vector $\boldsymbol{\omega}$, so \begin{align} \left(\frac{\text{d}\mathbf{A}}{\text{d}t}\right)_{K} &= \dot{A'}_i\mathbf{e}'_i +A'_i\dot{\mathbf{e}}'_i\\ & = \dot{A'}_i\mathbf{e}'_i + A'_i \boldsymbol{\omega}\times\mathbf{e}'_i\\ & = \dot{A'}_i\mathbf{e}'_i+ \boldsymbol{\omega}\times(A'_i \mathbf{e}'_i)\\ & = \dot{A'}_i\mathbf{e}'_i +\boldsymbol{\omega}\times\mathbf{A}, \end{align} and thus $$ \left(\frac{\text{d}\mathbf{A}}{\text{d}t}\right)_{K'} = \left(\frac{\text{d}\mathbf{A}}{\text{d}t}\right)_{K} - \boldsymbol{\omega}\times\mathbf{A}. \tag{2} $$ This is the crux of the problem: to an observer in the moving frame, every vector appears to have an additional term in its time derivative, when in fact it is the frame itself that is changing.

With this, Euler's equations follow quite naturally. Let $\mathbf{T}$ be the torque on the body, and let $\underline{\mathbf{I}}\boldsymbol{\omega}$ be the angular momentum of the body, where $\underline{\mathbf{I}}$ is the moment of inertia tensor. The key point is that as measured in $K'$, the moment of inertia tensor is constant. (This is not true in $K$, as the distribution of the mass of the body with respect to the $\mathbf{e}_i$ basis changes with time.) Using equation (2), this means that $$ \left(\frac{\text{d}}{\text{d}t}\right)_{K'}\underline{\mathbf{I}}\boldsymbol{\omega} = \underline{\mathbf{I}}\left(\frac{\text{d}}{\text{d}t}\right)_{K'}\boldsymbol{\omega} = \underline{\mathbf{I}}\left[\left(\frac{\text{d}}{\text{d}t}\right)_{K}\boldsymbol{\omega}-\underbrace{\boldsymbol{\omega}\times\boldsymbol{\omega}}_{\mathbf{0}}\right] = \underline{\mathbf{I}}\left(\frac{\text{d}}{\text{d}t}\right)_{K}\boldsymbol{\omega}.\tag{$\star$} $$ Finally, the rate of change of angular momentum in the body is equal to the applied torque: $$ \mathbf{T}= \left(\frac{\text{d}}{\text{d}t}\right)_{K}\underline{\mathbf{I}}\boldsymbol{\omega} = \left(\frac{\text{d}}{\text{d}t}\right)_{K'}\underline{\mathbf{I}}\boldsymbol{\omega}+\boldsymbol{\omega}\times(\underline{\mathbf{I}}\boldsymbol{\omega}) =\underline{\mathbf{I}}\left(\frac{\text{d}}{\text{d}t}\right)_{K}\boldsymbol{\omega} +\boldsymbol{\omega}\times(\underline{\mathbf{I}}\boldsymbol{\omega}). $$ This is Euler's equation! It holds from the perspective of the inertial frame, but is derived using the basis vectors of the non-inertial frame.

All that remains is to prove equation (1). To do so, it is best to forget about the body frame $K'$, and to think of the $\mathbf{e}'_i$ as simply a set of vectors which move with respect to the $\mathbf{e}_i$, as seen from $K$. The moving basis vectors $\mathbf{e}'_j(t)$ are related to the fixed basis vectors $\mathbf{e}_i$ by a time dependent rigid rotation, which can be described by a matrix $\underline{\mathbf{R}}(t)$. Using Einstein notation, we have $$ \mathbf{e}'_i(t) = R_{ij}(t) \mathbf{e}_j, \qquad\text{and}\qquad \mathbf{e}_i = R_{ij}^T(t) \mathbf{e}'_j(t). \tag{1} $$ Since $\underline{\mathbf{R}}$ is a rotation matrix, it is orthogonal, so for all times $t$ it satisfies $$ \underline{\mathbf{R}}(t)\underline{\mathbf{R}}^T(t) = \underline{\mathbf{I}}, $$ where $\underline{\mathbf{I}}$ is the identity matrix. Differentiating this with respect to time we find $$ \underline{\dot{\mathbf{R}}}(t)\underline{\mathbf{R}}^T(t)+\underline{\mathbf{R}}(t)\underline{\dot{\mathbf{R}}}^T(t)= \underline{\dot{\mathbf{R}}}(t)\underline{\mathbf{R}}^T(t) + \left(\underline{\dot{\mathbf{R}}}(t)\underline{\mathbf{R}}^T(t)\right)^T = \underline{\mathbf{0}}, $$ so the matrix $\underline{\dot{\mathbf{R}}}(t)\underline{\mathbf{R}}^T(t)$ must be antisymmetric. This implies that there exists some vector $\boldsymbol{\omega} = (\omega_1, \omega_2, \omega_3)$ such that $$ \underline{\dot{\mathbf{R}}}(t)\underline{\mathbf{R}}^T(t) = [\boldsymbol{\omega}\times] $$ where the matrix $$ [\boldsymbol{\omega}\times] = \begin{pmatrix}0 & -\omega_3 & \omega_2 \\ \omega_3 & 0 & -\omega_1 \\ -\omega_2 & \omega_1 &0 \end{pmatrix} $$ acts on vectors like a cross product as $[\boldsymbol{\omega}\times]\mathbf{A} = \boldsymbol{\omega}\times\mathbf{A}$ (see here if you've not seen this before). For our purposes, it will be most useful to know the components of this matrix in the moving basis $\mathbf{e}'_j$. Using $\boldsymbol{\omega} = \omega'_j\mathbf{e}'_j$, we have $$ \left(\underline{\dot{\mathbf{R}}}(t)\underline{\mathbf{R}}^T(t)\right)'_{ij} = \mathbf{e}'^T_i\left(\underline{\dot{\mathbf{R}}}(t)\underline{\mathbf{R}}^T(t)\right)\mathbf{e}'_j = \epsilon_{ijk}\omega'_k. $$ With this, we can calculate the time derivatives of the primed basis vectors: \begin{align} \dot{\mathbf{e}}'_i(t) &= \dot{R}_{ij}(t) \mathbf{e}_j \\ &= \dot{R}_{ij} R^T_{jk}\mathbf{e}'_k\\ &= \epsilon_{ikm}\omega'_m \mathbf{e}'_k\\ &= \epsilon_{ikm}(\boldsymbol{\omega}\cdot\mathbf{e}'_m)\mathbf{e}'_k\\ &= \frac{1}{2} \epsilon_{ikm}(\boldsymbol{\omega}\cdot\mathbf{e}'_m)\mathbf{e}'_k - \frac{1}{2}\epsilon_{ikm}(\boldsymbol{\omega}\cdot\mathbf{e}'_k)\mathbf{e}'_m\\ &=\frac{1}{2}\epsilon_{ikm}\boldsymbol{\omega}\times(\mathbf{e}'_k\times\mathbf{e}'_m)\\ &=\frac{1}{2}\epsilon_{ikm}\boldsymbol{\omega}\times(\epsilon_{kml}\mathbf{e}'_l)\\ &=\delta_il\boldsymbol{\omega}\times\mathbf{e}'_l\\ &=\boldsymbol{\omega}\times\mathbf{e}'_i(t).\tag{2} \end{align} Thus, as seen from the inertial frame, there exists an instantaneous angular velocity vector $\boldsymbol{\omega}$ around which the primed basis vectors $\mathbf{e}'_i(t)$ rotate.

Finally, here is an alternative derivation of Euler's equation that I am particularly fond of. We proceed by directly by differentiating the moment of inertia tensor. To do this, we align the $\mathbf{e}'_i$ with the principal axes of the body. We denote the principal moment of the body along the $\mathbf{e}'_i$ axis as $I_i$. Since the body is rigid, the pricipal moments are constants, and so we can explicitly write the moment of inertial tensor as $$ \underline{\mathbf{I}} = \sum_i I_i \mathbf{e}'_i \otimes \mathbf{e}'_i $$ (summation convention not used here). The time derivative of $\underline{\mathbf{I}}$ (in $K$) is then $$ \left(\frac{\text{d}}{\text{d}t}\right)_{K} \underline{\mathbf{I}} = \sum_i I_i\left[\dot{\mathbf{e}}'_i\otimes\mathbf{e}'_i+\mathbf{e}'_i\otimes\dot{\mathbf{e}}'_i\right] = \sum_i I_i\left[(\boldsymbol{\omega}\times\mathbf{e}'_i)\otimes\mathbf{e}'_i+\mathbf{e}'_i\otimes(\boldsymbol{\omega}\times\mathbf{e}'_i)\right]. $$ Now, let $\mathbf{b}$ and $\mathbf{c}$ be arbitrary vectors, and $\underline{\mathbf{A}}$ be a linear operator. Using the definition of the tensor product, we have $$ \left[(\underline{\mathbf{A}}\mathbf{b})\otimes\mathbf{c}\right]_{ij}=\left(\underline{\mathbf{A}}\mathbf{b}\right)_i c_j = A_{ik}b_k c_j = \left(\underline{\mathbf{A}}\right)_{ik}\left(\mathbf{b}\otimes\mathbf{c}\right)_{kj} = \left[\underline{\mathbf{A}}(\mathbf{b}\otimes\mathbf{c})\right]_{ij} $$ and $$ \left[\mathbf{b}\otimes(\underline{\mathbf{A}}\mathbf{c})\right]_{ij}=b_i\left(\underline{\mathbf{A}}\mathbf{c}\right)_j = b_i A_{jk} c_k = \left(\mathbf{b}\otimes\mathbf{c}\right)_{ik}\left(\underline{\mathbf{A}}^T\right)_{kj} = \left[(\mathbf{b}\otimes\mathbf{c})\underline{\mathbf{A}}^T\right]_{ij}. $$ Therefore, \begin{align} \left(\frac{\text{d}}{\text{d}t}\right)_{K} \underline{\mathbf{I}}&= \sum_i I_i\left[(\boldsymbol{\omega}\times\mathbf{e}'_i)\otimes\mathbf{e}'_i+\mathbf{e}'_i\otimes(\boldsymbol{\omega}\times\mathbf{e}'_i)\right] \\ &= \sum_i I_i\left[[\boldsymbol{\omega}\times](\mathbf{e}'_i\otimes\mathbf{e}'_i)+(\mathbf{e}'_i\otimes\mathbf{e}'_i)[\boldsymbol{\omega}\times]^T\right]\\ &=[\boldsymbol{\omega}\times]\underline{\mathbf{I}}-\underline{\mathbf{I}}[\boldsymbol{\omega}\times] \end{align} where $[\boldsymbol{\omega}\times]$ is the operator defined in the previous section, and in the last line I have used that this operator is anti-symmetric. Thus, the time derivative of the angular momentum is \begin{align} \mathbf{T} &= \left(\frac{\text{d}}{\text{d}t}\right)_{K} (\underline{\mathbf{I}}\boldsymbol{\omega}) \\ &= \underline{\mathbf{I}}\left(\frac{\text{d}}{\text{d}t}\right)_{K} \boldsymbol{\omega}+\boldsymbol{\omega}\left(\frac{\text{d}}{\text{d}t}\right)_{K} \underline{\mathbf{I}}\\ &=\underline{\mathbf{I}}\left(\frac{\text{d}}{\text{d}t}\right)_{K} \boldsymbol{\omega} + [\boldsymbol{\omega}\times]\underline{\mathbf{I}}\boldsymbol{\omega}-\underline{\mathbf{I}}\underbrace{[\boldsymbol{\omega}\times]\boldsymbol{\omega}}_{\boldsymbol{\omega}\times\boldsymbol{\omega}=\mathbf{0}}\\ &=\underline{\mathbf{I}}\left(\frac{\text{d}}{\text{d}t}\right)_{K} \boldsymbol{\omega} + \boldsymbol{\omega}\times(\underline{\mathbf{I}}\boldsymbol{\omega}) \end{align} as before.

The word frame is intended to refer to physical matter. For example we talk of the Earth frame, and if you are travelling by car you will naturally use the car as your reference frame when considering objects in the car. A reference frame is the matter relative to which coordinates are defined. I have defined these in The Large and the Small:

  • A reference frame consists of the reference matter, the apparatus, and the procedures, required to determine a spacetime coordinate system.
  • A coordinate system is a mapping from physical events to coordinates with the form $(t, x, y, z)$ where $t$ is the time of the event and $(x, y, z)$ describes the position of the event (for example, in terms of distances, north, east and up, from some chosen fixed point, or origin).

(this is more strictly a spacetime coordinate system. In Newtonian mechanics you can consider space coordinates and take time as a parameter).

Thus the frame and the coordinate system are closely related, and in many cases (such as you describe) it is not necessary to distinguish between them. We may talk of motion in a reference frame, meaning that coordinates are implicit (or known) or we may talk of motion in coordinates, knowing that coordinates have been derived from an implicit frame.

if it's still not clear what I'm struggling with, it is the statement that Euler's equation for example, is with respect to an inertial frame of reference, but with respect to coordinate axes fixed with a rotating body. How is it not a contradiction? Aren't the axes variant with time as rotation, making the reference point from there non-inertial?

You are right, accounts of Euler's equation are confusing. It would not make much sense to work in the frame of the body (which is what many accounts seem to suggest) because then you would simply have $\mathbf\omega = \mathbf 0$. Euler's equation does apply relative to an inertial frame of reference, in which we can write $$ \mathbf T = \dot {\mathbf L}.$$ But this is a difficult equation to use, because ${\mathbf L} = I\omega$. Hence

$$\dot {\mathbf L} = \dot {\mathbf I}\omega + {\mathbf I}\dot\omega$$

and $\dot {\mathbf I}$ is not easy to work with. Instead we rewrite $ \mathbf I$ in the body frame, determined in general from the eigenvectors of $ \mathbf I$. Then $\dot {\mathbf I} =0 $, and $ \mathbf T = \dot {\mathbf L}$ is rewritten as $$ \mathbf T = \dot {\mathbf L} + \mathbf \omega \times \mathbf L.$$

i.e. Euler's equation. IOW we do not write the equation in the rotating frame, but only adapt the equation so as to use the components of the Moment of Inertia Tensor, $\mathbf I$, in the rotating frame, because in this frame the components are constant.

In other words, we are not really describing the motion in the rotating frame, but simply writing vectors in terms of vectors parallel to the principle axes, while actually remaining in the inertial frame.

To make this clearer, consider the simplest case of circular motion in a plane about the origin. In the inertial frame we can describe any vector in terms the unit vectors $\mathbf i$ and $\mathbf j$, but it is often more convenient to use the unit vectors $\mathbf {\hat r}$ and $\mathbf {\hat {\theta}}$ (this is directly applicable in the case where the principle axes are aligned with $\mathbf {\hat r}$ and $\mathbf {\hat {\theta}}$). We have, for any vector $\mathbf a$,

$$ \mathbf a = a_i \mathbf i + a_j \mathbf j = a_r \mathbf {\hat r} + a_{\theta} \mathbf {\hat {\theta}} $$

Then when we differentiate $\mathbf a$ with respect to time, terms appear in the inertial frame from the derivatives of $\mathbf {\hat r}$ and $\mathbf {\hat {\theta}}$ because $\mathbf {\hat r}$ and $\mathbf {\hat {\theta}}$ are varying in the inertial frame.

This is made confusing in rigid body treatments, because $a_r$ and $a_{\theta}$ are also the components of $\mathbf a$ in the rotating frame. Imv, it is not actually helpful to think of them like that. In the rotating frame $\mathbf {\hat r}$ and $\mathbf {\hat {\theta}}$) are constant vectors, and we would need to introduce inertial (fictitious) forces to describe the motion.