Angular velocity of a moving frame and the rotating-frame acceleration identity

Use one-based labels $z_i:=z(i-1)$ and $A_{ij}:=A(i-1,j-1)$ for $1\le i,j\le 3$, so the displayed components correspond in order to the imported coordinates $0,1,2$. Matrix-vector products mean $(Az)(k):={\sum }_{j<3}A(k,j)z(j)$, obtained from the rectangular product by representing $z$ as the column $C_z(j,0)=z(j)$ (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes).

Let $I\subseteq \mathbb{R}$ be a nondegenerate interval, let $R:I\to SO(3)$ and $c:I\to {\mathbb{R}}^3$ be twice continuously differentiable in the sense that all entries of $R$ and all components of $c$ have continuous second derivatives, and put

$\mathrm{\Omega }(t):=R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)(t\in I)\mathrm{.}$

1. $\mathrm{\Omega }(t)$ is skew-symmetric for every $t\in I$, that is $\mathrm{\Omega }(t{)}^{\mathsf{T}}=-\mathrm{\Omega }(t)$.
2. For every $t\in I$ there is exactly one vector $\omega (t)\in {\mathbb{R}}^3$ such that

$\mathrm{\Omega }(t)z=\omega (t)\times z(z\in {\mathbb{R}}^3),$

the cross product being that of The cross product in ${\mathbb{R}}^3$. Its components in the standard basis are $\omega (t)=(-{\mathrm{\Omega }}_{23}(t),{\mathrm{\Omega }}_{13}(t),-{\mathrm{\Omega }}_{12}(t))$. Since $R$ is $C^2$, $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ and hence $\omega$ are $C^1$; thus ${\omega }^{\mathrm{\prime }}(t)$ in clause 3 is defined. 3. If $y:I\to {\mathbb{R}}^3$ is twice continuously differentiable and

$x(t):=R(t)y(t)+c(t)(t\in I),$

then for every $t\in I$

$R(t{)}^{\mathsf{T}}{x}^{\mathrm{\prime }\mathrm{\prime }}(t)=y^{\mathrm{\prime }\mathrm{\prime }}(t)+2\omega (t)\times y^{\mathrm{\prime }}(t)+{\omega }^{\mathrm{\prime }}(t)\times y(t)+\omega (t)\times (\omega (t)\times y(t))+R(t{)}^{\mathsf{T}}{c}^{\mathrm{\prime }\mathrm{\prime }}(t)\mathrm{.}$

4. For every $t\in I$ and $z\in {\mathbb{R}}^3$, $\mathrm{\parallel }R(t)z{\mathrm{\parallel }}_2=\mathrm{\parallel }z{\mathrm{\parallel }}_2$: the rotation part of the change of coordinates preserves Euclidean norms (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

• The vector $\omega (t)$ is the angular velocity of the frame in the body-fixed convention $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}$. The vector attached to the other convention $R^{\mathrm{\prime }}{R}^{\mathsf{T}}$ is the spatial angular velocity, and the two differ by a rotation by $R$; this item records only the convention used in clause 3, and its consumers must state which convention they use.

• The identity of clause 3 is a coordinate identity for twice differentiable curves. It asserts nothing about forces: reading the terms $2\omega \times y^{\mathrm{\prime }}$ and $\omega \times (\omega \times y)$ as "Coriolis" and "centrifugal" accelerations is an interpretation made by the physical items, under their own mass and force hypotheses.