Moving spatial charts of an inertial frame: rotating and translating coordinates, angular velocity and inertial terms

Fix one inertial frame $F=(o,u,(e_1,e_2,e_3))$ of the model with adapted coordinates $(t,x)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames); lengths and times are read in metres and seconds (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). A moving spatial chart of this frame is the following kinematic data:

(a) Data. A nondegenerate interval $I\subseteq \mathbb{R}$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length) together with maps $c:I⟶{\mathbb{R}}^3,R:I⟶SO(3),$ the matrix $R(t)$ being identified with its nine entries, such that all components of $c$ and all entries of $R$ have continuous second derivatives on $I$ ($C^k$ Euclidean maps and diffeomorphisms, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral); we then say that $c$ and $R$ are of class $C^2$. The components $c_j(t)$ are spatial coordinates, with $[c_j]=\mathrm{\ell }$, and the entries of $R(t)$ are dimensionless, with $[R_{ij}]=1$; the moving coordinates $y_j$ also have dimension $\mathrm{\ell }$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). The map $\mathrm{\Phi }:I\times {\mathbb{R}}^3⟶I\times {\mathbb{R}}^3,\mathrm{\Phi }(t,y):=(t,R(t)y+c(t)),$ is the coordinate change of the chart: for $t\in I$ it identifies the triple $y\in {\mathbb{R}}^3$ with the point of the inertial chart whose coordinates are $x=R(t)y+c(t)$. The curve $t\mapsto c(t)$ is the reference curve of the chart and the family $R(t)$ its rotation; the chart is attached to the fixed inertial frame, and its data are kinematic data of the description, not new physical objects.

(b) Coordinate curves, velocities and accelerations. Let $J\subseteq I$ be an interval with nonempty interior and let $x:J\to {\mathbb{R}}^3$ be a curve of class $C^2$ in the inertial chart (for instance the coordinate curve of a worldline in the sense of Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors). The coordinate curve of $x$ in the moving chart is $y(t):=R(t{)}^{\mathsf{T}}(x(t)-c(t))(t\in J);$ its coordinate velocity and coordinate acceleration in the moving chart are the componentwise derivatives $\dot{y}:=y^{\mathrm{\prime }}$ and $\ddot{y}:=y^{\mathrm{\prime }\mathrm{\prime }}$ (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral), of dimensions $\mathrm{\ell }{\tau }^{-1}$ and $\mathrm{\ell }{\tau }^{-2}$. These are chart-local notions: because the chart is generally time-dependent, these need not be the coordinate velocity and acceleration of an inertial frame. If $R$ is constant and $c$ is affine on $I$, they agree on $I$ with the corresponding quantities in the inertial frame whose adapted chart restricts to this chart. Their relation to the fixed frame $F$ is the kinematic identity of the cited mathematical supplier, not an identification with the quantities read in $F$.

(c) Angular velocity of the chart. Put $\mathrm{\Omega }(t):=R(t{)}^{\mathsf{T}}{R}^{\mathrm{\prime }}(t)$. By Angular velocity of a moving frame and the rotating-frame acceleration identity clause 1, $\mathrm{\Omega }(t)$ is skew-symmetric for every $t$, and by clause 2 there is a unique $\omega (t)\in {\mathbb{R}}^3$ with $\mathrm{\Omega }(t)z=\omega (t)\times z(z\in {\mathbb{R}}^3),$ the cross product being that of The cross product in ${\mathbb{R}}^3$. The vector $\omega (t)$ is the angular velocity of the chart in the convention $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ (the body convention), of dimension ${\tau }^{-1}$ read in radians per second; its components are read off the three entries of $\mathrm{\Omega }$ above the diagonal. With the opposite convention ${\mathrm{\Omega }}_s:=R^{\mathrm{\prime }}{R}^{\mathsf{T}}$ one has ${\mathrm{\Omega }}_s=R\mathrm{\Omega }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ by clause 3 of The skew-matrix/cross-product correspondence $[\omega {]}_{\times }z=\omega \times z$ and its conjugation law $R[\omega {]}_{\times }{R}^{\mathsf{T}}=[R\omega {]}_{\times }$ and The orthogonal group $O(3)$ and the rotation group $SO(3)$, so the representing vector of the opposite convention is $R(t)\omega (t)$: the two conventions differ by the rotation, and every statement of this framework names the one it uses. This definition uses the body convention throughout.

(d) Inertial terms. For $(t,y,\dot{y})\in I\times {\mathbb{R}}^3\times {\mathbb{R}}^3$ define $\tau (t,y,\dot{y}):=-(2\omega (t)\times \dot{y}+{\omega }^{\mathrm{\prime }}(t)\times y+\omega (t)\times (\omega (t)\times y)+R(t{)}^{\mathsf{T}}{c}^{\mathrm{\prime }\mathrm{\prime }}(t)),$ of dimension $\mathrm{\ell }{\tau }^{-2}$; $\tau$ is the field of inertial terms of the chart, per unit mass in the sense that the moving-chart equation of the next item carries the factor $m\tau$ for a label of mass $m$. Its four additive contributions are named, in order: the Coriolis term $-2\omega \times \dot{y}$, the Euler term $-{\omega }^{\mathrm{\prime }}\times y$, the centrifugal term $-\omega \times (\omega \times y)$, and the translational term $-R^{\mathsf{T}}{c}^{\mathrm{\prime }\mathrm{\prime }}$. The names are conventions attached to the displayed terms; the definition attaches no force to them, and clause (f) records their kinematic status.

(e) Curves at rest in the chart. For fixed $y_0\in {\mathbb{R}}^3$ let $x_{y_0}(s):=R(s)y_0+c(s)$ be the spatial coordinate curve in the inertial chart. Its spacetime lift is the curve ${\gamma }_{y_0}:I\to A$ defined by ${\gamma }_{y_0}(s):=o+su+{\sum }_{j=1}^3(x_{y_0}(s){)}_j{e}_j\mathrm{.}$ It is $C^2$ and satisfies $t({\gamma }_{y_0}(s))=s$, so it has the parameter condition of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses. This lifted curve is the path of a point at rest in the chart; whether it is assigned as the worldline of a particle label is a separate declaration of a model and is not made here.

(f) Status. A moving chart is a time-dependent coordinate description attached to one inertial frame. In general it is not an inertial frame of the model: Inertial frames, stationary observers and the adopted Galilean transformations between frames defines inertial charts using a fixed spatial basis and an affine reference curve. When $R$ is constant and $c$ is affine on $I$, the coordinate change has the form of an allowed Galilean frame change and is the restriction to $I$ of an adapted chart. The inertial-forces theorem of this page proves that these are exactly the data for which the inertial terms of clause (d) vanish identically. A moving chart is not an observer congruence: a congruence is a family of curves of the spacetime data, whereas a chart is a coordinate map. The identity relating the inertial-chart acceleration to the moving-chart acceleration of the same curve is the mathematical lemma cited in clause (c); the equation of motion of a force model in the moving chart, with the inertial terms entering as the mass multiples $m\tau$, is the inertial-forces theorem of this page, with its own hypotheses. Nothing in this definition asserts a force law, an equation of motion, a transformation law for physical interactions or a measured value.

• Convention warning. The single most common error in reading the literature is to mix the two angular-velocity conventions. This framework fixes $\mathrm{\Omega }=R^{\mathsf{T}}{R}^{\mathrm{\prime }}$ and $\omega$ as above; a computation using the opposite convention must replace $\omega$ by $R\omega$ or state its convention explicitly. The identity of clause (c) is the exact conversion.

• Time coordinate unchanged. The chart changes spatial coordinates only: the time coordinate of the chart is the inertial frame's time coordinate $t$, and no time shift, time dilation or reparametrization is introduced. The charts of this item use the same time coordinate as $F$. Constant rotation and an affine origin give a Galilean inertial chart; more general time dependence of the rotation or origin produces nonzero inertial terms.

• Accelerated reference curves. The translational term involves $c^{\mathrm{\prime }\mathrm{\prime }}$, the inertial-chart acceleration of the chart's reference point; a uniformly accelerating chart has vanishing Coriolis, Euler and centrifugal terms but a nonzero translational term. The companion example verifies the four terms in a chart with all of them nonvanishing.