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The equation of motion in a moving chart: inertial terms, inertial forces, and when it restricts to an inertial chart

Physical scope

The conditional description of the motions of a force-branch model in a rigidly moving and rotating spatial chart: the exact identity for the coordinate acceleration with the four inertial terms, the reading of those terms as mass-proportional inertial forces, and the characterization of exactly inertial charts as the charts whose inertial terms vanish, in which case the framework's force-covariance law is recovered. The inertial terms are properties of the chart's motion, not forces of the model's force map; the item asserts no equivalence between a moving chart and a gravitational field, and no measured value.

Statement

Adopt the force branch of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance: $N \geq 1$ labels with masses $m_{1}, \dots, m_{N} > 0$ , time window $I_{0}$ , open configuration domain $Q \subseteq R^{3 N}$ and force map $F$ with the standing regularity; fix one inertial frame $F$ with adapted coordinates $(t, x)$ (Inertial frames, stationary observers and the adopted Galilean transformations between frames, Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework). Let a moving spatial chart of $F$ be given by data $(I, c, R)$ with $I \subseteq I_{0}$ (Moving spatial charts of an inertial frame: rotating and translating coordinates, angular velocity and inertial terms), with angular velocity $ω$ and inertial-terms field $τ (t, y, \dot{y}) : = - (2 ω (t) \times \dot{y} + ω^{'} (t) \times y + ω (t) \times (ω (t) \times y) + R (t)^{T} c^{''} (t)) .$

Let $x : J \to Q$ be a motion of the model on an interval $J \subseteq I$ with nonempty interior (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model clause (c)): $m_{i} x_{i}^{''} (t) = F_{i} (t, x (t), w (t)) (t \in J),$ with $w = x^{'}$ the coordinate velocity of the labels, and for each label define the moving-chart coordinate curve and its derivatives $y_{i} (t) : = R (t)^{T} (x_{i} (t) - c (t)), {\dot{y}}_{i} = y_{i}^{'}, {\ddot{y}}_{i} = y_{i}^{''} .$ Then:

moving-chart equation. For every $i$ and every $t \in J$ , $m_{i} {\ddot{y}}_{i} (t) = R (t)^{T} F_{i} (t, x (t), w (t)) + m_{i} τ (t, y_{i} (t), {\dot{y}}_{i} (t)) .$
inertial forces. Define the inertial force on label $i$ along the motion by $F_{i}^{i n} (t) : = m_{i} τ (t, y_{i} (t), {\dot{y}}_{i} (t))$ , with the four additive components $- 2 m_{i} ω \times {\dot{y}}_{i}$ , $- m_{i} ω^{'} \times y_{i}$ , $- m_{i} ω \times (ω \times y_{i})$ and $- m_{i} R^{T} c^{''}$ (Coriolis, Euler, centrifugal and translational). Then clause 1 reads $m_{i} {\ddot{y}}_{i} = R^{T} F_{i} + F_{i}^{i n}$ . The inertial forces are determined by the chart's motion and the label's mass; they are not contributions of the model's force map $F$ , and a general moving chart is not an inertial frame of the model; clause 3 characterizes the case in which it is the restriction of another inertial chart.
criterion for an inertial chart. The inertial terms vanish identically, $τ (t, y, \dot{y}) = 0$ for every $(t, y, \dot{y}) \in I \times R^{3} \times R^{3}$ , if and only if $R$ is constant on $I$ and $c$ is affine on $I$ . In that case the moving chart $(t, y)$ on $I$ is the restriction of the adapted coordinate chart of an inertial frame $F_{*}$ of the model related to $F$ by a Galilean transformation with vanishing time shift, and clause 1 reduces for each label to $m_{i} {\ddot{y}}_{i} = R^{T} F_{i} (t, x (t), w (t))$ , which is the transformed equation of motion of clause (vi) of Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance read in $F_{*}$ . Conversely, if the moving chart on $I$ is the restriction of the adapted chart of an inertial frame related to $F$ by a Galilean transformation with vanishing time shift, then $R$ is constant and $c$ is affine and $τ \equiv 0$ .

The identity of clause 1 is a conditional consequence of the adopted force branch and of the chart's kinematics; clause 3 is its consistency check against the frame class of Inertial frames, stationary observers and the adopted Galilean transformations between frames. No claim is made that a moving chart is physically equivalent to an inertial frame with an additional force field, and no measured value is asserted.

Facts & Assumptions

Given: The force-branch model, one inertial frame $F$ , a moving chart $(I, c, R)$ with $I \subseteq I_{0}$ , a motion $x : J \to Q$ of the model with velocity $w = x^{'}$ , and the curves $y_{i} (t) = R (t)^{T} (x_{i} (t) - c (t))$ .

[L1]

The moving chart (Moving spatial charts of an inertial frame: rotating and translating coordinates, angular velocity and inertial terms): $c$ and $R$ are $C^{2}$ with $R (t) \in S O (3)$ ; $Ω = R^{T} R^{'}$ is skew with unique representing vector $ω$ , $Ω z = ω \times z$ ; $τ$ is the inertial-terms field of clause (d); $y_{i}$ is the moving-chart coordinate curve of $x_{i}$ with velocity ${\dot{y}}_{i}$ and acceleration ${\ddot{y}}_{i}$ .

[L2]

The rotating-frame acceleration identity (Angular velocity of a moving frame and the rotating-frame acceleration identity clause 3): for $C^{2}$ maps $R : I \to S O (3)$ , $c : I \to R^{3}$ and $y : I \to R^{3}$ with $x = R y + c$ , $R^{T} x^{''} = y^{''} + 2 ω \times y^{'} + ω^{'} \times y + ω \times (ω \times y) + R^{T} c^{''} .$

[L3]

The motion and the force branch (The associated first-order system, motions, maximal motions and the Newtonian flow of a force model clause (c), Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clauses (iii),(iv)): $m_{i} x_{i}^{''} (t) = F_{i} (t, x (t), w (t))$ on $J$ , with $F$ of the standing regularity and $Q$ open; the coordinate curves $x_{i}$ are $C^{2}$ .

[L4]

Frame changes (Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes clause 1, Transformation between adapted charts and its kinematic consequences, Inertial frames, stationary observers and the adopted Galilean transformations between frames): two inertial frames are related by $t^{'} = t + b$ , $x^{'} = Q x + v t + a_{0}$ with $b \in R$ , $a_{0}, v \in R^{3}$ , $Q \in S O (3)$ . An inertial frame is given by an event origin, a time direction of time component 1, and a fixed positively oriented orthonormal spatial basis.

[L5]

The force-covariance law (Adopted force model: force maps, the equation of motion, additive composition of force contributions and frame covariance clause (vi)): for a spatial rotation $Q \in S O (3)$ , boost $υ \in R^{3}$ , translation $a \in R^{3}$ and time shift $b$ , define $Q_{N} = diag (Q, \dots, Q)$ and repeated tuples $\overset{ˉ}{υ} = (υ, \dots, υ)$ , $\overset{ˉ}{a} = (a, \dots, a)$ . The tuple describing the same worldlines is $η (t^{'}) = Q_{N} x (t^{'} - b) + \overset{ˉ}{υ} (t^{'} - b) + \overset{ˉ}{a}$ , with $m_{i} η_{i}^{''} (t^{'}) = Q F_{i} (t^{'} - b, x (t^{'} - b), w (t^{'} - b))$ . On the transferred domain, where the arguments of $F$ lie in its original domain, the force map is $F_{*} (t^{'}, y, u) = Q_{N} F (t^{'} - b, Q_{N}^{T} (y - \overset{ˉ}{υ} (t^{'} - b) - \overset{ˉ}{a}), Q_{N}^{T} (u - \overset{ˉ}{υ}))$ .

[L6]

Calculus of constant and affine maps: a differentiable map on an interval with identically vanishing derivative is constant; a twice differentiable map with identically vanishing second derivative is affine (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant, Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[L7]

Matrix and cross-product algebra: $R^{T} R = I_{3}$ and $R$ is invertible for $R \in S O (3)$ ; $Ω = 0$ if and only if $ω = 0$ ; and $[R ω]_{\times} = R [ω]_{\times} R^{T}$ (The orthogonal group $O (3)$ and the rotation group $S O (3)$ , $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ , The skew-matrix/cross-product correspondence $[ω]_{\times} z = ω \times z$ and its conjugation law $R [ω]_{\times} R^{T} = [R ω]_{\times}$ , The cross product in $R^{3}$ , The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

Proof

technique · direct; apply the rotating-frame acceleration identity to each label of the motion and read off the inertial terms, then characterize the vanishing of those terms

1.1L1L2L3givenalgebra

Clauses 1 and 2: fix $i$ and apply [L2] to the $C^{2}$ curves $R$ , $c$ and $y_{i}$ with $x_{i} = R y_{i} + c$ : $R^{T} x_{i}^{''} = {\ddot{y}}_{i} + 2 ω \times {\dot{y}}_{i} + ω^{'} \times y_{i} + ω \times (ω \times y_{i}) + R^{T} c^{''} .$ Multiplying by $m_{i}$ and substituting $m_{i} x_{i}^{''} = F_{i}$ from [L3], $m_{i} {\ddot{y}}_{i} = R^{T} F_{i} - m_{i} (2 ω \times {\dot{y}}_{i} + ω^{'} \times y_{i} + ω \times (ω \times y_{i}) + R^{T} c^{''}) = R^{T} F_{i} + m_{i} τ (t, y_{i}, {\dot{y}}_{i}),$ the last equality being the definition of $τ$ in [L1]; this is clause 1. Writing $F_{i}^{i n} : = m_{i} τ$ and multiplying the four components of $τ$ by $m_{i}$ gives the Coriolis, Euler, centrifugal and translational terms of clause 2, and by [L1] and [L2] the field $τ (t, y, \dot{y})$ is built from the chart data and its position and velocity arguments, and the label mass enters through $F_{i}^{i n} = m_{i} τ (t, y_{i}, {\dot{y}}_{i})$ ; the force map enters the transformed equation through $R^{T} F_{i}$ ; this is clause 2.

1.2L1L7algebra

Clause 3, forward: assume $τ (t, y, \dot{y}) = 0$ for all $(t, y, \dot{y})$ . Fix $t \in I$ . First set $y = 0$ and $\dot{y} = 0$ ; then $τ (t, 0, 0) = - R (t)^{T} c^{''} (t) = 0$ , so $c^{''} (t) = 0$ because $R (t)$ is invertible by [L7]. Now set $y = 0$ and $\dot{y} = e_{j}$ for each standard basis vector $e_{j}$ ; since $c^{''} (t) = 0$ , the identity $τ (t, 0, e_{j}) = 0$ reduces to $- 2 ω (t) \times e_{j} = 0$ . Thus $ω (t) \times e_{j} = 0$ for $j = 1, 2, 3$ , which by the coordinate formula of the cross product implies $ω (t) = 0$ . As $t$ was arbitrary, $ω \equiv 0$ and $c^{''} \equiv 0$ on $I$ .

2.1step 1.2L6L7

With $ω \equiv 0$ one has $Ω = [ω]_{\times} = 0$ by [L7], so $R^{'} = R Ω = 0$ and $R$ is constant on $I$ by [L6]; and $c^{''} \equiv 0$ with the interval $I$ nondegenerate gives $c$ affine on $I$ by [L6], $c (t) = c (t_{0}) + c^{'} (t_{0}) (t - t_{0})$ for a fixed $t_{0} \in I$ .

2.2L4L5L6L7step 1.1construct

Clause 3, inertial-frame identification: assume $R$ is constant and $c (t) = a_{0} + v t$ on $I$ . For the original frame $F = (o, u, (e_{j}))$ , set $o_{*} : = o + \sum_{j} (a_{0})_{j} e_{j}$ , $u_{*} : = u + \sum_{j} v_{j} e_{j}$ and $e_{* k} : = \sum_{j} e_{j} R_{j k}$ . The time component of $u_{*}$ is 1 and the fixed spatial basis $(e_{* k})$ is positively oriented and orthonormal because $R \in S O (3)$ ; thus [L4] defines an inertial frame $F_{*}$ . Its adapted coordinates satisfy $t_{*} = t$ and $x = R y + a_{0} + v t$ for all times, so the moving chart on $I$ is their restriction. Equivalently $y = Q x + v_{*} t + a_{0 *}$ with $Q = R^{T}$ , $v_{*} = - R^{T} v$ and $a_{0 *} = - R^{T} a_{0}$ . Apply [L5] to the tuple $η = (y_{1}, \dots, y_{N})$ on $J$ with $b = 0$ : $m_{i} η_{i}^{''} = R^{T} F_{i} (t, x (t), w (t))$ . Since $R$ is constant, $ω \equiv 0$ and $c^{''} \equiv 0$ , hence $τ \equiv 0$ and clause 1 reduces to this covariance identity.

3.1L4step 2.2algebra

Clause 3, converse: if the moving chart on $I$ is the restriction of the adapted coordinates of an inertial frame related to $F$ by a Galilean transformation with vanishing time shift, then $y = Q x + v t + a_{0}$ on $I$ with constant $Q \in S O (3)$ , $v, a_{0}$ (Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes clause 1), so $x = Q^{T} y - Q^{T} v t - Q^{T} a_{0}$ ; comparison with $x = R (t) y + c (t)$ for all $y$ shows that $R = Q^{T}$ is constant and $c (t) = - Q^{T} v t - Q^{T} a_{0}$ is affine on $I$ . Step 2.2 then gives $τ \equiv 0$ .

4.1step 1.1step 1.2step 2.1step 2.2step 3.1∎

Clauses 1-3: clauses 1 and 2 are step 1.1, and clause 3 is the equivalence assembled from steps 1.2, 2.1, 2.2 and 3.1.

Remarks

A chart, not a frame. The moving chart is attached to the inertial frame $F$ ; its coordinate triple $y$ is not the coordinate of a physical rotation of matter. The inertial forces depend on the chart's reference acceleration $c^{''}$ and rotation data $(ω, ω^{'})$ and are unaffected by the interactions of the model; two charts with different data produce different inertial terms for the same motion.
No equivalence principle. Clause 2 permits the inertial terms to be carried along in computations in a moving chart, which is what makes the description of terrestrial and rotating-apparatus experiments convenient; it does not assert that a chart's inertial terms are a gravitational field or that a true force law of the model has been transformed into a fictitious one. Whether a term in a concrete computation is an interaction force of the model or an inertial term of the chart is decided by the model's declaration of $F$ and of the chart data.
Where the criterion is used. Clause 3 identifies exactly the charts for which the framework's force covariance applies unchanged; the remaining charts have nonzero inertial terms, and the companion example exhibits all four of them on explicit data.

The equation of motion in a moving chart: inertial terms, inertial forces, and when it restricts to an inertial chart

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources