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Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes

Physical scope

The kinematic theorem of the framework: the transformation law between two inertial frames of the Newtonian model, the induced laws for the coordinate velocity and acceleration of a worldline, the frame-independence of the property of zero coordinate acceleration, and the group structure of the Galilean transformations. The result is a conditional deduction from the adopted spacetime, frame and worldline assumptions and the cited mathematical transformation law; it asserts no force law, no conservation law and no empirical claim, and it does not by itself identify free motion with zero coordinate acceleration.

Statement

Adopt the spacetime assumptions of Adopted Newtonian spacetime structure: absolute time, Euclidean spatial slices and their smooth structure, the frame assumptions of Inertial frames, stationary observers and the adopted Galilean transformations between frames and the worldline assumptions of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses. Let

$F = (o, u, (e_{1}, e_{2}, e_{3})), F^{'} = (o^{'}, u^{'}, (e_{1}^{'}, e_{2}^{'}, e_{3}^{'}))$

be two inertial frames of the model with adapted coordinates $(t, x)$ and $(t^{'}, x^{'})$ , and let $γ : I \to A$ be a worldline of a particle label described in $F$ , with coordinate curve $x : I \to R^{3}$ and coordinate velocity and acceleration $w = x^{'}$ and $a = x^{''}$ in the sense of Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors. Then:

There are $b \in R$ , $a_{0}, v \in R^{3}$ and $R \in S O (3)$ (The orthogonal group $O (3)$ and the rotation group $S O (3)$ ) such that for every event $p \in A$

$t^{'} (p) = t (p) + b, x^{'} (p) = R x (p) + v t (p) + a_{0} .$

The parameters have dimensions $[b] = τ$ , $[a_{0, j}] = ℓ$ and $[v_{j}] = ℓ τ^{- 1}$ , and $R$ is dimensionless (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework); they are the ones that occur in the adopted class of frame changes of Inertial frames, stationary observers and the adopted Galilean transformations between frames.

Let $η : I + b \to R^{3}$ be the coordinate curve of the same label in $F^{'}$ , defined by $η (t^{'}) : = x^{'} (γ (t^{'} - b))$ for $t^{'} \in I + b$ . Then $η$ is of class $C^{2}$ and

$η^{'} (t^{'}) = R w (t^{'} - b) + v, η^{''} (t^{'}) = R a (t^{'} - b) (t^{'} \in I + b) .$

The coordinate acceleration vanishes identically in one frame exactly when it does in the other:

$a \equiv 0 on I ⟺ η^{''} \equiv 0 on I + b .$

Equivalently, the property "the coordinate curve is an affine function of the frame time" is an inertial-frame-independent property of the motion.

Write $G (b, a_{0}, v, R)$ for the transformation $(t, x) \mapsto (t + b, R x + v t + a_{0})$ of the coordinate space. Then

$G (b_{2}, a_{2}, v_{2}, R_{2}) \circ G (b_{1}, a_{1}, v_{1}, R_{1}) = G (b_{1} + b_{2}, R_{2} a_{1} + v_{2} b_{1} + a_{2}, R_{2} v_{1} + v_{2}, R_{2} R_{1})$

and

$G (b, a_{0}, v, R)^{- 1} = G (- b, - R^{T} (a_{0} - v b), - R^{T} v, R^{T}),$

with all parameters again in $R \times R^{3} \times R^{3} \times S O (3)$ . Consequently the Galilean transformations form a group of transformations of the coordinate space, and the relation on inertial frames "the two charts differ by a Galilean transformation" is an equivalence relation.

Facts & Assumptions

Given: Two inertial frames $F, F^{'}$ of the model with coordinates $(t, x)$ and $(t^{'}, x^{'})$ , a worldline $γ : I \to A$ described in $F$ with coordinate curve $x$ , coordinate velocity $w$ and coordinate acceleration $a$ .

[L1]

Transformation law and kinematic consequences for adapted charts: there are $b \in R$ , $a_{0}, v \in R^{3}$ and $R \in O (3)$ with $t^{'} = t + b$ and $x^{'} = R x + v t + a_{0}$ on $A$ ; if both bases are positively oriented then $R \in S O (3)$ ; and for a curve $γ$ with $t (γ (s)) = s$ the map $y (s) : = x^{'} (γ (s))$ satisfies $y^{'} (s) = R x^{'} (s) + v$ and $y^{''} (s) = R x^{''} (s)$ , with $x^{''} = 0$ if and only if $y^{''} = 0$ (Transformation between adapted charts and its kinematic consequences).

[L2]

The set $S O (3)$ is closed under products and transposes, $R^{T} R = I_{3}$ , and $R^{T} \in S O (3)$ for $R \in S O (3)$ ( $S O (3)$ is a subgroup of the general linear group, with $R^{- 1} = R^{T}$ , The orthogonal group $O (3)$ and the rotation group $S O (3)$ ).

[L3]

A worldline described in $F$ is a $C^{2}$ curve with $τ (γ (s) - o) = s$ , that is, $t (γ (s)) = s$ , for every $s \in I$ ; the description in a second frame is the reparametrized curve of clause (ii) of Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses (Particle labels, worldlines, the frame-time parameter condition and fixed inertial masses).

[L4]

Coordinate velocity and acceleration are $w = x^{'}$ and $a = x^{''}$ , and the description of the motion in the second frame is $η (t^{'}) = x^{'} (γ (t^{'} - b))$ (Coordinate velocity and acceleration of a worldline in an inertial frame, and the spatial velocity and acceleration vectors, clause (e)).

[L5]

Chain rule for total derivatives: if $f$ is totally differentiable at $a$ and $g$ at $f (a)$ , then $g \circ f$ is totally differentiable at $a$ with $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ; applied to the one-variable composition $t^{'} \mapsto x (t^{'} - b)$ with the affine inner map, whose derivative is the identity, it gives the derivative $x^{'} (t^{'} - b)$ of the composition (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

technique · direct

1.1L1given

By clause 1 of [L1] applied to the two adapted charts $F, F^{'}$ , there are $b \in R$ , $a_{0}, v \in R^{3}$ and $R \in O (3)$ with $t^{'} (p) = t (p) + b$ and $x^{'} (p) = R x (p) + v t (p) + a_{0}$ for all $p \in A$ ; both bases of an inertial frame are positively oriented by definition, so the orientation clause of [L1] gives $R \in S O (3)$ ; this proves clause 1.

1.2L2algebra

For the composition, apply $G (b_{1}, a_{1}, v_{1}, R_{1})$ and then $G (b_{2}, a_{2}, v_{2}, R_{2})$ to a point $(t, x)$ : the intermediate point is $(s, y) = (t + b_{1}, R_{1} x + v_{1} t + a_{1})$ , and the final point is $(s + b_{2}, R_{2} y + v_{2} s + a_{2}) = (t + b_{1} + b_{2}, R_{2} R_{1} x + (R_{2} v_{1} + v_{2}) t + R_{2} a_{1} + v_{2} b_{1} + a_{2})$ , which is the displayed composite by associativity of matrix multiplication and bilinearity of the matrix–vector product.

1.3L2algebra

For the inverse, let $(s, y) = G (b, a_{0}, v, R) (t, x)$ with $R \in S O (3)$ . Then $t = s - b$ and $x = R^{- 1} (y - v t - a_{0}) = R^{T} (y - v (s - b) - a_{0}) = R^{T} y - R^{T} v s + R^{T} (v b - a_{0})$ , so the inverse map is $(s, y) \mapsto (s - b, R^{T} y + (- R^{T} v) s + R^{T} (v b - a_{0}))$ , the displayed transformation, and substituting it into the forward transformation returns $(s, y)$ .

2.1step 1.1L3algebra

By [L3], $t (γ (s)) = s$ for all $s \in I$ ; hence $t^{'} (γ (s)) = s + b$ by step 1.1, and substituting $p = γ (s)$ into the second display of step 1.1 gives $x^{'} (γ (s)) = R x (s) + v s + a_{0}$ for all $s \in I$ .

2.2step 1.2step 1.3L2algebra

In the composites of steps 1.2 and 1.3 the rotation parameters are $R_{2} R_{1}$ and $R^{T}$ , both in $S O (3)$ by [L2], and the other parameters lie in $R \times R^{3} \times R^{3}$ ; the identity transformation $G (0, 0, 0, I_{3})$ is present, so the set of Galilean transformations is closed under composition and inversion and the frame-change relation is reflexive, symmetric and transitive; this proves clause 4.

3.1step 2.1L4L5algebra

Define $η (t^{'}) : = x^{'} (γ (t^{'} - b))$ on $I + b$ as in [L4]. By step 2.1, $η (s + b) = x^{'} (γ (s)) = R x (s) + v s + a_{0}$ , so substituting $s = t^{'} - b$ gives $η (t^{'}) = R x (t^{'} - b) + v (t^{'} - b) + a_{0}$ there; a composition of the $C^{2}$ curve $x$ with the affine map $t^{'} \mapsto t^{'} - b$ is $C^{2}$ , so $η$ is of class $C^{2}$ , and differentiating the display with the chain rule [L5], whose inner derivative is $1$ , gives $η^{'} (t^{'}) = R x^{'} (t^{'} - b) + v = R w (t^{'} - b) + v$ and $η^{''} (t^{'}) = R x^{''} (t^{'} - b) = R a (t^{'} - b)$ ; this proves clause 2.

4.1step 3.1L2algebra∎

If $a \equiv 0$ on $I$ , then $η^{''} (t^{'}) = R a (t^{'} - b) = 0$ for every $t^{'} \in I + b$ by step 3.1; conversely, if $η^{''} \equiv 0$ on $I + b$ , then for every $s \in I$ step 3.1 gives $R a (s) = η^{''} (s + b) = 0$ , and multiplying by $R^{T}$ on the left and using $R^{T} R = I_{3}$ from [L2] gives $a (s) = R^{T} η^{''} (s + b) = 0$ ; this proves clause 3.

Galilean transformation of worldlines: velocity and acceleration laws, frame-independence of zero coordinate acceleration, and the group of frame changes

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources