Kinematics

Contents

Foundations
Galilean relativity
Circular motion

Kinematics is the description of motion before any inquiry into its causes: it fixes the arena — space, time, frames of reference — and the quantities — position, velocity, acceleration — with which every later dynamical statement of this treatise is phrased. The chapter closes with the itinerary equation for constant acceleration, the first law of motion in this book to be confronted directly with experiment, in Experiment: Free Fall and Projectile Motion.

Foundations

Fundamental concepts

Motion

Motion is the subject of this part, and it is worth fixing at the outset what kind of statement a statement about motion is. When a body is said to move, what is asserted is never a property of the body by itself: it is a relation between that body and a second one, chosen — freely, and for convenience alone — to count as fixed. A passenger asleep in an aircraft seat is at rest with respect to the cabin, moving at some $250\,\mathrm{m}/\mathrm{s}$ with respect to the ground, and moving at about $3\times 10^{4}\,\mathrm{m}/\mathrm{s}$ with respect to the Sun; no experiment settles which of the three is the truth, because all three are. The second body, together with the rulers and clocks tied to it, is the frame of reference of Definition 18.7, and every kinematic quantity in this chapter carries an unwritten “with respect to $K$”.

Definition 18.1 (Motion and rest).

A particle is in motion with respect to a given frame of reference when the point of space it occupies is not the same at every instant, and is at rest in that frame otherwise. Quantitatively, motion is the statement that the itinerary map Equation (18.1) is not constant. Rests on Definitions 18.7 and 18.9.

Kinematics answers the question how a particle moves and declines the question why: it takes the trajectory as given and extracts from it the quantities — velocity, acceleration — in which the dynamics of Newtonian Dynamics is afterwards phrased. The division is not merely pedagogical. A kinematic statement is a statement about a curve, and is settled by measurements of positions and times alone; a dynamical statement asserts in addition what produces the curve, and needs a law.

The mathematics that the description requires is already in place and is not rebuilt here. A trajectory is a differentiable curve in $\R^{3}$ in the sense of Definition 13.8; the instantaneous quantities are its derivatives with respect to the parameter, in the sense of Definition 7.26; and the tangent, principal normal and binormal of Definitions 13.12, 13.14 and 13.15 supply the natural frame in which an acceleration is read, as Section 18.3.4 shows.

Remark 18.2 (What is assumed here, and what is measured).

That motion is relational is an assumption of this chapter, not a theorem of it. Newton took the opposite view, holding that there exists an absolute space with respect to which a body is really at rest [Newton:1687]; Mach objected that no experiment of Newton's own mechanics can identify that space — Postulate 18.16 makes every inertial frame equivalent — and that an unobservable should be struck from the theory [Mach:1883]. This treatise follows Mach on the evidence, with one qualification the reader should carry forward: rotation is not relational in the same way. A rotating frame is detectable from inside it, by the tilt of a plumb line, by the precession of a pendulum [Foucault:1851], or by the surface of water in a spinning bucket, and Phenomenon 29.53 records the measurements. What uniform translation lacks, and rotation possesses, is an acceleration. Rests on Postulate 18.16 and Phenomenon 29.53.

Space and time

To study nature and the motions that occur in it, we may take the universe and regard it as a three-dimensional Euclidean manifold; that is, we shall consider the universe to be $\R^3$, to which we shall refer simply as space.

We may further consider a parameter with respect to which the state of a physical system evolves. We shall refer to this parameter simply as time: a real parameter that indicates the state of a system at any instant.

To determine the location of objects of the macrocosm, such as stars and galaxies, and of the microcosm, such as atoms and electrons, we shall employ physical and mathematical resources to be defined in the course of this treatise, always within the context of space and time.

Particle

Definition 18.3 (Particle).

A particle is a point of space that can be isolated for its study; that is, a particle may be studied in $\R^3$ as if it were alone in the universe.

For example, to study the motion of a ball we may consider that the ball, together with the Earth, is alone in the universe, since the interactions due to a galaxy located millions of light-years away can be entirely negligible.

[figure: particle.pdf]

A particle: a point of space isolated for its study.

Body

Definition 18.4 (Body).

A body is a particle with a definite volume in space. Rests on Definition 18.3.

Remark 18.5.

This provisional notion is refined in Newtonian Dynamics, where a body is described as a system of many particles occupying a region of space. Rests on Definition 18.4.

Frames of reference

Free particle

Definition 18.6 (Free particle).

A free particle is a particle free of any interaction of nature. Rests on Definition 18.3.

Frame of reference

Definition 18.7 (Frame of reference).

A frame of reference is the choice of a particle with respect to which the motion of another is observed. Rests on Definition 18.3.

In order to study distinct frames of reference we shall denote them by $K$, $K'$, $K''$, and so on. The choice of a frame of reference allows us to establish the location of the origin of $\R^3$ so as to describe a system mathematically — that is, to introduce distances and angles, which the Euclidean metric permits us to do.

[figure: sisref.pdf]

A frame of reference: a particle chosen as origin, with respect to which the motion of another particle is observed.

Inertial frame of reference

Definition 18.8 (Inertial frame of reference).

An inertial frame of reference is a frame of reference located on a free particle. Rests on Definitions 18.6 and 18.7.

Kinematics of a particle

On the time parameter

We may say that a motion began at the instant $t_0$ and that, at an arbitrary instant $t\in\R$, the total time of the motion is $\Delta t=t-t_0$.

Just as in space, we may isolate a portion of time in order to study the motion of a particle.

Position

Definition 18.9 (Position).

Consider a particle at the point $(x,y,z)\in\R^3$. The position of that particle is the vector (as a bilocal object) joining $(x,y,z)$ and the origin, directed towards the particle, which we denote by $\vect{x}$. Rests on Definitions 18.3 and 18.7.

[figure: position.pdf]

The position vector of a particle, joining the origin of the frame of reference to the point occupied by the particle.

We may choose time as a parameter with respect to which the position of a particle varies, that is,

\begin{equation}\tag{18.1} \vect{x}=\vect{x}(t)\ep \end{equation}

This equation is called the itinerary equation of the particle; it describes a curve in $\R^3$, which we shall call the trajectory. In the figure we can see the trajectory of a particle during the time interval $[t_a,t_b]$.

[figure: trayec.pdf]

The trajectory of a particle: the curve in $\R^3$ described by the itinerary equation $\vect{x}=\vect{x}(t)$ over the interval $[t_a,t_b]$.

Velocity

Consider a particle of position $\vect{x}(t)$ with respect to a given frame of reference. How can we evaluate a change of position between two distinct times $t_a$ and $t_b$? Define $\Delta\vect{x}=\vect{x}(t_b)-\vect{x}(t_a)$ and $\Delta t=t_b-t_a$. Then the quantity

\begin{equation}\tag{18.2} \frac{\Delta\vect{x}}{\Delta t} =\frac{\vect{x}(t_b)-\vect{x}(t_a)}{t_b-t_a} \end{equation}

is a density, called the mean velocity, which indicates how much the position has changed in a given interval of time.

So far we have compared two times. But, given the position of a particle at every time, what quantity can we introduce that indicates the change of position at each instant? To this end, let us recall the definitions of derivative and of vector. When defining a vector on a manifold we concluded that it could not be a bilocal object, and for that purpose we resorted to the definition of the derivative. Something similar happens here: we wish to define a quantity that indicates the change of position at a single point of the time interval — we do not want to compare two points.

We may thus anticipate that the quantity we seek is a vector tangent to a point of the trajectory (conceived as a curve in $\R^3$). The trick that removes the bilocality while retaining the information corresponding to a single point is to compare two points with the techniques already known, but infinitesimally separated — that is, the derivative.

Definition 18.10 (Velocity).

The velocity of the particle with respect to the given frame of reference is

\begin{equation}\tag{18.3} \vect{v}=\dv{\vect{x}}{t}\ec \end{equation}

which is indeed a vector in $\R^3$ tangent to the curve $\vect{x}$, a known result of the theory of curves (Definition 13.9). Rests on Definitions 7.26, 13.9 and 18.9.

[figure: velocity.pdf]

The velocity vector: tangent to the trajectory at each of its points.

Acceleration

By the same reasoning as above, we may define a quantity that measures the change of the velocity of a particle in time.

Definition 18.11 (Acceleration).

The acceleration of the particle with respect to the given frame of reference is

\begin{equation}\tag{18.4} \vect{a}=\dv{\vect{v}}{t}\ep \end{equation}

Rests on Definitions 7.26 and 18.10.

The itinerary equation for constant acceleration

Consider a particle with constant acceleration $\vect{a}_0$. One has

\begin{equation}\tag{18.5} \frac{\dd^2\vect{x}}{\dd t^2}=\vect{a}_0\ec \end{equation}

which is a differential equation from which the itinerary equation is obtained. Considering the initial conditions

\begin{align*} \vect{x}(t_0)&=\vect{x}_0\ec & \vect{v}(t_0)&=\vect{v}_0\ec \end{align*}

and integrating once, we have

\[ \dv{\vect{x}}{t}=\vect{v}_0+\vect{a}_0(t-t_0)\ec \]

that is, we obtain

\begin{equation}\tag{18.6} \boxed{\vect{v}-\vect{v}_0=\vect{a}_0(t-t_0)}\ep \end{equation}

Integrating once more, we obtain

\begin{equation}\tag{18.7} \boxed{\vect{x}(t)=\vect{x}_0+\vect{v}_0(t-t_0) +\frac{1}{2}\vect{a}_0(t-t_0)^2}\ec \end{equation}

which is known as the itinerary equation of motion with constant acceleration.

Phenomenon 18.12 (Uniformly accelerated motion).

A body released near the surface of the Earth, with the resistance of the air removed, gains speed in proportion to the elapsed time and falls through a distance proportional to its square; and one and the same constant governs bodies of every mass and composition [Galilei:1638]. The regularity is not a rough one. In a modern ballistic gravimeter the fitted drop reproduces Equation (18.7) at the nanometre level over some $20\,\mathrm{cm}$ of fall, and the constant is thereby fixed to about two parts in $10^{9}$ [Niebauer:1995]; the experiment is examined in Experiment: Free Fall and Projectile Motion. Rests on Equation (18.7) and Definition 18.11.

Derivation. Derives Phenomenon 18.12. The kinematical content is the pair of integrations just performed: Equation (18.5) with the stated initial conditions yields Equation (18.6), linear in $t-t_0$, and Equation (18.7), quadratic in it. Nothing else is possible, since prescribing a second derivative fixes a function up to a polynomial of first degree, and the two initial conditions fix that polynomial. What kinematics cannot supply is the observed universality of $\vect{a}_0$: that every body released at the same place receives the same $\vect{a}_0$ is an experimental input at this stage, and it becomes a statement about mass only in Newtonian Dynamics, where the inertial mass of Newton's second law cancels against the gravitational mass in the weight [Newton:1687].

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Galilean relativity

Newton's postulates of absolute space and time

Newton's Scholium to the definitions asserts two absolutes, and this treatise treats them very differently. Absolute space, in Newton's words, remains always similar and immovable without relation to anything external, so that a body has a true motion in it as well as the relative motions we measure [Newton:1687].

Remark 18.13 (Absolute space is stated here, and not adopted).

No postulate of absolute space is made in this book. The reason is recorded as Phenomenon 18.17, which is Newton's own fifth corollary: within his mechanics, nothing distinguishes rest in absolute space from uniform motion through it, so the quantity is unmeasurable and the theory is complete without it. What replaces it is Postulate 18.16, which asserts the equivalence of the inertial frames and nothing more. Remark 18.2 states the position and its one qualification — that rotation, unlike uniform translation, is locally detectable, which is what makes Newton's bucket a real argument and not a rhetorical one. Rests on Postulate 18.16 and Phenomenon 18.17.

The postulate on time is a different matter: it is used, constantly and essentially, and it is stated as a postulate of this part.

Postulate 18.14 (Absolute time).

Time is an absolute parameter for the whole universe, with respect to any frame of reference.

That is, time elapses equally for every physical system that we can describe in nature [Newton:1687].

[figure: reloju.pdf]

Absolute time: a single universal clock, ticking equally for every frame of reference.

Phenomenon 18.15 (A common time, and the speed at which it fails).

Clocks of every construction — pendulum, quartz, atomic — that are at rest with respect to one another are observed to keep a common rate, so that a single parameter $t$ serves them all. That, and not more, is the measured content of Postulate 18.14. Clocks in relative motion do not keep a common rate: a moving clock runs slow by the factor $\gamma=\left(1-v^2/c^2\right)^{-1/2}$, as seen in the decay of fast muons [Rossi:1941] [Frisch:1963], in the transverse Doppler shift of a canal-ray beam [Ives:1938], and in caesium clocks flown around the world [Hafele:1972b]. Postulate 18.14 is therefore an approximation with a measurable domain of validity, not an identity. Rests on Postulate 18.14.

Derivation. Derives Phenomenon 18.15. The postulate itself is not derived — it is postulated — but the domain over which it is safe can be quantified. Expanding the factor quoted above for $v\ll c$,

\begin{equation}\tag{18.8} \gamma-1=\frac{v^2}{2c^2} +O\!\left(\frac{v^4}{c^4}\right)\ec \end{equation}

so the fractional disagreement between two clocks is second order in their relative speed. At a laboratory speed of $10\,\mathrm{m}/\mathrm{s}$ it is about $6\times10^{-16}$; at the $300\,\mathrm{m}/\mathrm{s}$ of an airliner it is about $5\times10^{-13}$, which over a two-day circumnavigation accumulates to some tens of nanoseconds — the order of the effect reported in [Hafele:1972b]. The factor $\gamma$ is derived, rather than quoted, in Lorentz Transformations, and the experiments that exhibit it are collected in Experiment: Time Dilation and Relativistic Kinematics.

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Galileo's principle of relativity

We enunciate as a fundamental principle:

Postulate 18.16 (Principle of Galilean relativity).

All inertial frames of reference are equivalent.

This is known as Galileo's principle of relativity. A consequence is the following: a physical law takes the same form with respect to any inertial frame of reference — or, put differently, the laws of physics do not change according to the observer — a notion known as Galilean invariance.

Phenomenon 18.17 (Uniform motion is undetectable from within).

No mechanical experiment performed entirely inside a closed cabin discloses whether the cabin is at rest or moving uniformly in a straight line: bodies dropped inside it fall along the same vertical, thrown bodies describe the same arcs, and pendulums keep the same period. Newton records the fact as the fifth corollary to the laws of motion [Newton:1687]. Whether the same holds of optical experiments is a separate question, and an experimental one; the null result of aether-drift interferometry [Michelson:1887] answers it in the affirmative, and it is that extension — not the mechanical statement above — which forces the kinematics of Lorentz Transformations. Rests on Postulate 18.16 and Definition 18.8.

Derivation. Derives Phenomenon 18.17. Let $K'$ translate with constant velocity $\vect{V}$ with respect to $K$, so that the positions the two frames assign to one and the same particle differ by

\begin{equation}\tag{18.9} \vect{x}\,'(t)=\vect{x}(t)-\vect{V}t-\vect{x}_0\ep \end{equation}

Differentiating once gives $\vect{v}\,'=\vect{v}-\vect{V}$, and differentiating a second time, $\vect{V}$ being constant,

\begin{equation}\tag{18.10} \vect{a}\,'=\vect{a}\ep \end{equation}

Separations between particles are likewise unchanged, since the terms $\vect{V}t$ and $\vect{x}_0$ cancel in a difference $\vect{x}_1-\vect{x}_2$. Every quantity that enters Equation (18.5) — accelerations and relative positions — is therefore identical in the two frames, so an experiment confined to the cabin returns the same numbers whichever frame it is described in. The argument used $\dv{\vect{V}}{t}=\vect{0}$, and it fails at once for a cabin that rotates or accelerates; such motion is detectable from inside, as Rigid Bodies and Rotating Frames records.

∎

Galilean transformations

Postulate 18.16 asserts that all inertial frames are equivalent; it does not say how the coordinates assigned by two of them are related. That relation is fixed by three inputs, each a physical assumption and none of them a mathematical identity: absolute time (Postulate 18.14); the homogeneity of space and of time, that is, that no place and no instant is distinguished; and the isotropy of space, that no direction is distinguished.

Definition 18.18 (Galilean transformation).

A Galilean transformation is a map of the coordinates $(\vect{x},t)$ assigned by an inertial frame $K$ to the coordinates $(\vect{x}\,',t')$ assigned by an inertial frame $K'$ of the form

\begin{equation}\tag{18.11} \vect{x}\,'=R\vect{x}-\vect{V}t+\vect{a}\ec\qquad t'=t+b\ec \end{equation}

with $R\in\SO(3)$ a rotation, $\vect{V}\in\R^{3}$ the velocity of $K'$ with respect to $K$, $\vect{a}\in\R^{3}$ a displacement of the spatial origin and $b\in\R$ a shift of the origin of time. The parameters number $3+3+3+1=10$. The pure boost Equation (18.9) is the case $R=\identity$, $b=0$, $\vect{a}=-\vect{x}_{0}$. Rests on Postulate 18.14, Postulate 18.16 and Definition 18.8.

Remark 18.19 (Why the map is affine, and what is assumed to make it so).

The linear-plus-constant form of Equation (18.11) is assumed here, not derived, and the assumption should be visible rather than buried. What would derive it is a purely mathematical statement — that a bijection of $\R^{3}\times\R$ carrying every uniform rectilinear motion to a uniform rectilinear motion is necessarily affine, the fundamental theorem of affine geometry — and Part II — Mathematical Methods does not at present carry that theorem, so this treatise records the affine form where it is used instead of appealing to a result it has not proved. Physically the assumption is exactly the homogeneity of space and time: were the map not affine, its derivative would differ from place to place or from instant to instant, which would single out a place or an instant. The same assumption, made in the same place and for the same reason, is what licenses the linear ansatz Equation (38.5) from which the Lorentz transformation is obtained; the two derivations differ in their second input, absolute time here and the invariance of $c$ there, and in nothing else. Rests on Definition 18.18.

Proposition 18.20 (The Galilean group).

The transformations Equation (18.11) form a group of ten parameters under composition, with composition law

\begin{equation}\tag{18.12} \left(R_{2},\vect{V}_{2},\vect{a}_{2},b_{2}\right)\ast \left(R_{1},\vect{V}_{1},\vect{a}_{1},b_{1}\right) =\left(R_{2}R_{1},\;\vect{V}_{2}+R_{2}\vect{V}_{1},\; \vect{a}_{2}+R_{2}\vect{a}_{1}-\vect{V}_{2}b_{1},\; b_{1}+b_{2}\right)\ec \end{equation}

neutral element $\left(\identity,\vect{0},\vect{0},0\right)$ and inverse

\begin{equation}\tag{18.13} \left(R,\vect{V},\vect{a},b\right)^{-1} =\left(R^{-1},\;-R^{-1}\vect{V},\; -R^{-1}\left(\vect{a}+\vect{V}b\right),\;-b\right)\ep \end{equation}

It is the iterated semidirect product

\begin{equation}\tag{18.14} \mathrm{Gal}(3) =\R^{4}\rtimes\left(\R^{3}\rtimes\SO(3)\right)\ec \end{equation}

whose normal factor $\R^{4}$ consists of the translations of space and time and whose homogeneous factor $\R^{3}\rtimes\SO(3)$ — boosts and rotations — is isomorphic, as a group, to the Euclidean group $\mathrm{SE}(3)$ of Example 4.74. Rests on Definition 18.18, Definition 4.69 and Example 4.74.

Proof.

Derives Proposition 18.20. Apply the transformation labelled $1$ and then the one labelled $2$. The first gives $\vect{x}_{1}=R_{1}\vect{x}-\vect{V}_{1}t+\vect{a}_{1}$ and $t_{1}=t+b_{1}$; substituting these into the second,

\begin{align} \vect{x}_{2} &=R_{2}\vect{x}_{1}-\vect{V}_{2}t_{1}+\vect{a}_{2}\nn\\ &=R_{2}R_{1}\vect{x} -\left(\vect{V}_{2}+R_{2}\vect{V}_{1}\right)t +\left(\vect{a}_{2}+R_{2}\vect{a}_{1} -\vect{V}_{2}b_{1}\right)\ec \tag{18.15} \end{align}

and $t_{2}=t+b_{1}+b_{2}$. The result is again of the form Equation (18.11), with $R_{2}R_{1}\in\SO(3)$ since $\SO(3)$ is a group, and reading off the four slots gives Equation (18.12). Composition of maps is associative, $\left(\identity,\vect{0},\vect{0},0\right)$ is plainly neutral, and substituting Equation (18.13) into Equation (18.12) returns the neutral element in each slot, which establishes the group axioms.

For the structure, note first that the translations $T=\set{\left(\identity,\vect{0},\vect{a},b\right)}$ close under Equation (18.12) into $\left(\vect{a}_{1} +\vect{a}_{2},b_{1}+b_{2}\right)$, so $T\cong\left(\R^{4},+\right)$ and is abelian. It is normal: conjugating $\left(\identity,\vect{0},\vect{a},b\right)$ by a general $g=\left(R,\vect{V},\vect{c},\beta\right)$ with Equations (18.12) and (18.13) gives

\begin{equation}\tag{18.16} g\left(\identity,\vect{0},\vect{a},b\right)g^{-1} =\left(\identity,\vect{0},\,R\vect{a}-\vect{V}b,\,b\right)\in T\ep \end{equation}

The complementary subgroup $H=\set{\left(R,\vect{V},\vect{0},0\right)}$ closes as $\left(R_{2}R_{1},\vect{V}_{2}+R_{2}\vect{V}_{1}\right)$, meets $T$ only in the identity, and together with $T$ generates the whole group, since $\left(R,\vect{V},\vect{a},b\right) =\left(\identity,\vect{0},\vect{a},b\right) \ast\left(R,\vect{V},\vect{0},0\right)$. By Theorem 4.72 the group is therefore $T\rtimes H=\R^{4}\rtimes H$, which is Equation (18.14) once $H$ is identified. That identification is a comparison of two composition laws: writing the element of $H$ as the pair $\left(\vect{V},R\right)$, its law reads $\left(\vect{V}_{2},R_{2}\right)\ast\left(\vect{V}_{1},R_{1}\right) =\left(\vect{V}_{2}+R_{2}\vect{V}_{1},R_{2}R_{1}\right)$, which is Equation (4.49) verbatim with the boost velocity in the slot the Euclidean translation occupies. Hence $H\cong\mathrm{SE}(3)=\R^{3}\rtimes\SO(3)$.

∎

Proposition 18.21 (Galilean invariants).

Under every transformation Equation (18.11):

the time interval between two events is unchanged, $\Delta t'=\Delta t$;
simultaneity is absolute, that is, $\Delta t=0$ if and only if $\Delta t'=0$;
the distance between two simultaneous events is unchanged, $\abs{\Delta\vect{x}\,'}=\abs{\Delta\vect{x}}$.

Rests on Definition 18.18 and Postulate 18.14.

Proof.

Derives Proposition 18.21. For two events labelled $A$ and $B$, subtract the second equation of Equation (18.11) at $A$ from the same equation at $B$: the constant $b$ cancels and $\Delta t'=\Delta t$, which is (i), and (ii) is the special case $\Delta t=0$. Subtracting the first equation likewise gives $\Delta\vect{x}\,'=R\,\Delta\vect{x}-\vect{V}\Delta t$, so when $\Delta t=0$ we have $\Delta\vect{x}\,'=R\,\Delta\vect{x}$ and, $R$ being orthogonal, $\abs{R\,\Delta\vect{x}}=\abs{\Delta\vect{x}}$, which is (iii). The qualification “simultaneous” in (iii) is not decoration: for $\Delta t\neq0$ the separation $\abs{\Delta\vect{x}\,'}$ depends on $\vect{V}$, as it must, since the two events then happen at places whose relative position is itself changing.

∎

Remark 18.22 (Superseded, and in exactly what sense).

Equation (18.11) is not the transformation law of nature. The measured law is the Poincaré transformation of Minkowski Space and Its Symmetries, whose homogeneous part is the Lorentz group and under which Proposition 18.21 fails in all three of its parts: time intervals, simultaneity and lengths are all frame dependent. The Galilean group is recovered from the Poincaré group as an Inönü–Wigner contraction in the limit $c\to\infty$ (Example 14.89), which is the precise sense in which the two are related: not as rival descriptions, but as a group and its limit. Since $c$ is finite — $c=299792458\,\mathrm{m}/\mathrm{s}$ exactly, by the definition of the metre [BIPM:2019] — the contraction is never exact, and the size of the error is set by $v^{2}/c^{2}$. At the Earth's orbital speed of $2.98\times 10^{4}\,\mathrm{m}/\mathrm{s}$ that is about $10^{-8}$, which is why the whole of Part III — Classical Mechanics may use Equation (18.11) without apology and why the optical experiments of Experiments: Light, the Aether, and Time, sensitive at precisely that order, could not. Rests on Propositions 18.20 and 18.21.

Transformation of velocities and accelerations

Proposition 18.23 (Transformation of velocity and acceleration).

Let $K$ and $K'$ be related by the Galilean transformation Equation (18.11). Then the velocity and the acceleration that the two frames assign to one and the same particle are related by

\begin{equation}\tag{18.17} \vect{v}\,'=R\vect{v}-\vect{V}\ec\qquad \vect{a}\,'=R\vect{a}\ep \end{equation}

The acceleration is therefore unaffected by a boost or by a translation of either origin, and is carried by a rotation exactly as a vector is. So too is the relative velocity of two particles, since $\vect{v}_{1}'-\vect{v}_{2}'=R\left(\vect{v}_{1}-\vect{v}_{2}\right)$ with $\vect{V}$ cancelling in the difference. Rests on Definitions 18.10, 18.11 and 18.18.

Proof.

Derives Proposition 18.23. By the second equation of Equation (18.11), $\dd t'/\dd t=1$, so differentiation with respect to $t'$ and with respect to $t$ are the same operation — this is where Postulate 18.14 enters, and it is the only place it is needed. Differentiating the first equation once, with $R$, $\vect{V}$ and $\vect{a}$ constant,

\[ \vect{v}\,'=\dv{\vect{x}\,'}{t'}=\dv{\vect{x}\,'}{t} =R\dv{\vect{x}}{t}-\vect{V}=R\vect{v}-\vect{V}\ec \]

and differentiating a second time removes the constant $\vect{V}$, leaving $\vect{a}\,'=R\vect{a}$. Both statements are Equation (18.17). The claim about the relative velocity follows by writing Equation (18.17) for each particle and subtracting.

∎

Corollary 18.24 (Galilean composition of velocities).

If $K''$ moves with constant velocity $\vect{V}_{2}$ with respect to $K'$, and $K'$ with $\vect{V}_{1}$ with respect to $K$, all three sets of axes being parallel, then $K''$ moves with

\begin{equation}\tag{18.18} \vect{V}=\vect{V}_{1}+\vect{V}_{2} \end{equation}

with respect to $K$: Galilean velocities compose by vector addition, and boosts commute with one another. Rests on Proposition 18.20.

Proof.

Derives Corollary 18.24. Set $R_{1}=R_{2}=\identity$, $\vect{a}_{i}=\vect{0}$ and $b_{i}=0$ in Equation (18.12); the boost slot returns $\vect{V}_{2}+\identity\,\vect{V}_{1}=\vect{V}_{1}+\vect{V}_{2}$ and every other slot returns its neutral value. The pure boosts are thus a copy of $\left(\R^{3},+\right)$ inside $\mathrm{Gal}(3)$, and addition in $\R^{3}$ is commutative, so any two boosts commute.

∎

Remark 18.25 (Where the additivity goes).

Two features of Corollary 18.24 fail relativistically, and they fail separately. Additivity is replaced by the composition law Equation (38.29), which returns Equation (18.18) only to leading order and never lets the result exceed $c$; and commutativity is lost as well, two non-collinear boosts composing to a boost and a rotation. Neither failure is visible in the mechanics of this part: the fractional discrepancy in Equation (18.18) is of order $V_{1}V_{2}/c^{2}$, which for two speeds of $1\,\mathrm{km}/\mathrm{s}$ is about $10^{-11}$. Rests on Corollary 18.24.

Phenomenon 18.26 (Composition of velocities, and its limit).

Two observers in uniform relative motion assign to one and the same body velocities that differ by their relative velocity and by nothing else, and assign to it the same acceleration. Everyday mechanics — a ball thrown on a moving ship, a walker on a train — shows no departure from the rule. Light does not obey it. The speed of light in vacuum is found to be the same in frames whose relative velocity is the Earth's orbital velocity of about $29.8\,\mathrm{km}/\mathrm{s}$ [Michelson:1887], and light in a moving transparent medium is carried along by only a fraction of the medium's speed, not by all of it [Fizeau:1851]. Rests on Proposition 18.23 and Corollary 18.24.

Derivation. Derives Phenomenon 18.26. The first claim is Proposition 18.23 with $R=\identity$: differentiating Equation (18.9) once and twice gives

\begin{equation}\tag{18.19} \vect{v}\,'=\vect{v}-\vect{V}\ec\qquad \vect{a}\,'=\vect{a}\ec \end{equation}

the second of which is the invariance of the acceleration already recorded as Equation (18.10).

The second claim is not derived here; it is measured, and the two measurements cited are worth separating. Fizeau sent the two arms of an interferometer through water flowing in opposite directions and found the light dragged not by the full speed $u$ of the water, as Equation (18.19) requires, but by a fraction of it close to Fresnel's $1-1/n^{2}$, which for water at $n=1.33$ is $0.43$ [Fizeau:1851]. Michelson and Morley looked for the motion of the Earth through the supposed medium and found no shift at all where Equation (18.19) predicted one of order $v/c\approx10^{-4}$ in the arrival times [Michelson:1887]. Against those two results, the departure of the exact law of Lorentz Transformations from Equation (18.19) can be stated once and for all: it is of relative order $vV/c^{2}$, which for two speeds of $1\,\mathrm{km}/\mathrm{s}$ is about $10^{-11}$ and for light, $v=c$, is of order unity. The failure is thus invisible throughout the mechanics of this part and total in optics; the historical situation this created is the subject of Experiments: Light, the Aether, and Time.

∎

Circular motion

Circular motion is the simplest motion that is not uniform, and it is the one the rest of mechanics leans on most: a planet held in orbit, a pendulum bob on its arc, an electron in a magnetic field and a point of a spinning body all move, instantaneously, on circles. Everything in this section is kinematics and nothing in it is dynamics — what is computed is the acceleration a circular path requires, never the agent that supplies it.

The angular description

Definition 18.27 (Angular position, velocity and acceleration).

Let a particle move on a circle of fixed radius $r$ centred on the origin, in the plane $z=0$, and let $\theta(t)$ be the angle its position makes with a fixed axis of that plane, measured in radians. The angular velocity and angular acceleration are

\begin{equation}\tag{18.20} \omega=\dv{\theta}{t}\ec\qquad \alpha=\dv{\omega}{t}=\frac{\dd^{2}\theta}{\dd t^{2}}\ec \end{equation}

of SI units $\mathrm{rad}/\mathrm{s}$ and $\mathrm{rad}/\mathrm{s}^{2}$. The arc travelled is $s=r\theta$ and the speed is

\begin{equation}\tag{18.21} v=\abs{\dv{s}{t}}=r\abs{\omega}\ep \end{equation}

Rests on Definitions 18.9, 18.10 and 18.11.

Remark 18.28 (The radian carries no dimension).

The radian is the coherent SI unit of plane angle and is equal to the number one, being the ratio of two lengths [BIPM:2019]. It is written in Equation (18.20) because it identifies what is being counted, not because it contributes a dimension: $\omega r$ comes out in $\mathrm{m}/\mathrm{s}$ and $\omega^{2}r$ in $\mathrm{m}/\mathrm{s}^{2}$ with no conversion factor. This is also why Equation (18.21) fails if $\theta$ is measured in any other angular unit. Rests on Definition 18.27.

When the motion is periodic it is convenient to name its period $T$, the time of one complete revolution, and its frequency $f=1/T$ in $\mathrm{Hz}$. For constant $\omega$ these are

\begin{equation}\tag{18.22} T=\frac{2\pi}{\abs{\omega}}\ec\qquad f=\frac{1}{T}=\frac{\abs{\omega}}{2\pi}\ep \end{equation}

Definition 18.29 (Angular velocity vector).

Let $\hat{\vect{n}}$ be the unit normal to the plane of the motion, oriented by the right-hand rule about the sense of increasing $\theta$. The angular velocity vector is $\vect{\omega}=\omega\,\hat{\vect{n}}$, and it reproduces the velocity as

\begin{equation}\tag{18.23} \vect{v}=\vect{\omega}\times\vect{x}\ep \end{equation}

Rests on Definitions 18.10 and 18.27.

Remark 18.30 (The angular velocity is an element of $\mathfrak{so}(3)$).

Writing Equation (18.23) in components with the basis Equation (14.32) of $\mathfrak{so}(3)$,

\begin{equation}\tag{18.24} \left(\vect{\omega}\times\vect{x}\right)_{i} =\epsilon_{ijk}\omega_{j}x_{k} =\omega_{k}\left(L_{k}\right)_{ij}x_{j}\ec \end{equation}

so the cross product with $\vect{\omega}$ is the antisymmetric matrix $\omega_{k}L_{k}$ acting on $\vect{x}$. The angular velocity is thus not a vector by nature but an element of the Lie algebra of the rotation group, which the three-dimensionality of space allows us to label by three numbers; the finite rotation it generates is $\exp\left(t\,\omega_{k}L_{k}\right)$, evaluated in closed form by Proposition 14.30. Nothing in this chapter depends on the distinction, but Section 29.1.1 does, and so does every statement about how $\vect{\omega}$ behaves under a reflection. Rests on Definition 18.29 and Proposition 14.27.

Uniform circular motion

Phenomenon 18.31 (Uniform circular motion is accelerated motion).

A body carried at constant speed $v$ around a circle of radius $r$ is not in uniform motion. It has to be held to the circle by something — a string, a rail, a gravitational attraction — and what is required of that agent is an acceleration of constant magnitude $v^2/r$ directed at every instant towards the centre. Withdraw the agent and the body departs along the tangent, as a stone released from a sling. The rule $v^{2}/r$ is Huygens', stated among the theorems on centrifugal force appended to the Horologium Oscillatorium [Huygens:1673]; the observation is Newton's point of departure for orbital motion [Newton:1687]. Rests on Definitions 18.11 and 18.27.

Derivation. Derives Phenomenon 18.31. Take the circle in the plane $z=0$, with the angular speed $\omega=v/r$ constant, so that the itinerary equation Equation (18.1) reads

\begin{equation}\tag{18.25} \vect{x}(t)=r\left(\cos\omega t,\;\sin\omega t,\;0\right)\ep \end{equation}

Differentiating once, by Equation (18.3),

\[ \vect{v}(t)=r\omega\left(-\sin\omega t,\;\cos\omega t,\;0\right)\ec \]

whose magnitude is the constant $r\omega=v$ and whose scalar product with $\vect{x}$ vanishes — the velocity is tangent to the circle, as it must be. Differentiating again, by Equation (18.4),

\begin{equation}\tag{18.26} \vect{a}(t)=-r\omega^2\left(\cos\omega t,\;\sin\omega t,\;0\right) =-\omega^2\vect{x}(t)\ec \end{equation}

so the acceleration points from the body straight at the centre and has the constant magnitude

\begin{equation}\tag{18.27} \abs{\vect{a}}=\omega^2 r=\frac{v^2}{r}\ep \end{equation}

The result is purely kinematical: it says what acceleration a circular path at constant speed requires, and says nothing whatever about what supplies it. Supplying it is the business of Newtonian Dynamics, and identifying the supplier for the planets is the business of Central Forces and Statics.

∎

Two features of Equation (18.26) deserve to be named, because both are constantly misread. First, a motion at constant speed is nevertheless an accelerated motion, since it is the velocity vector that Definition 18.11 differentiates and that vector is turning. Second, the acceleration Equation (18.26) is directed towards the centre, and there is no outward acceleration anywhere in the calculation: the “centrifugal” term of everyday speech is an artefact of describing the motion in a frame rotating with the body, which is not inertial, and it is derived — with its sign and its companion Coriolis term — in Section 29.5.1.

Non-uniform circular motion

Proposition 18.32 (Tangential and centripetal components).

Let a particle move on the circle of radius $r$ in the plane $z=0$ with an arbitrary angle $\theta(t)$ of class $C^{2}$, and write $\hat{\vect{r}}=\left(\cos\theta,\sin\theta,0\right)$ and $\hat{\vect{t}}=\left(-\sin\theta,\cos\theta,0\right)$ for the outward radial and the tangential unit vectors. Then

\begin{equation}\tag{18.28} \vect{v}=r\omega\,\hat{\vect{t}}\ec\qquad \vect{a}=r\alpha\,\hat{\vect{t}}-r\omega^{2}\,\hat{\vect{r}}\ec \end{equation}

with $\omega$ and $\alpha$ as in Equation (18.20). The tangential component $r\alpha=\dd v/\dd t$ changes the speed; the radial component $-r\omega^{2}\hat{\vect{r}}$, of magnitude $v^{2}/r$, changes only the direction. Rests on Definition 18.27 and Phenomenon 18.31.

Proof.

Derives Proposition 18.32. The itinerary equation is $\vect{x}(t)=r\hat{\vect{r}}(\theta(t))$. Since $\dd\hat{\vect{r}}/\dd\theta=\hat{\vect{t}}$ and $\dd\hat{\vect{t}}/\dd\theta=-\hat{\vect{r}}$, the chain rule (Proposition 7.31) gives $\dd\hat{\vect{r}}/\dd t=\omega\hat{\vect{t}}$ and $\dd\hat{\vect{t}}/\dd t=-\omega\hat{\vect{r}}$. Hence $\vect{v}=r\omega\hat{\vect{t}}$, and differentiating once more,

\[ \vect{a}=\dv{}{t}\left(r\omega\hat{\vect{t}}\right) =r\alpha\,\hat{\vect{t}}+r\omega\dv{\hat{\vect{t}}}{t} =r\alpha\,\hat{\vect{t}}-r\omega^{2}\,\hat{\vect{r}}\ec \]

which is Equation (18.28). Setting $\alpha=0$ recovers Equation (18.26), and $\abs{\vect{a}} =\sqrt{r^{2}\alpha^{2}+r^{2}\omega^{4}}$ reduces to Equation (18.27) in that case.

∎

The intrinsic decomposition of the acceleration

The split of Equation (18.28) into a part along the motion and a part across it is not special to a circle. It holds for every trajectory, with the radius of the circle replaced by the radius of curvature of the curve at the point in question, and it is the reason the curve theory of Section 13.3.1 was set up.

Proposition 18.33 (Intrinsic decomposition of the acceleration).

Let $\vect{x}(t)$ be a trajectory of class $C^{2}$ whose speed $v=\abs{\dd\vect{x}/\dd t}$ is nowhere zero, let $s$ be its arc length Equation (13.35), and let $\hat{T}$, $\hat{N}$, $\hat{B}$ be the Frenet frame of Definitions 13.12, 13.14 and 13.15, with curvature $\kappa>0$. Then

\begin{equation}\tag{18.29} \vect{v}=v\,\hat{T} \end{equation}

and

\begin{equation}\tag{18.30} \vect{a}=\dv{v}{t}\,\hat{T}+\kappa v^{2}\,\hat{N} =\dv{v}{t}\,\hat{T}+\frac{v^{2}}{\rho}\,\hat{N}\ec \end{equation}

where $\rho=1/\kappa$ is the radius of curvature. In particular $\vect{a}\cdot\hat{B}=0$: the acceleration always lies in the osculating plane, its component across the motion is directed towards the centre of curvature, and it never has a component out of that plane. Rests on Definitions 13.12, 13.14 and 18.11.

Proof.

Derives Proposition 18.33. By Equation (13.36) the arc length satisfies $\dd s/\dd t=v$, and by Equation (13.38) the unit tangent is $\hat{T}=\dd\vect{x}/\dd s$. The chain rule (Proposition 7.31) therefore gives Equation (18.29),

\[ \vect{v}=\dv{\vect{x}}{t} =\dv{\vect{x}}{s}\dv{s}{t}=v\,\hat{T}\ep \]

Differentiating that product and using the chain rule once more on $\hat{T}$,

\begin{equation}\tag{18.31} \vect{a}=\dv{}{t}\left(v\hat{T}\right) =\dv{v}{t}\,\hat{T}+v\,\dv{\hat{T}}{s}\dv{s}{t} =\dv{v}{t}\,\hat{T}+v^{2}\,\dv{\hat{T}}{s}\ec \end{equation}

From Equations (13.36) and (13.38) (differentiating $\vect{v}=v\hat{T}$ and converting $\dd\hat{T}/\dd t$ to $\dd\hat{T}/\dd s$ with $\dd s/\dd t=v$). and the first Frenet equation Equation (13.43), $\dd\hat{T}/\dd s=\kappa\hat{N}$, turns Equation (18.31) into Equation (18.30). The last claim follows because $\hat{T},\hat{N},\hat{B}$ are orthonormal by Definition 13.15, so the expansion Equation (18.30) has no $\hat{B}$ component to begin with.

∎

Remark 18.34 (The two ways an acceleration can be zero).

Equation (18.30) makes the classification immediate. The tangential term vanishes when the speed is constant, and the normal term vanishes when the curvature is zero, that is when the path is straight; only when both vanish is the motion unaccelerated, which is uniform motion in a straight line. Uniform circular motion is the second case made non-trivial: $\dd v/\dd t=0$ and $\kappa=1/r$ constant, so Equation (18.30) returns $\abs{\vect{a}}=v^{2}/r$, which is Equation (18.27) again, now derived without ever choosing coordinates. Where $\kappa=0$ the principal normal Equation (13.42) is undefined and the statement degenerates to $\vect{a}=\left(\dd v/\dd t\right)\hat{T}$, which is correct as it stands. Rests on Proposition 18.33.

Magnitudes

Centripetal accelerations in nature and in the laboratory span some ten orders of magnitude, and it is worth having a few of them in mind.

motion	$r$	$\omega$	$\omega^{2}r$
Earth, spin at the equator	$6.378\times 10^{6}\,\mathrm{m}$	$7.292\times 10^{-5}\,\mathrm{rad}/\mathrm{s}$	$3.39\times 10^{-2}\,\mathrm{m}/\mathrm{s}^{2}$
Earth, orbit about the Sun	$1.496\times 10^{11}\,\mathrm{m}$	$1.991\times 10^{-7}\,\mathrm{rad}/\mathrm{s}$	$5.93\times 10^{-3}\,\mathrm{m}/\mathrm{s}^{2}$
Car on a highway curve	$3.0\times 10^{2}\,\mathrm{m}$	$9.3\times 10^{-2}\,\mathrm{rad}/\mathrm{s}$	$2.6\,\mathrm{m}/\mathrm{s}^{2}$
Ultracentrifuge rotor	$5.0\times 10^{-2}\,\mathrm{m}$	$1.05\times 10^{4}\,\mathrm{rad}/\mathrm{s}$	$5.5\times 10^{6}\,\mathrm{m}/\mathrm{s}^{2}$

Centripetal accelerations $\omega^{2}r$ for four circular motions, computed from Equation (18.27). The Earth's spin rate is the sidereal one and $r$ its equatorial radius; the orbital row uses the mean Sun–Earth distance; the car travels at $28\,\mathrm{m}/\mathrm{s}$ on a curve of $300\,\mathrm{m}$ radius; the rotor is a laboratory ultracentrifuge of $5\,\mathrm{cm}$ radius. For comparison, the conventional standard acceleration of free fall is $g_{0}=9.80665\,\mathrm{m}/\mathrm{s}^{2}$ exactly [BIPM:2019].

The first row is not a curiosity: it is measured, and measured easily. A body at rest on the equator is in circular motion, so the free-fall acceleration read by a gravimeter there is smaller than the gravitational attraction by the $3.39\times 10^{-2}\,\mathrm{m}/\mathrm{s}^{2}$ of Table 18.1 — against an instrumental accuracy of about $2\times 10^{-8}\,\mathrm{m}/\mathrm{s}^{2}$ [Niebauer:1995], which resolves the effect a million times over. The observed equator-to-pole difference is larger still, $5.2\times 10^{-2}\,\mathrm{m}/\mathrm{s}^{2}$, the remainder coming from the equatorial bulge that the same rotation produces; Phenomenon 29.55 treats both together, and Equation (29.68) is its quantitative form.

The itinerary equation for constant acceleration, Equation (18.7), is the first quantitative law of this treatise, and it is not left as a formal statement: in Experiment: Free Fall and Projectile Motion it is confronted with experiment — from Galileo's inclined planes to the modern ballistic gravimeters — with every assumption behind it declared and tested.

motion	$r$	$\omega$	$\omega^{2}r$
Earth, spin at the equator	\(6.378\times 10^{6}\,\mathrm{m}\)	\(7.292\times 10^{-5}\,\mathrm{rad}/\mathrm{s}\)	\(3.39\times 10^{-2}\,\mathrm{m}/\mathrm{s}^{2}\)
Earth, orbit about the Sun	\(1.496\times 10^{11}\,\mathrm{m}\)	\(1.991\times 10^{-7}\,\mathrm{rad}/\mathrm{s}\)	\(5.93\times 10^{-3}\,\mathrm{m}/\mathrm{s}^{2}\)
Car on a highway curve	\(3.0\times 10^{2}\,\mathrm{m}\)	\(9.3\times 10^{-2}\,\mathrm{rad}/\mathrm{s}\)	\(2.6\,\mathrm{m}/\mathrm{s}^{2}\)
Ultracentrifuge rotor	\(5.0\times 10^{-2}\,\mathrm{m}\)	\(1.05\times 10^{4}\,\mathrm{rad}/\mathrm{s}\)	\(5.5\times 10^{6}\,\mathrm{m}/\mathrm{s}^{2}\)