Rigid body mechanics in Galilean spacetimesandrew/papers/pdf/2002f.pdf · Rigid body mechanics in Galilean spacetimes 3 in a Galilean spacetime are thought of as observer dependent

Rigid body mechanics in Galilean spacetimes∗

Ajit Bhand†

2002/09/10

“There is no subject so old that something new cannot be said about it.”

–Fyodor Mikhailovich Dostoyevsky.

Abstract

An observer–independent formulation of rigid body dynamics is provided in the gen-eral setting of an abstract spacetime. In particular, definitions of (body and spatial)angular and linear velocities and momenta are provided independent of an observer.Rigid motions are defined and their properties are studied in detail. A rigid body isdefined in this general setting and its various properties are explored. The equationsgoverning the motion of a rigid body undergoing a rigid motion in a Galilean spacetimeare derived using the conservation of spatial linear and spatial angular momenta. Thecanonical Galilean group is shown to decompose into rotations, space translations, veloc-ity boosts and time translations and its properties are discussed. The abstract Galileangroup is studied in detail and is found to be isomorphic to the canonical Galilean groupin the presence of an observer. Various subgroups of the abstract Galilean group arecharacterized. Total velocities corresponding to a rigid motion are defined. The for-mulation of rigid body dynamics is then studied in the presence of an observer. It isshown that in this case, the velocities and momenta defined previously project to thewell known quantities. Finally, it is seen that the equations of motion derived earlierdescribe the usual motion given by the solutions of the Euler equations for a rigid bodyundergoing rigid motion.

Contents

1. Introduction 2

2. Preliminaries 32.1 Affine spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Time and distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Observers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

World lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Galilean mappings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

∗Report for project in fulfilment of requirements for MSc†Assistant Professor, Mathematics Department, Indian Institute of Science Education and Re-search, Bhopal, IndiaEmail: [email protected], URL: http://home.iiserbhopal.ac.in/~abhand/Work performed while a graduate student at Queen’s University.Research supported in part by a grant from the Natural Sciences and Engineering Research Council ofCanada.

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2 A. Bhand

3. Observer-independent formulation of rigid body dynamics 103.1 Rigid motions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 Body and spatial velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Angular velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Linear velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Rigid bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13The inertia tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Eigenvalues of the inertia tensor. . . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Dynamics of rigid bodies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Spatial momenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Body momenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Galilean–Euler equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4. The structure of the Galilean group 214.1 The canonical Galilean group. . . . . . . . . . . . . . . . . . . . . . . . . . . 21

The structure of the canonical Galilean group. . . . . . . . . . . . . . . . . 21Subgroups of the canonical Galilean group. . . . . . . . . . . . . . . . . . . 23Properties of the canonical Galilean group. . . . . . . . . . . . . . . . . . . 24

4.2 The abstract Galilean group. . . . . . . . . . . . . . . . . . . . . . . . . . . 26The structure of the Galilean group. . . . . . . . . . . . . . . . . . . . . . . 26Subgroups of the abstract Galilean group. . . . . . . . . . . . . . . . . . . . 30

4.3 Total velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5. Dynamics of rigid bodies in the presence of an observer 355.1 Angular and linear velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Angular velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Linear velocities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.2 Spatial and body momenta. . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3 Euler equations for a rigid body. . . . . . . . . . . . . . . . . . . . . . . . . 38

Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

6. Conclusions and future scope 396.1 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396.2 Future scope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1. Introduction

The main aim of this project is to understand the dynamics of a rigid body in the generalframework of an abstract Galilean spacetime. Underlying the spacetime is an affine spacewhich is thought of as a “translation” of a vector space. We wish to study how the motionof a rigid body is related to rigid motions– the set of mappings belonging to the groupof Galilean transformations from an affine space to itself, called the Galilean group. Ourobjective is also to formulate rigid body dynamics in an observer–independent manner. Wenote that the angular and linear momenta associated with a rigid body undergoing motion

Rigid body mechanics in Galilean spacetimes 3

in a Galilean spacetime are thought of as observer dependent quantities in the literature(c.f. [Artz 1981]). In this project we take the view that momentum is an inherent propertyof a rigid body in motion and that it is possible to define it without using any externalstructure. The observer merely affects the way the momentum is measured.

The role of Galilean structure of the Newtonian spacetime has been understood in thecase of the dynamics of a particle [Arnol′d 1978, Artz 1981] and of course, the dynamicsof a rigid body in a fixed Galilean frame has been investigated quite throughly [Arnol′d1978]. However, to our knowledge, the role played by the Galilean structure has not beenexplained for rigid body mechanics. In particular, it is not entirely clear what quantities inthe description of the dynamics of a rigid body are dependent on the choice of a Galileanobserver, and in what manner the dependence arises.

An observer in a Galilean spacetime, apart from providing a reference frame for ob-serving Newton’s laws, also plays its role by providing an isomorphism from the abstractGalilean group to the “standard” or canonical Galilean group which consists of rotations,spatial translations, uniform velocity boosts and time translations. This isomorphism is thekey to understanding the structure of the Galilean group.

Our treatment is more general also because we allow arbitrary velocity boosts in ouranalysis. This facilitates a better understanding of the “generalized” Euler equations fora rigid body. In the presence of an observer, the solutions to these equations are found toagree with the motion of a rigid body in the classical setup.

This report is organized as follows. In Chapter 2, we present the mathematical pre-liminaries relevant to our investigation. Several important terms like affine spaces andsubspaces, observers, Galilean spacetimes and the Galilean group are introduced and theirvarious properties are described. With this background, we first look at rigid motions inChapter 3. Various quantities associated with a rigid motion, such as the body and spa-tial linear and angular velocities are defined. Throughout this chapter, the treatment isobserver–independent. The notion of a rigid body along with its attendant features is in-troduced. In particular, the inertia tensor of a rigid body is defined and and its propertiesare thoroughly explained. The discussion then focuses on angular and spatial momenta andfinally the equations of motion for a rigid body are derived. In Chapter 4, the structure ofthe canonical as well as the abstract Galilean group is investigated in detail. It is shown thatan observer induces an isomorphism between the Galilean group to the canonical group.Various subgroups of the abstract Galilean group are characterized Next, “total velocities”are defined and the Lie algebra of the canonical Galilean group is also described. In Chapter5, the formulation presented in chapter 3 is studied in the presence of an observer. It isshown that in such a case, we recover the familiar quantities associated with the classicaltreatment of rigid body mechanics. The Euler equations are also studied and, as mentionedearlier, are found to have solutions consistent with the motion of a rigid body in the classicalframework. Finally, in Chapter 6, we summarize our results and discuss some directions forfurther investigations.

2. Preliminaries

In this chapter, we present the mathematical background and introduce the notationto be used in the following chapters. In section 2.1, we define affine spaces and subspaces–the principle objects of interest in this project. In Section 2.2, we introduce the idea of

4 A. Bhand

a Galilean spacetime and describe the affine spaces naturally associated with it. We alsointroduce the set of Galilean velocities. Next, we define observers in a Galilean spacetimeand discuss their properties. Finally, in Section 2.4, we define the Galilean group andintroduce the fundamental maps associated with a Galilean mapping.

2.1. Affine spaces. In this section, we define affine spaces and subspaces and record someof their properties. We refer to [Berger 1987] for more details. We start by defining groupactions.

2.1 Notation: Given a set X, we represent by inv(X), the set of invertible maps from Xto itself. Given vector spaces V and U , we denote by L(V ;U), the set of linear maps fromV to U .

2.2 Definition: Let G be a group and X a set. A G–action on X is a homomorphismΘ : G→ inv(X).

We shall identify the map Θ with the associated map from G×X → X.Given x ∈ X, the orbit of x under the group action is given by

orbG(x) = Θ(g, x) | g ∈ G.

The isotropy group of x is given by

isoG(x) = g ∈ G | Θ(g, x) = x

2.3 Definition: Let Θ be a G–action on a set X.

1. Θ is transitive if there is only one orbit.

2. Θ is faithful if Θ(g, x) = x for every x ∈ X implies g = e

3. Θ is free if Θ(g, x) = x for any x ∈ X implies g = e. In this case, isoG(x) = e

4. Θ is simply–transitive if it is free and transitive.

Next we introduce the concept of an affine space. It may be thought of as a vector spacewithout an origin. Thus, it makes sense only to consider the “difference” of two elementsof an affine space as being a vector. The elements themselves are not to be regarded asvectors.

2.4 Definition: Let A be a set and V a real vector space. An affine space modeled on V isa triple (A, V,Θ) where Θ is a faithful, transitive V –action on A.

We shall now cease to use the map Θ and instead use the more suggestive notation Θ(v, x) =v + x. By definition, if x, y ∈ A, there exists v ∈ V such that y = x+ v.

2.5 Notation: In this case we shall denote v = y − x.


2.6 Remark: The minus sign here is simply notation; we have not really defined “subtrac-tion” in A. The idea is that to any two points in A, we may assign a unique vector in Vand we notationally write this as the difference between the two elements.

All this leads to the following result which is easily proved once all the symbols are under-stood. By a slight abuse of notation, we shall denote an affine space (A, V,Θ) simply byA.

2.7 Proposition: Let A be a R- affine space modeled on V . For fixed x ∈ A define vectoraddition on A by

y1 + y2 = ((y1 − x) + (y2 − x)) + x, ∀y1, y2 ∈ A

and scalar multiplication on A by

ay = (a(y − x)) + x, ∀y ∈ A.

These operations make A a R–vector space and y 7→ y − x is an isomorphism of thisR–vector space with V .

2.8 Notation: We shall denote the vector space introduced in the above result by Ax.

Let A and B be affine spaces modeled on vector spaces V and U respectively and letf : A → B be a map. Given x0 ∈ A, let Ax0 and Bf(x0) be the corresponding vectorspaces defined in Proposition 2.7. Ax0 is isomorphic to V with the isomorphism x 7→ x−x0

and Bf(x0) is isomorphic to U with the isomorphisms y 7→ y − f(x0). We denote theseisomorphism by Φx0 : Ax0 → V and Φf(x0) : Bf(x0) → U , respectively. The next propositionprovides a definition of an affine map between two affine spaces.

2.9 Proposition: The following conditions are equivalent:

(i) f ∈ L(Ax0 ;Bf(x0)) for some x0 ∈ A;

(ii) f ∈ L(Ax0 ;Bf(x0)) for all x0 ∈ A;

(iii) Φf(x0) f Φ−1x0∈ L(V ;U) for some x0 ∈ A;

(iv) Φf(x0) f Φ−1x0∈ L(V ;U) for all x0 ∈ A

Moreover, if these conditions are satisfied, Φf(x0)f Φ−1x0∈ L(V ;U) depends only on f , and

will be denoted by fV . A map satisfying the properties above is called an (affine) morphism,or an affine map, between V and U . We say that f is an (affine) isomorphism if f is amorphism and is bijective; f is an automorphism if f is an isomorphism from A to itself.

Next, we define affine subspaces.

2.10 Definition: Given an affine space (A, V,Θ) and U a subspace of V , a subset B ⊂ A

is called an affine subspace if the triple (B, U,Θ|U ) is an affine space modeled on U.

The following result characterizes affine subspaces.

2.11 Proposition: Let A be a R–affine space modeled on V and let B ⊂ A. The followingare equivalent:

(i) B is an affine subspace of A;

(ii) there exists a subspace U of V so that for some fixed x ∈B,

B = u+ x| u ∈ U

(iii) if x ∈B then y − x| y ∈B ⊂ V is a subspace.

6 A. Bhand

2.2. Time and distance. We begin by giving the basic definition of a Galilean spacetimeand by providing meaning to the intuitive notions of time and distance.

2.12 Definition: A Galilean spacetime is a quadruple G = (E, V, g, τ) where

GSp1. V is a 4-dimensional R–vector space.

GSp2. τ : V → R is a surjective linear map called the time map.

GSp3. g is an inner product on ker(τ), and

GSp4. E is an affine space modeled on V .

Points in E are called events–thus E is a model for the spatio-temporal world of Newtonianmechanics. With the time map, we may measure the time between two events x1, x2 ∈ E

as τ(x2 − x1). Note, however, that it does not make sense to talk about the “time” of aparticular event x ∈ E. Events x1, x2 ∈ E are called simultaneous if τ(x2 − x1) = 0; thatis, if x2 − x1 ∈ ker(τ)).

Using the definition, we may define the distance between simultaneous events x1, x2 ∈Eto be

√g(x2 − x1, x2 − x1). Note that this method for defining distance does not allow us

to measure distances between events that are not simultaneous. In particular, it doesn’tmake sense to talk about two non-simultaneous events as occurring in the same place.

Simultaneity is an equivalence relation on E and the quotient we denote by IG = E/ ∼,with ∼ denoting the relation of simultaneity. IG is simply the collection of equivalenceclasses of simultaneous events. We call it the set of instants. We denote, by πG : E → IGthe canonical projection.

For s ∈ IG, we denote by E(s) the collection of events x ∈ E with the property thatπG(x) = s. Thus events in E(s) are simultaneous. The following result shows that E(s) isan affine space modeled on ker(τ).

2.13 Lemma: For each s ∈ IG, E(s) is a 3-dimensional affine space modeled on ker(τ).

Proof: The affine action of ker(τ) on E(s) is that obtained by restricting the affine actionof V on E. We must show that this restriction is well defined. That is, given v ∈ ker(τ) weneed to show that v+x ∈E(s) for every x ∈E(s). If x ∈E(s) then τ((v+x)−x) = τ(v) = 0which means that v + x ∈ E(s) as claimed. Also, for x1, x2 ∈ E(s), by definition, thereexists w ∈ ker(τ) such that x2 = w + x1. This shows that the action is transitive whichtherefore means that for a fixed x ∈E(s), we have

E(s) = u+ x| u ∈ ker(τ)

It now follows from Proposition 2.11 that E(s) is an affine subspace of E modeled on ker(τ).

Just as a single set of simultaneous events is an affine space, so too is the set of instants.


2.14 Lemma: IG is a 1-dimensional affine space modeled on R.

Proof: The affine action of R on IG is defined as follows. For t ∈ R and s1 ∈ IG, we definet + s1 to be s2 = πIG(x2) where τ(x2 − x1) = t for some x1 ∈ E(s1) and x2 ∈ E(s2).We need o show that this definition does not depend on the choice of x1 and x2. Letx1 ∈E(s1) and x2 ∈E(s2). Since x1 ∈E(s1), we have x1 − x1 = v1 ∈ ker(τ) and similarlyx2 − x2 = v2 ∈ ker(τ). Therefore we have

τ(x2 − x1) = τ((v2 + x2)− (v1 + x1)) = τ((v2 − v1) + (x2 − x1)) = τ(x2 − x1),

where we have used the associativity of affine addition. Therefore the condition that τ(x2−x1) = t does not depend on the choice of x1 and x2.

We next denote by VG the vectors v ∈ V for which τ(v) = 1. We call vectors in VG Galileanvelocities. Elementary linear algebra shows that VG is an affine space modeled over ker(τ)with the affine action given by vector addition in V .

2.3. Observers. An observer is to be thought of intuitively as someone who is presentat each instant. Such an observer should be moving at a uniform velocity. Note that ina Galilean spacetime, the notion of “stationarity” makes no sense. We now provide ourdefinition of an observer.

2.15 Definition: An inertial observer in a Galilean spacetime G = (E, V, g, τ). is a 1-dimensional affine subspace O of E with the property that πG(O) consists of more than onepoint.

The definition thus requires that O not be comprised entirely of simultaneous events.It is easy to see that the definition implies that πG(O) = IG. As a consequence of thedefinition, we have the following result.

2.16 Proposition: If O is an observer in a Galilean spacetime G = (E, V, g, τ) then for eachs0 ∈ IG there exists a unique point x ∈ O ∩E(s0).

Proof: Since O is a 1-dimensional affine subspace of E whose projection consists of morethan one point, given s0 ∈ IG there exists x0 ∈ O such that πG(x0) = s0. By Proposition2.11, there exists a 1-dimensional subspace W ⊂ V such that

O = w + x0| w ∈W

Clearly, x0 ∈E(s0)∩O. This means that for each s0 ∈ IG, the intersection E(s0)∩O is non-empty. Next, assume that there exists y0 ∈ E(s0) with y0 6= x0 such that y0 ∈ E(s0) ∩O.Now, y0 ∈ E(s0) implies that y0 − x0 ∈ ker(τ). On the other hand, y0 ∈ O implies thaty0 − x0 ∈W . This means that y0 = x0 which proves the uniqueness of x0.

This means that an observer does exactly what it should: it resides in exactly one placeat each instant. By requiring that O be an affine subspace, we ensure that it has a uniformvelocity, and so is an appropriate reference for observing Newton’s laws. We shall oftenrefer to an inertial observer as merely an “observer” when there’s no danger in doing so.We shall denote by Os the unique point in the intersection O ∩E(s).

8 A. Bhand

Since an observer O is a 1-dimensional affine subspace, there is a unique 1-dimensionalsubspace U of V upon which O is modeled. Therefore, there exists a unique vector vO ∈ VGwith the property that τ(vO) = 1. Conversely, given v ∈ VG and x ∈ E, there exists aunique observer O with the properties (1) x ∈ O (2) v = vO. We call vO the Galileanvelocity of the observer O. It provides a reference velocity with which we can measure othervelocities. Indeed, given an observer O we may define an associated map PO : V → ker(τ)by

PO(v) = v − (τ(v))vO

In particular, if v ∈ VG, we note that v = vO +PO(v). Thus PO can be thought of as givingthe velocity of v relative to the observer’s Galilean velocity vO. Note that such velocitiesalways live in the 3-dimensional vector space ker(τ) that is to be thought of as the space ofvelocities that are familiar in mechanics. Such velocities are, however, only defined in thepresence of an observer.

World lines. Intuitively, a world line is to be thought of as being the spatio-temporal historyof something experiencing the spacetime. We make the following definition.

2.17 Definition: Let G = (E, V, g, τ) be a Galilean spacetime. A world line in G is a sectionof πG : E → IG.

A world line c : IG →E is differentiable at s0 ∈ IG if the limit

c′(s0) := limt→0

c(t+ s0)− c(s0)

t

exists. Since c is a section of πG, τ(c(t + s0) − c(s0)) = t and so c′(s0) ∈ VG, provided itexists. Similarly, for a differentiable world line, if the limit

limt→0

c′(t+ s0)− c′(s0))

t

exists, we denote it by c′′(s0), the acceleration of the world line at the instant s0. Since

τ(c′′(s0)) = limt→0

τ(c′(t+ s0)− c′(s0))

t= lim

t→0

1− 1

t= 0,

we have c′′(s0) ∈ ker(τ).

2.4. Galilean mappings. If Gi = (Ei, Vi, gi, τi), i = 1, 2, are two Galilean spacetimes, aGalilean mapping from G1 to G2 is a map ψ : E1 →E2 with the following properties:

GM1. ψ is an affine map.

GM2. τ2(ψ(x1)− ψ(x2)) = τ1(x1 − x2) for x1, x2 ∈E1;

GM3. g2(ψ(x1)−ψ(x2), ψ(x1)−ψ(x2)) = g1(x1−x2, x1−x2) for simultaneous events x1, x2 ∈E1.

A Galilean mapping φ : G → R3 × R into the canonical Galilean spacetime is a coordinatesystem for G. A Galilean mapping from G to itself is a Galilean transformation, and theseform a Lie group under composition. We shall denote this group by Gal(G) and its Liealgebra by gal(G). If ψ ∈ Gal(G) then there are induced natural mappings of V, IG, and Ras follows.


2.18 Lemma: Let G = (E, V, g, τ) be a Galilean spacetime with ψ ∈ Gal(G). The followingmappings are well defined:

(i) the mapping ψV : V → V defined by ψV (v) = ψ(v + x0)− ψ(x0) where x0 ∈E;

(ii) the mapping ψIG : IG → IG defined by ψIG(s) = πG(ψ(x)) where x ∈E(s);

(iii) the mapping ψτ : R→ R defined by ψτ (t) = t+ ψIG(s)− s for s ∈ IG.

Furthermore,

(iv) ψV (v) = ψ(x1)− ψ(x2) where x2 − x2 = v, and

(v) there exists tψ ∈ R so that ψIG(s) = s+ tψ and ψτ (t) = t+ tψ.

Proof: Let x0, x0 ∈E. We have

ψV (v) = ψ(v + x0)− ψ(x0)

= ψ((v + (x0 − x0)) + x0)− ψ((x0 − x0) + x0)

= ψ((x0 − x0) + x0) + ψ(v + x0)− ψ(x0)− ψ((x0 − x0) + x0)

= ψ(v + x0)− ψ(x0).

where we have used the property

ψ((v1 + v2) + x) = (ψ(v1 + x) + ψ(v2 + x))− ψ(x)

since ψ is an affine map. This property is readily verified using the definition of an affinemap.

We will now show that ker(τ) is an invariant submanifold for ψV . We let x, x ∈E havethe property that x− x = u ∈ ker(τ). Then

τ(ψV (u)) = τ(ψ(x)− ψ(x)) = τ(u) = 0,

where we have used the property GM2.(ii) Let x, x ∈E(s). There exists u ∈ ker(τ) so that x = u+ x. Now we compute

πG(ψ(x))− πG(ψ(x)) = τ(ψ(x)− ψ(x))

= τ(ψV (x− x))

= τ(ψV (u)) = 0,

using the fact that ker(τ) is an invariant submanifold for ψV .(iii) We wish to show that the definition is independent of the choice of s ∈ IG. For

s ∈ IG, we compute

t+ ψIG(s)− s = t+ ψIG(s+ (s− s))− s+ (s− s)= t+ ψIG(s+ (s− s))− ψIG(s) + ψIG(s)− s+ (s− s)= t+ ψIG(s)− s

10 A. Bhand

(iv) We have

ψ(x1)− ψ(x2) = ψ((x1 − x2) + x2)− ψ(x2)

= ψV (x1 − x2),

as desired.(v) Let x0 ∈E and let tψ = τ(ψ(x0)− x0). For s ∈ IG, let x ∈E(s). We then have

ψIG(s)− s = ψIG(s)− πG(x)

= πG(ψ(x))− πG(x)

= τ(ψ(x)− ψ(x0)) + τ(ψ(x0)− x0) + τ(x0 − x)

= τ(ψ(x0)− x0) = tψ,

where we have used the property GM2 of Galilean mappings. This shows that the definitionof tψ is independent of x0 and that ψIG(s) = tψ + s, as desired. From (iii) it also followsthat ψτ (t) = tψ + t.

2.19 Remarks: (i) The above lemma shows that given ψ ∈ Gal(G), ψV leaves ker(τ)invariant. We shall see in the next chapter that ψV |ker(τ) has a special meaning, atleast in the context of rigid motions.

(ii) Using the definition of a Galilean mapping, it is easy to see that VG is also invariantunder ψV .

3. Observer-independent formulation of rigid body dynamics

In this chapter, we provide an observer-independent formulation of rigid body dynamics.To our knowledge, such a formulation is yet to be done in the general setting of a Galileanspacetime although dynamics of a particle are well understood when an observer is present[Artz 1981]. In Section 3.1, we define rigid motions in a Galilean spacetime and list theirfundamental properties. In the next section we define the body and spatial velocities asso-ciated with a rigid motion. Here again, the treatment is completely observer- independent.In Section 3.3 we provide our definition of a rigid body and discuss in it’s implications indetail. Finally, in section 3.4 we we define momenta and derive the equations of motion fora rigid body undergoing a rigid motion in a Galilean spacetime.

3.1. Rigid motions. A rigid motion in a Galilean spacetime G = (E, V, g, τ) is a smoothmapping Ψ : R→ Gal(G) with the property that Ψt(x) ∈E(t+ πG(x)) for each x ∈E(s).Thus a rigid motion maps points in E(s) to E(s) at t = 0 and for t 6= 0 the points getshifted by the affine action of R on IG. A rigid motion is proper when Ψ0 : E(s) → E(s)preserves orientation for a given ( and hence for all) s ∈ IG. Note that we do not demandthat Ψ0 = idE although it makes sense that this will often be the case.

Let us give some of the basic properties of rigid motions. To state these properties, letO(ker(τ)) denote the g-orthogonal linear mappings of ker(τ), with SO(ker(τ)) ⊂ O(ker(τ))those mappings with determinant 1. The Lie algebra of O(ker(τ)) we denote by o(ker(τ)),recalling that it is the collection of g-skew symmetric linear mappings of ker(τ). Thefollowing result is immediate.


3.1 Lemma: Given a rigid motion Ψ, for each x ∈ E the map IG 3 s 7→ Ψs−πG(x)(x) ∈ E

is a section of πG : E → IG.

Proof: We need only show that Ψs−πG(x)(x) ∈E(s). However, this follows directly from thedefinition of a rigid motion.

In view of Definition 2.17, we may think of the map s 7→ Ψs−πG(x)(x) as a world line of apoint x ∈E in the Galilean spacetime.

3.2 Proposition: Let Ψ be a rigid motion in a Galilean spacetime G = (E, V, g, τ). Then,for each t ∈ R, Ψt,V |ker(τ) ∈ O(ker(τ)).

Proof: From Lemma 2.18, it is clear that Ψt,V maps ker(τ) to itself. Next for simultaneousevents x1 and x2, we compute

g(Ψt,V |ker(τ)(x1 − x2),Ψt,V |ker(τ)(x1 − x2)) = g(Ψt,V (x1 − x2),Ψt,V (x1 − x2))

= g(Ψt(x1)−Ψt(x2),Ψt(x1)−Ψt(x2))

= g(x1 − x2, x1 − x2),

where we have used the properties of the rigid motion and Lemma 2.18. This shows thatΨt,V |ker(τ) ∈ O(ker(τ)) as desired.

This proposition shows that given a rigid motion in a Galilean spacetime, we can associateto this rigid motion, a unique map from R→ O(ker(τ)). We denote this map by RΨ.

The next proposition gives a geometrical interpretation of the map Ψt,V .

3.3 Proposition: Let Ψ be a rigid motion in a Galilean spacetime G = (E, V, g, τ). Forx ∈E, the tangent map at x, denoted TxΨt : Ex →EΨt(x) is given by

TxΨt(v) = Ψt,V (v)

Proof: Let us choose a curve c(t) = vt + x where v ∈ V such that c′(t)∣∣t=0

= v. Thedefinition of the tangent map then gives

TxΨt(v) =d

dt

∣∣∣∣t=0

Ψt(vt+ x)

= limt→0

Ψt(vt+ x)−Ψt(x)

t

= limt→0

Ψt,V (tv)

t

= Ψt,V (v)

as desired.

12 A. Bhand

3.4 Remark: Although we have proved it for a rigid motion Ψ, this result is true for ageneral Galilean mapping. This is reflected in the fact that we do not use any propertiesof the rigid motion in the proof.

3.2. Body and spatial velocities. In this section we provide our observer–independentdefinitions for the body and spatial velocities corresponding to a rigid motion in a Galileanspacetime.

Angular velocities. To define the body and spatial angular velocities, we require the nota-tion of the map from ker(τ) to o(ker(τ)) given by ω 7→ ω. This is a generalization of of themap from R3 to o(3) defined by ω1

ω2

ω3

7→ 0 −ω3 ω2

ω3 0 −ω1

−ω2 ω1 0

and may be explicitly defined by choosing an orthonormal basis for ker(τ) and then apply-ing this transformation to the components in this basis. Since the vector product in R3

commutes with orthogonal transformations, this definition is independent of the choice oforthonormal basis. In like manner one can define u1 × u2 for any u1, u2 ∈ ker(τ) as thegeneralization of the R3 vector product.

We may now define the body and spatial angular velocities associated with a given rigidmotion.

3.5 Definition: Let Ψ be a rigid motion in a Galilean spacetime G = (E, V, g, τ) with RΨ

as defined in Proposition 3.2.

(i) The body angular velocity is the map ΩΨ : R→ ker(τ) satisfying

ΩΨ(t) = R−1Ψ (t) RΨ(t),

(ii) The spatial angular velocity is the map ωΨ : R→ ker(τ) satisfying

ωΨ(t) = RΨ(t) R−1Ψ (t).

Linear velocities. It is possible to define linear velocities associated with a rigid motiononce a point x ∈ E is chosen. This establishes a reference point with respect to which the“translations” can be measured. We now define body linear and spatial linear velocities inthis setting.

3.6 Definition: Let Ψ be a rigid motion in a Galilean spacetime G = (E, V, g, τ).

(i) The body linear velocity, is the map V bΨ : E × R→ VG satisfying

V bΨ(x, t) = Ψ−1

t,V (d

dtΨt(x)).

(ii) The spatial linear velocity, is the map vsΨ : E × R→ VG satisfying

vsΨ(x, t) = −Ψt,V (d

dt(Ψ−1

t (x))).


3.7 Remarks: (i) Notice that, by the definition of a rigid motion Ψ in a Galilean space-time, the quantity d

dt(Ψt(x)) ∈ VG. Also, Ψt,V leaves VG invariant.

(ii) The body linear velocity is obtained by first computing the velocity of the pointundergoing the motion Ψ at time t and then pulling this quantity back through Ψ−1

t,V .

(iii) The interpretation of spatial linear velocity is less intuitive. It will become clearerin chapter 5 when we deal with this situation in the presence of an observer. Insuch a case, we shall recover the familiar expressions for the body and spatial linearvelocities.

3.3. Rigid bodies. In order to talk about momenta, we need the notion a of a rigid body.In this section we provide our definition for a rigid body and provide the implications ofthis definition. We begin by proving in our Galilean setting some of the basic properties ofthe inertia tensor of a rigid body.

Definitions. Let G = (E, V, g, τ) be a Galilean spacetime. A rigid body is a pair (B, µ)where B ⊂E(s0) is a compact subset of simultaneous events, and µ is a mass-distributionon E(s0) with support equal to B. Our definition thus allows such degenerate rigid bodiesas point masses, and bodies whose mass distribution is contained in a line in E(s0). Wedenote

µ(B) =

∫dµ

as the mass of the body.The center of mass of the body (B, µ) is the point

xc =1

µ(B)(

∫(x− x0)dµ) + x0.

Note that the integrand is in ker(τ) and so too will be the integral. The following lemmagives some of the basic properties of this definition. If S ⊂ A is a subset of an affine spaceA. we let conv(S) denote the convex hull of S and aff(S) denote the affine hull of S. If Xis a topological space with subsets T ⊂ S ⊂ X, intS(T ) denotes the interior of T relativeto the induced topology on S.

3.8 Lemma: Let (B, µ) be a rigid body in a Galilean spacetime with B ⊂ E(s0). Thefollowing statements hold:

(i) the expression

xc =1

µ(B)(

∫(x− x0)dµ) + x0

is independent of the choice of x0 ∈E(s0);

(ii) xc is the unique point in E(s0) with the property that∫(x− xc)dµ = 0

(iii) xc ∈ intaff(B)(conv(B)).

14 A. Bhand

Proof: (i) To check that the definition of xc is independent of x0 ∈E(s0), we let x0 ∈E(s0)and compute

1

µ(B)(

∫(x− x0)dµ) + x0 =

1

µ(B)(

∫(x− x0)dµ) +

1

µ(B)(

∫(x0 − x0)dµ)

+ (x0 − x0) + x0

=1

µ(B)(

∫(x− x0)dµ) + x0

(ii) By definition of xc and by part (i) we have

xc =1

µ(B)(

∫(x− xc)dµ) + xc.

from which it follows that ∫(x− xc)dµ = µ(B)(xc − xc) = 0

Now suppose that xc ∈E(s0) is an arbitrary point with the property that∫(x− xc)dµ = 0

Then, by (i),

xc =1

µ(B)(

∫(x− xc)dµ) + xc.

from which we conclude that xc = xc.

(iii) If xc is on the relative boundary of conv(B) or not in B at all, then there exists ahyperplane P in E(s0) passing through xc so that there are points in B which lie on oneside of P , but there are no points in B on the opposite side. In other words, there existsλ ∈ ker(τ)? so that the set

x ∈ B| λ(x− xc) > 0is non-empty, but the set

x ∈ B| λ(x− xc) < 0is empty. But this would imply that∫

λ(x− xc)dµ > 0

contradicting (ii).

The inertia tensor. The properties of a rigid body are characterized by three things: (1)itsmass, (2) its center of mass and (3) its inertia tensor. We now define the latter. Letx0 ∈ E(s0). We define the inertia tensor about x0 of a rigid body (B, µ) to be the linearmap Ix0 : ker(τ)→ ker(τ) defined by

Ix0(u) =

∫(x− x0)× (u× (x− x0))dµ.

We denote the inertia tensor about the center of mass of (B, µ) by Ic. Next, we record somebasic properties of the inertia tensor.


3.9 Proposition: The inertia tensor Ix0 of a rigid body (B, µ) is symmetric with respect tothe inner product g.

Proof: Using the vector identity

g(u, v × w) = g(w, u× v)

we compute

g(Ix0(u1), u2) =

∫g((x− x0)× (u1 × (x− x0)), u2)dµ

=

∫g(u1 × (x− x0), (u2 × (x− x0))dµ

=

∫g(u1, (x− x0)× (u2 × (x− x0)))dµ

= g(u1, Ix0(u2)),

which is what we wished to show.

It is often useful to be able to compute the inertia tensor about a general point by firstcomputing it about the center of mass. The following result shows how this is done.

3.10 Proposition: Ix0(u) = Ic(u) + µ(B)(xc − x0)× (u× (xc − x0)).

Proof: We compute

Ix0 =

∫(x− x0)× (u× (x− x0))dµ

=

∫(x− xc) + (xc − x0))× (u× ((x− xc) + (xc − x0)))dµ

=

∫(x− xc)× (u× (x− xc))dµ+

∫(xc − x0)× (u× (xc − x0))dµ

+

∫(x− xc)× (u× (xc − x0))dµ+

∫(xc − x0)× (u× (x− xc))dµ

It then follows from Lemma 3.8 that the last two terms vanish, and from this the resultfollows.

Eigenvalues of the inertia tensor. Since Ix0 is symmetric, its eigenvalues are real. Fur-thermore, they are non-negative. The following result demonstrates this, as well as othereigenvalue related assertions.

3.11 Proposition: Let (B, µ) be a rigid body with B ∈ E(s0) and let x0 ∈ E(s0). Let Ix0

denote the inertia tensor of (B, µ) about x0. The following statements hold:

(i) the eigenvalues of the inertia tensor Ix0 of a rigid body are never negative;

(ii) if Ix0 has a zero eigenvalue, then the other two eigenvalues are equal.

(iii) if Ix0 has two zero eigenvalues, then Ix0 = 0.

16 A. Bhand

Proof: (i) Since Ix0 is symmetric, it’s eigenvalues will be non-negative if and only if thequadratic form u 7→ g(Ix0(u), u) is positive semi-definite. For u ∈ ker(τ) we compute

g(Ix0(u), u) =

∫g(u, (x− x0)× (u× (x− x0)))dµ

=

∫g(u× (x− x0), u× (x− x0))dµ.

Since the integrand is non-negative, so too will be the integral.

(ii) Let I1 be the zero eigenvalue with v1 a unit eigenvector. We claim that the supportof the mass distribution µ must be contained in the line

`v1 = sv1 + x0| s ∈ R.

To see that this must be so, suppose that the support of µ is not contained in `v1 . Thenthere exists a Borel set S ⊂E(s0) \ `v1 so that µ(S) > 0. This would imply that

g(Ix0(v1), v1) =

∫g(v1 × (x− x0), v1 × (x− x0))dµ

≥∫Sg(v1 × (x− x0), v1 × (x− x0))dµ.

Since S ∩ `v1 = ∅ it follows that for all points x ∈ S, the vector x− x0 is not collinear withv1. Therefore

g(v1 × (x− x0), v1 × (x− x0)) > 0

for all x ∈ S, and this would imply that Ix0(v1, v1) > 0. But this contradicts v1 being aneigenvector with zero eigenvalue, and so the support of B must be contained in the line `v1 .

To see that this implies the remaining two eigenvectors are equal, we shall show thatany vector that is g-orthogonal to v1 is an eigenvector for Ix0 . First write

x− x0 = f1(x)v1 + f2(x)v2 + f3(x)v3

for functions f i : E(s0)→ R, i = 1, 2, 3. Since the support of µ is contained in the line `v1

we have ∫(x− x0)× (u× (x− x0))dµ = v1 × (u× v1)

∫(f1(x))2dµ

for all u ∈ ker(τ). Now recall the property of the cross product that v1 × (u × v1) = uprovided u is orthogonal to v1 and that v1 has unit length. Therefore we see that for anyu that is orthogonal to v1 we have

Ix0(u) =

(∫(f1(x))2dµ

)u,

meaning that all such vectors u are eigenvectors with the same eigenvalue, which is whatwe wished to show.

(iii) It follows from the above arguments that if two eigenvalues I1 and I2 are zero, thenthe support µ must lie in the intersection of the lines `v1 and `v2 (here vi is an eigenvectorfor Ii, i = 1, 2), and this intersection is a single point, that must therefore be x0. From thisand the definition of Ix0 , it follows that Ix0 = 0.

Note that in proving the result we have proved the following corollary.


3.12 corollary: Let (B, µ) be a rigid body with inertia tensor Ix0. The following statementsare true:

(i) Ix0 has a zero eigenvalue if and only if B is contained in a line through x0;

(ii) if Ix0 has two zero eigenvalues then B = x0, i.e., B is a particle located at x0;

(iii) if there is no line through x0 that contains the support of µ, then the inertia tensor isan isomorphism.

In coming to an understanding of the “appearance” of a rigid body, it is most convenientto refer to its inertia tensor Ic about it’s center of mass. Let I1, I2, I3 be the eigenvalues ofIc that we call the principal inertias of (B, µ). If v1, v2, v3 are orthonormal eigenvectorsassociated with these eigenvalues, we call these the principal axes of (B, µ). Related tothese is the the inertial ellipsoid which is the ellipsoid in ker(τ) given by

E(B) =x1v1 + x2v2 + x3v3 ∈ ker(τ)| I1(x1)2 + I2(x2)2 + I3(x3)2 = 1

,

provided that none of the eigenvalues of Ix0 are zero. If one of the eigenvalue does vanish,then by Proposition 3.11, the other two eigenvalues are equal. If we suppose that I1 = 0and I2 = I3 = I then in the case of a single zero eigenvalue, the inertial ellipsoid is

E(B) =

x1v1 + x2v2 + x3v3 ∈ ker(τ)| x2 = x3 = 0, x1 ∈

− 1√

I,

1√I

.

in the most degenerate case, when all eigenvalues are zero, we define E(B) = 0. Theselatter two inertia ellipsoids, correspond to cases (i) and (ii) in Corollary 3.12.

To relate these properties of the eigenvalues of Ic with the inertial ellipsoid E(B. itis helpful to introduce the notion of an axis of symmetry for a rigid body. We let Ic bethe inertia tensor about the center of mass, and denote by I1, I2, I3 its eigenvalues andv1, v2, v3 it’s orthogonal eigenvectors. A vector v ∈ ker(τ) \ 0 is an axis of symmetryfor (B, µ) if for every R ∈ O(ker(τ)) which fixes v we have R(E(B)) = E(B). The followingresult gives the relationship between axes of symmetry and the eigenvalues of Ic.

3.13 Proposition: Let (B, µ) be a rigid body with inertia tensor Ic about its center of mass.Let I1, I2, I3 be the eigenvalues of Ic with orthonormal eigenvectors v1, v2, v3. If I1 = I2

then v3 is an axis of symmetry of B.Conversely, if v ∈ ker(τ) is an axis of symmetry, then v is an eigenvector of Ic. If I is

the eigenvalue for which v is an eigenvector, then the other two eigenvalues of Ic are equal.

Proof: Write I1 = I2 = I3. We then see that any vector v ∈ spanRv1, v2 will have theproperty that

Ic(v) = Iv

Now let R ∈ O(ker(τ)) fix the vector v3. Because R is orthogonal, if we have v ∈spanRv1, v2 then R(v) ∈ spanRv1, v2. Also, if

v = a1v1 + a2v2,

then,R(v) = (cos θa1 + sin θa2)v1 + (− sin θa1 + cos θa2)v2 (3.1)

18 A. Bhand

for some θ ∈ R since R is simply a rotation in the plane spanned by v1, v2. Now letu ∈ E(B). We then write u = x1v1 + x2v2 + x3v3 and note that

I(x1)2 + I(x2)2 + I(x3)3 = 1

It is now a straightforward but tedious calculation to verify that R(u) ∈ E(B) using (3.1)and the fact that R fixes v3. This shows that R(E(B)) = E(B), and so v3 is an axis ofsymmetry for B.

For the second part of the proposition, note that R ∈ O(ker(τ)) has the property thatR(E(B)) = E(B) if and only if R maps the principal axes of the ellipsoid E(B) to theprincipal axes. Since R is a rotation about some axis, this means that R fixes a principalaxis of E(B). Thus if v ∈ ker(τ) is an axis of symmetry, then V must lie along a principalaxis of the ellipsoid E(B). By our definition of E(B), this means that v is an eigenvector ofIc. Let I be the associated eigenvalue and I1, I2, I3 = I be the collection of all eigenvaluesof Ic with eigenvectors v1, v2, v3 = v. Since v is an axis of symmetry, any rotation aboutv must map principal axes of E(B) to principal axes. This means that for every θ ∈ R thevectors

v′1 = cos θv1 − sin θv2, v′2 = sin θv1 + cos θv2

are eigenvectors for Ic. This means that all non-zero vectors in spanRv1, v2 are eigen-vectors for Ic. This means that the restriction of Ic to spanRv1, v2s diagonal in everyorthonormal basis for spanRv1, v2. Therefore, if v1, v2 are chosen to be orthonormalthen v′1, v′2 as defined previously are also orthonormal. Our conclusions assert the exis-tence of I ′1, I

′2 ∈ R so that

Ic(v′i) = Iiv′i, i = 1, 2.

but by the definition of v′1 and v′2 we also have

Ic(v′1) = cos θIc(v1)− sin θIc(v2)

= cos θI1v1 − sin θv2

= cos θI1(cos θv′1 + sin θv′2)− sin θI2(− sin θv′1 + cos θv′2).

Therefore,I ′1v′1 = (cos2 θI1 + sin2 θI2)v′1 + sin θ cos θ(I1 − I2)v′2

for every θ ∈ R. Since v′1, v′2 are orthogonal, this means that choosing θ so that sin θ cos θ 6= 0

implies that I1 − I2 = 0. This is what we wished to show.

3.4. Dynamics of rigid bodies. In this section, we derive the “equations of motion” for arigid body undergoing a rigid motion in a Galilean spacetime. To do so, we need to definebody and spatial momenta.

Spatial momenta. We let G = (E, V, g, τ) be a Galilean spacetime with Ψ a rigid motionand (B, µ) a rigid body with B ⊂ E(s0). We wish to see how to define spatial linearmomentum. To motivate our definition, we recall how it might be defined for a particle ofmass m. If the particle is moving in R3 following the curve t 7→ x(t), its linear momentumis taken to be mx(t). The key point here is that one measures the linear momentum ofthe center of mass. Motivated by the classical definition, the angular momentum could bedefined as the image of the angular velocity under the inertia tensor.


3.14 Definition: Let (B, µ) be a rigid body in a Galilean spacetime G = (E, V, g, τ) and letΨ be a rigid motion.

(i) The spatial linear momentum is the map msΨ,B : R→ VG given by

msΨ,B(t) = µ(B)

d

dt(Ψt(xc)).

(ii) The spatial angular momentum is the map `sΨ,B : R→ ker(τ) given by

`sΨ,B(t) = Ic(t)ωΨ(t)

where

Ic(t)(u(t)) =

∫B(t)

(Ψt(x)−Ψt(xc))× (u(t)× (Ψt(x)−Ψt(xc))dµ(t).

Notice that since the integrand is in ker(τ), so too will be the integral. The following resultshows what the spatial angular momentum looks like in terms of the inertia tensor of thebody about its center of mass.

3.15 Lemma: `sΨ,B(t) = Ic(R−1Ψ (t)ωΨ(t)).

Proof: We represent, by B(t) the rigid body after it has undergone the transformation Ψt

and the corresponding mass distribution by dµ(t). We compute

`sΨ,B(t) = Ic(t)(ωΨ(t)) =

∫B(t)

(Ψt(x)−Ψt(xc))× (ωΨ(t)× (Ψt(x)−Ψt(xc))) dµ(t)

=

∫B(t)

(Ψt,V (x− xc))× (ωΨ(t)× (Ψt,V (x− xc))) dµ(t)

=

∫B(t)

(RΨ(t)(x− xc))× (ωΨ(t)× (RΨ(t)(x− xc))) dµ(t)

=

∫B(t)

(x− xc)×(R−1

Ψ (t)ωΨ(t)× (x− xc))dµ(t)

=

∫(x− xc)×

(R−1

Ψ (t)ωΨ(t)× (x− xc))dµ

= Ic(R−1Ψ (t)ωΨ(t))

where we have used the fact that x−xc ∈ ker(τ) and therefore Ψt,V (x−xc) = RΨ(t)(x−xc).

Body momenta. We now proceed to define the body momenta.

3.16 Definition: Let (B, µ) be a rigid body in a Galilean spacetime G = (E, V, g, τ) and letΨ be a rigid motion.

(i) The body linear momentum is the map MΨ,B : R→ VG satisfying

MΨ,B(t) = Ψ−1t,V (ms

Ψ,B(t))

(ii) The body angular momentum is the map LΨ,B : R→ ker(τ) satisfying

LΨ,B(t) = R−1Ψ (t)(`sΨ,B(t))

20 A. Bhand

3.17 Remarks: (i) Notice that MΨ,B(t) = µ(B)V bΨ(t).

(ii) The idea behind the definitions of body momenta is intuitively clear–they are themomenta “seen” by the body in it’s reference frame.

Galilean–Euler equations. In this subsection, we derive the so-called “Euler equations” forthe rigid body. These are the equations governing the motion of a rigid body undergoinga rigid motion in a Galilean spacetime obtained by using the principle of conservation ofmomenta. As we have remarked earlier, these equations are considerably general than thosefor the classical case of a rigid body in R3 × R with rigid motions coming from the specialEuclidean group SE(3). We have the following result.

3.18 proposition: Let G = (E, V, g, τ) be a Galilean spacetime with (B, µ) a rigid body withB ∈E(s0) and let Ψ be a rigid motion. The following statements are equivalent:

1. The spatial linear and spatial angular momenta are conserved.

2. The motion of the rigid body satisfies the following differential equations.

RΨ(t) = RΨ(t) ΩΨ(t)

xc(t) = 0

LΨ,B(t) = LΨ,B(t)× ΩΨ(t)

where xc(t) = d2

dt2(Ψt(xc)).

Proof: The first equation says that Ω(t) = R−1Ψ (t) RΨ(t) which is just the definition of Ω.

Next, conservation of spatial linear momentum implies that

d

dtmΨ,B(t) =

d

dt(µ(B)xc(t)) = 0

This gives us the second equation. To see the body angular momentum equation, we write

`Ψ,B(t) = RΨ(t)(LΨ,B(t))

Differentiating this with respect to t and setting it equal to zero (since the spatial angularmomentum is conserved), we get

0 = RΨ(t)LΨ,B(t) +RΨ(t)LΨ,B(t)

= RΨ(t)ΩΨ(t)LΨ,B(t) +RΨ(t)LΨ,B(t)

This gives usLΨ,B(t) = LΨ,B(t)× ΩΨ(t)

which is what we wanted to show.


Discussion. We call the equations given in part 2 of Proposition 3.18 the Galilean–Eulerequations. These equations are quite formidable because they have been derived in thegeneral setting of abstract Galilean spacetimes without requiring an observer. However,the generality of the treatment makes certain things less obvious. In particular, we arenot able to recover an equation governing the evolution of the body angular momentumwith time. It is implicit in these Galilean–Euler equations. We shall see that we come tothe same conclusions in Chapter 5 when we look at these equations in the presence of anobserver.

4. The structure of the Galilean group

As defined previously, the Galilean group of a Galilean spacetime G = (E, V, g, τ) is theset of affine maps from E to itself that preserve simultaneity of events and the distancebetween simultaneous events. In this chapter, we shall examine the Galilean group closelyand describe its properties. We shall also look at its various subgroups. In Section 4.1we study the canonical Galilean group and show that it consists of rotations, translations,velocity boosts and temporal origin shifts. We also look at it’s subgroups and describethe various fundamental objects associated with it. In section 4.2, we study the abstractGalilean group Gal(G). We show that that in the presence of an observer, the Galilean groupis isomorphic to the canonical Galilean group. We characterize the various subgroups of theabstract Galilean group and mention how these subgroups are related to the subgroups ofthe canonical group. Finally, in section 4.3, we introduce total velocities and describe theirimages under the isomorphism of the Lie algebras induced by the isomorphism constructedpreviously.

4.1. The canonical Galilean group. In this section, we study the Galilean group of acanonical Galilean spacetime which is a generalization of the “standard” Galilean spacetimeR3×R. To be precise, given a Galilean spacetime G = (E, V, g, τ), the canonical spacetimeof G is the Galilean spacetime Gcan := (Ecan := ker(τ)⊕R, V, gcan, τ). We wish to investigatethe structure of the canonical Galilean group Gal(Gcan).

The structure of the canonical Galilean group. The next proposition shows that Gal(Gcan)decomposes into rotations, spatial translations, Galilean velocity boosts and temporal trans-lations. We need some terminology and notation first. Recall that a subgroup H of a groupG is normal if gHg−1 ⊂ H for every g ∈ G. Next, a group G is a semi-direct product of Hby N if

(i) N is a normal subgroup of G

(ii) h · n : h ∈ H,n ∈ N = G

(iii) H ∩N = e.

4.1 Notation: We denote the semi-direct product of H by N as H nN .

22 A. Bhand

4.2 proposition: The Galilean group Gal(Gcan) of the canonical spacetime Gcan admits thedecomposition

Gal(Gcan) = (O(ker(τ)) n ker(τ)) n (ker(τ)× R)

such that for (Ri, ri, ui, ti) ∈ Gal(Gcan), i = 1, 2, the group operation on Gal(Gcan) is givenby

((R1, r1, u1, t1) · (R2, r2, u2, t2) = (R1 R2, r1 +R1(r2) + t2u1, u1 +R1(u2), t1 + t2)

Proof: We first find the form of a Galilean transformation φ : Ecan →Ecan . Recall that sinceφ is an affine map, it has the form φ(x, t) = A(x, t)+(r, σ) where A : ker(τ)×R→ ker(τ)×Ris R-linear and where (r, σ) ∈ ker(τ)×R. Let us write A(x, t) = (A11x+A12t, A21x+A22t)where A11 ∈ L(ker(τ), ker(τ)), A12 ∈ L(R, ker(τ)), A21 ∈ L(ker(τ),R) and A22 ∈ L(R,R).By GM3, A11 is a g-orthogonal transformation of ker(τ). GM2 implies that

A22(t2 − t1) +A21(x2 − x1) = t2 − t1, t1, t2 ∈ R, x1, x2 ∈ ker(τ)

Thus, taking x1 = x2, we see that A22 = 1. This in turn implies that A21 = 0. Gatheringthis information shows that a Galilean transformation has the form

φ :

(xt

)7→[R u0t 1

](xt

)+

(rσ

)where R ∈ O(ker(τ)), σ ∈ R and r, u ∈ ker(τ). This proves the first part of the proposition.Now, it is easy to see that if φi, i = 1, 2 are Galilean transformations given by

φi :

(xt

)7→[Ri ui0t 1

](xt

)+

(riσi

), i = 1, 2.

then

φ1 φ2 :

(xt

)7→[R1 R2 u1 +R1(u2)

0t 1

](xt

)+

(r1 +R1(r2) + σ2u1

σ1 + σ2

).

which gives us the desired group operation.

4.3 Remarks: (1) It is clear from this proposition that Gal(Gcan) is a 10-dimensionalgroup. This is not altogether obvious from the definition.

(2) The meaning of the appearance of two semi-direct products in the decompositionof Gal(Gcan) should be understood correctly. They arise because ker(τ) × R is anormal subgroup of Gal(Gcan) and the quotient group itself is a semi-direct productof O(ker(τ)) by ker(τ).

(3) A canonical Galilean transformation may now be written as a composition of one ofthree basic classes of transformations.

(i) A shift of origin (xt

)7→(xt

)+

(rσ

)for r ∈ ker(τ), σ ∈ R.


(ii) A “rotation” of reference frame:(xt

)7→[R 00t 1

](xt

)for R ∈ O(ker(τ)).

(iii) A (Galilean) velocity boost(xt

)7→[

idker(τ) u

0t 1

](xt

)for u ∈ ker(τ).

The names we have given these fundamental transformations are suggestive. A shift of theorigin should be thought of as moving the origin to a new position, and resetting the clock,but maintaining the same orientation in space. A rotation of reference frame means theorigin stays in the same place, and uses the same clock but, but rotates the “point of view.”The final basic transformation, a velocity boost, means the origin maintains its orientationand uses the same clock, but now moves with a certain velocity with respect to the previousorigin.

Subgroups of the canonical Galilean group. In the previous section we showed that theelements of the canonical Galilean group Gal(Gcan) are composed of temporal and spatialtransformations, spatial rotations, and velocity boosts. In this section, we study some ofthe subgroups of Gal(Gcan) more carefully. Below we list the various subgroups and discusssome of their basic properties. Some of what we say here can be found in [Levy-Leblond1971].

(i) The set Gal0(Gcan) := (R, r, u, t) ∈ Gal(Gcan) : t = 0: This is a 9-dimensionalsubgroup sometimes called the isochronous Galilean group. This subgroup, which isthe maximal proper subgroup of Gal(Gcan) is also its commutator subgroup.

(ii) The set Galub(Gcan) := (R, r, u, t) ∈ Gal(Gcan) : u = 0: This is a 7-dimensionalsubgroup which we call the unboosted (or no-boost) subgroup.

(iii) The set (R, r, u, t) ∈ Gal(Gcan) : R = idker(τ): This is a 7-dimensional subgroupsometimes called the anisotropic Galilean group.

(iv) The set E(Gcan) := (R, r, u, t) ∈ Gal(Gcan) : u = 0, t = 0: This is a 6-dimensionalsubgroup called the Euclidean subgroup. It consists of space translations and rota-tions.

(v) The set Lin(Gcan) := (R, r, u, t) ∈ Gal(Gcan) : r = 0, t = 0: This is a 6-dimensionalsubgroup which we call the linear subgroup of Gal(Gcan). It is also called the homoge-neous canonical Galilean group. This subgroup is actually the quotient of Gal(Gcan)by the normal subgroup of the space-time translations mentioned below and can beidentified with the semi-direct product of O(ker(τ)) by ker(τ).

24 A. Bhand

(vi) The set (R, r, u, t) ∈ Gal(Gcan) : R = idker(τ), t = 0: This is a 6-dimensional sub-group identified with the set of straight world lines in Ecan which can be parametrizedby giving the intercept r0 of the line with a given time= constant plane, and the slopeu0 of the line, writing its equation

r = r0 + u0t,

the physical interpretation of which is clear.

(vii) ker(τ)×R: this is a 4-dimensional subgroup of space-time translations. It can be seenthat this is a normal subgroup.

(viii) The set (R, r, u, t) ∈ Gal(Gcan) : r = 0, u = 0: This is a 4-dimensional subgroupconsisting of rotations and time translations.

(ix) O(ker(τ)): This is a 3-dimensional subgroup, also called the subgroup of rotations orthe orthogonal subgroup.

(x) The set (R, r, u, t) ∈ Gal(Gcan) : R = idker(τ), r = 0, t = 0: This is a 3-dimensionalAbelian subgroup consisting of pure velocity boosts.

(xi) The set (R, r, u, t) ∈ Gal(Gcan) : R = idker(τ), u = 0, t = 0: This is a 3-dimensionalAbelian subgroup consisting of pure space translations.

(xii) R: This is a 1-dimensional subgroup consisting of pure time translations. It is thequotient of Gal(Gcan) by the isochronous Galilean group Gal0(Gcan) and is obviouslyAbelian.

Properties of the canonical Galilean group. In this section, we study some fundamentalobjects associated with the canonical group. We denote the Lie algebra of Gal(Gcan) bygal(Gcan) and note that as a vector space it is given by o(ker(τ))⊕ker(τ)⊕ker(τ)⊕R. Thefollowing result gives the Lie bracket on this Lie algebra.

4.4 Lemma: The following statements hold:

(i) the exponential map from gal(Gcan) to Gal(Gcan) is given by

exp(ω, β, ν, τ) = (exp(ω), Aω(β) + τBω(ν), Aω(ν), τ),

where

exp(ω) =

idker(τ) + sin ‖ω‖ ω

‖ω‖ + (1− cos ‖ω‖) ω2

‖ω‖2 , ω 6= 0

idker(τ), ω = 0

and

Aω =

idker(τ) +

(1−cos ‖ω‖‖ω‖

)ω‖ω‖ +

(1− sin ‖ω‖

‖ω‖

)ω2

‖ω‖2 , ω 6= 0

idker(τ), ω = 0

and

Bω =

12 idker(τ) +

(‖ω‖−sin ‖ω‖‖ω‖2

)ω‖ω‖ +

(‖ω‖2−2(1−cos ‖ω‖)

‖ω‖2

)ω2

‖ω‖2 , ω 6= 012 idker(τ), ω = 0


(ii) the Lie bracket in gal(Gcan) is given by

[(ω1, β1, ν1, τ1), (ω2, β2, ν2, τ2)] =

(ω1 × ω2, ω1 × β2 − ω2 × β1 + τ2ν1 − τ1ν2, ω1 × ν2 − ω1 × ν1, 0).

Proof: We prove the lemma by faithfully representing Gal(Gcan) in a vector space, in whichcase the exponential map is the usual one for linear mappings, and the Lie bracket is simplythe usual commutator of the linear mappings of this vector space. We let W = ker(τ)⊕R⊕Rand for (R, r, u, t) ∈ Gal(Gcan) define an isomorphism of W by

(µ, σ, ξ) 7→ (R(µ) + σu+ ξr, σ + ξt, ξ).

One readily verifies that this is a homomorphism from Gal(Gcan) to GL(W ). To see thatthe representation is faithful, suppose that

(µ, σ, ξ) 7→ (R(µ) + σu+ ξr, σ + ξt, ξ) = (µ, σ, ξ)

for all (µ, σ, ξ) ∈ W . Then we must have σ + ξt = 0, for all σ, ξ ∈ R, implying that t = 0.Similarly, R(µ)+σu+ξr = µ for all (µ, σ, ξ) ∈W implies that r = 0, u = 0, and R = idker(τ).Thus the representation is faithful. In block matrix form, the representation of (R, r, u, t)on W is R u r

0 1 t0 0 1

∈ GL(W ).

(i) A computation gives

exp

ω β ν0 0 τ0 0 0

=

exp(ω) Aων Aωβ + τBων0 0 τ0 0 1

,where

exp(ω) = idker(τ) + ω +ω2

2!+ω3

3!+ · · ·

Aω = idker(τ) +ω2

2!+ω3

3!+ω4

4!+ · · ·

Bω =idker(τ)

2!+ω3

3!+ω4

4!+ω5

5!+ · · · .

Thus our stated formulae for Aω and Bω hold when ω = 0. For ω 6= 0 we recall the followingreadily verified identities for ω ∈ o(ker(τ)):

ω2 = ω ⊗ ω[ − ‖ω‖2idker(τ)

ω3 = −‖ω‖2ω,

where ω ⊗ ω[ : ker(τ)→ ker(τ) is the linear map defined by

(ω ⊗ ω[)(u) = g(u, ω)ω.

26 A. Bhand

We use these relations to inductively compute the higher powers of ω and ω2:

ω2k+1 = (−1)k‖ω‖2kω, k ∈ N

ω2(k+1) = (−1)k‖ω‖2kω2, k ∈ N

and after some computations we arrive at

exp(ω) = idker(τ) + sin ‖ω‖ ω

‖ω‖+ (1− cos ‖ω‖) ω2

‖ω‖2

Aω = idker(τ) +

(1− cos ‖ω‖‖ω‖

)ω

‖ω‖+

(1− sin ‖ω‖

‖ω‖

)ω2

‖ω‖2

Bω =1

2idker(τ) +

(‖ω‖ − sin ‖ω‖‖ω‖2

)ω

‖ω‖+

(‖ω‖2 − 2(1− cos ‖ω‖)

‖ω‖2

)ω2

‖ω‖2.

(ii) The above computations show that the Lie subalgebra gl(W ) of GL(W ), generatedby gal(Gcan) using the given representation consists of those matrices of the form ω ν β

0 0 τ0 0 0

∈ gl(W ),

where (ω, β, ν, τ) ∈ gal(Gcan). The Lie bracket in gal(Gcan) is now readily computed usingthe commutator for linear maps on W , using the fact that ˆ : ker(τ) → o(ker(τ)) is a Liealgebra isomorphism if the Lie bracket on ker(τ) is ×.

4.2. The abstract Galilean group. In this section, we investigate the structure of theabstract Galilean group and its subgroups.

The structure of the Galilean group. In the usual presentation of Galilean invariant me-chanics (e.g. [Souriau 1997]) one considers a spacetime R3 × R and Galilean invariance isimposed by asking that the system admit the Galilean group as a symmetry group. In thiscase, the Galilean group naturally breaks down into rotations, Galilean boosts (constant ve-locity shifts) and temporal origin shifts. In our abstract setting, the Galilean group Gal(G)does not admit such a decomposition. Note that this is similar to what one sees in anaffine Euclidean space where a decomposition of an isometry into rotation and translationis not possible until one chooses an origin about which to measure rotations. However, thepresence of an observer in a Galilean spacetime defines, for each instant, an isomorphismfrom the abstract Galilean group Gal(G) into the canonical group Gal(Gcan).

4.5 Proposition: Let G = (E, V, g, τ) be a Galilean spacetime with O an observer. Thefollowing statements hold.

1. The mapping from V to ker(τ)⊕ R defined by v 7→ (PO(v), τ(v)) is an isomorphism.

2. For each s0 ∈ IG the observer at s0, Os0, induces a natural isomorphism ιOs0 fromGal(G) to the group Gal(Gcan). Explicitly, if ψ ∈ Gal(G) with tψ as defined in Lemma2.18, and if Rψ ∈ O(ker(τ)) and rψ,O ∈ ker(τ) satisfy

ψ(x) = (Rψ(x−Os0) + rψ,O) +Otψ+s0 , x ∈E(s0)


thenιOs(ψ) = (Rψ, rψ,O, uψ,O, tψ)

where uψ,O = PO(ψV (vO)).

Proof: (1) It suffices to show that the mapping v 7→ (PO(v), τ(v)) is injective. If τ(v) = 0then v ∈ ker(τ). Now, if we also have

PO(v) = v − (τ(v))vO = 0,

we must have v = 0, thus the mapping is injective as desired.(2) To prove this part of the proposition, we first assign to each (R, r, u, t) ∈ Gal(Gcan) a

Galilean mapping ψ, and we show that the construction implies that (R, r, u, t) = (Rψ, rψ, uψ, tψ),thus showing that ιOs0 is invertible. Given (R, r, u, t) ∈ Galcan(G), we define a mapψ : E →E by

ψ(x) = tvO + (R(x−OπG(x)) + (πG(x)− s0)u+ r) +OπG(x). (4.1)

We now show that this mapping is Galilean. First we show that it is affine. For v ∈ V wecompute

ψ(v +Os0)− ψ(Os0) = tvO + (R(v +Os0 −Oτ(v)+s0) + ((τ(v) + s0)− s0)u+ r)

+Oτ(v)+s − (tvO + r +Os0)

= R(v +Os0 − (τ(v)vO +Os0)) + τ(v)(u+ vO)

= R(v − τ(v)vO) + τ(v)(u+ vO)

= R(PO(v)) + τ(v)(u+ vO). (4.2)

Thus the map v 7→ ψ(v +Os0)− ψ(Os0) is linear, so ψ is affine. Similarly, we calculate

τ(ψ(x1)− ψ(x2)) = τ(tvO +OπG(x1))− τ(tvO +OπG(x2))

= t+ πG(x1)− (t+ πG(x2))

= τ(x1 − x2).

So GM2 is satisfied. Next, for s0 ∈ IG, consider y1, y2 ∈E(s). We compute

ψ(y1)− ψ(y2) = tvO +R(y1 −Os0) + (s0 − s0)u+ r +Os0

−(tvO +R(y2 −Os0) + (s0 − s0)u+ r +Os0)

= R(y1 − y2)

So ψ satisfies GM3. Next we show that (R, r, u, t) = (Rψ, rψ, uψ, tψ). By restricting ψ toE(s0) we get

(ψ|E(s0))(x) = tvO + (R(x−Os0) + r) +Os0

= (R(x−Os0) + r) +Ot+s0

However, the definition of Rψ and rψ gives

R(x−Os0) + r = Rψ(x−Os0) + rψ,O

28 A. Bhand

for each x ∈ E(s0). Taking x = Os0 gives r = rψ,O from which it follows that R = Rψ.Also, for x ∈E(s0), we have

ψIG(s0) = πG(ψ(x)) = t+ s0.

From Lemma 2.18, it follows that t = tψ. From (4.2) we also have

PO(ψV (vO)) = R(PO(vO)) + τ(vO)u = u

using the fact that PO(vO) = 0. This shows that u = uψ,O. We have now shown that forevery (R, r, u, t) ∈ Gal(Gcan) there is a Galilean mapping ψ such that ιOs0 (ψ) = (R, r, u, t).Thus we have shown that ιOs0 is surjective. Next we show that it is injective. For this, let

ψ ∈ Gal(G) be such that, for x ∈E,

ψ(x) = tvO + (R(x−Os0) + (πG(x)− s0)u+ r) +OπG(x).

that is, supposeιOs0 (ψ) = (R, r, u, t) = ιOs0 (ψ).

We shall show that ψ = ψ. Since ψ(Os0) = ψ(Os0), using (4.2) this will follow if we canshow that ψV = ψV . As in (i) we note that V ' ker(τ) ⊕ R and the pre-image of (u, t)under this isomorphism is u+ tvO. We also write

ψV (u+ tvO) = A11(u) +A12(t) + (A21(u) +A22(t))vO

for linear mappings A11 : ker(τ) → ker(τ), A12 : R → ker(τ), A21 : ker(τ) → R, andA22 : R → R. The property GM2 of Galilean mappings implies that ψV has ker(τ) as aninvariant subspace. Thus A21 = 0. We next calculate

τ(ψV (tvO)) = τ(ψ(tvO +Os0))− τ(ψ(Os0))

= τ(tvO +Os0)− τ(Os0) = t.

This gives A22 = t. With t = 0 property GM3 implies that A11 ∈ O(ker(τ)). Thus we have

ψV (u+ tvO) = R(u) + t(u+ vO) (4.3)

for some R ∈ O(ker(τ)) and u ∈ ker(τ). Since ψ|E(s) = ψ|E(s) we have

ψV (u) = ψ(u+Os0)− ψ(Os0) = Rψ(u),

giving R = R. From (4.2) we also have

PO(ψV (vO) = u

from which we get u = u. This shows that ψV = ψV thus showing that if ψ1, ψ2 ∈ Gal(G)satisfy ιOs0 (ψ1) = ιOs0 (ψ2), then ψ1 = ψ2. Therefore ιOs0 is injective.


Finally we show that ιOs0 is a homomorphism. We let ψ1, ψ2 ∈ Gal(G) and denoteιO(ψi) = (Ri, ri, ui, ti), i = 1, 2. We also let ιOs0 (ψ1 ψ2) = (R12, r12, u12, t12). First, wecompute

(ψ1 ψ2)V (v) = (ψ1 ψ2)(v +Os0)− (ψ1 ψ2)(Os0)

= ψ1(ψ2(v +Os0))− ψ1(ψ2(Os0))

= ψ1(tvO +R2(v +Os0 −Oτ(v)+s0) + τ(v)u2 + r2 +Oτ(v)+s0)

− ψ1(tvO + r +Os0)

= ψ1((tvO + r) +R2(PO(v)) + τ(v)(u2 + vO) +Os0)

− ψ1(tvO + r2 +Os0)

= ψ1,V (R2(PO(v)) + τ(v)(u2 + vO))

= ψ1,V (ψ2,V (v))

From this we deduce that

R12(u) + t(u12 + vO) = R1 R2(u) + t(u1 +R1(u2) + vO)

for each (u, t) ∈ ker(τ)× R. Thus we have

R12 = R1 R2, u12 = u1 +R1(u2).

Next we haveψ1 ψ2(Os0) = r12 +Ot12+s0 . (4.4)

Also,ψ2(Os0) = r2 +Ot2+s0

Therefore

ψ1 ψ2(Os0) = ψ1(r2 +Ot2+s0)

= R1(r2 +Ot2+s0 −Ot2+s0) + t2u1 + r1 +Os0+t2+t1

= R1r2 + t2u1 + r1 +Os0+t1+t2

comparing this to (4.2.3) we get

r12 = R1r2 + t2u1 + r1, t12 = t1 + t2.

Thus we have shown that the group action defined on Gal(Gcan) agrees with that on Gal(G)under the bijection ιOs0 .

Next, given a motion Ψ in a Galilean spacetime G = (E, V, g, τ), we explore how thequantities rΨ,O and uΨ,O change when we use a different observer.

30 A. Bhand

4.6 Proposition: Let Ψ be a motion in a Galilean spacetime G = (E, V, g, τ) and let O and O

be two observers. For s0 ∈ IG, , let (RΨ(t), rΨ,O(t), uΨ,O(t), tΨ) and (RΨ(t), rΨ,O(t), uΨ,O(t), tΨ)

be the images of Ψt under the isomorphisms between Gal(G) and Gal(Gcan) induced by O

and O respectively. Then,

rΨ,O(t) = rΨ,O(t) + (idker(τ) −RΨ(t))(Os0 − Os0) + tΨ(vO − vO)

uΨ,O(t) = uΨ,O(t) + (idker(τ) −RΨ(t))(vO − vO)

Proof: For x ∈E(s0) we have

Ψt(x) = RΨ(t)(x− Os0) + rΨ,O(t) + Os0+tΨ

= RΨ(t)(x−Os0) + rΨ,O(t) +Os0+tΨ

This means

RΨ(t)(x− Os0)−RΨ(t)(x−Os0) + (rΨ,O(t)− rΨ,O(t)) + (OtΨ+s0 −OtΨ+s0) = 0.

for all x ∈E(s0). Letting x = Os0 we get

rΨ,O(t) = rΨ,O(t) +RΨ(t)(Os0 −Os0) + (OtΨ+s0 − OtΨ+s0)

= rΨ,O(t) +RΨ(t)(Os0 −Os0) + (Os0 − Os0) + t(vO − vO)

which is the required expression.Now, by definition

uΨ,O(t) = PO

(Ψt,V (vO

))

= Ψt,V (vO

)− vO.

Similarly,uΨ,O(t) = Ψt,V (vO)− vO.

Therefore,

uΨ,O(t)− uΨ,O(t) = Ψt,V (vO− vO) + vO − vO

= RΨ(t)(vO− vO) + (vO − vO)

= (idker(τ) −RΨ(t))(vO − vO)

which gives the expression for uΨ,O.

Subgroups of the abstract Galilean group. In this section we wish to characterize thevarious subgroups of the abstract Galilean group Gal(G). First of all we recall that theaction of V on E is affine so GM1 is satisfied. It is easy to see that GM2 and GM3 are alsosatisfied and that V is a subgroup of Gal(G). The following result is easily verified.

4.7 Lemma: Let G = (E, V, g, τ) be a Galilean spacetime and let Gal(G) be the Galileangroup. The following are subgroups of Gal(G):

(i) The set Gal0(G) := ψ ∈ Gal(G) : ψIG = idIG(ii) The set Galan(G) := ψ ∈ Gal(G) : ψV |ker(τ) = idker(τ)

(iii) The set Gal0b(G) := Galan(G) ∩Gal0(G)

(iv) V .


4.8 Remarks: (i) Following the discussion on the subgroups of the canonical Galileangroup, we call Gal0(G) the isochronous Galilean group.

(ii) The notation used here is not altogether obvious. It is meant to be consistent with thethe one used for the subgroups of the canonical Galilean group. Galan(G) thereforedenotes the anisotropic Galilean group.

The following result shows that these subgroups have an additional property.

4.9 Lemma: Let G = (E, V, g, τ) be a Galilean spacetime and let Gal(G) be the Galileangroup. The subgroups V,Gal0(G) and Galan(G) and are normal subgroups of Gal(G).

Proof: First, let us show that V is a normal subgroup. For this, let ψ ∈ Gal(G) and φ ∈ V .We wish to show that ψ φ ψ−1 ∈ V . Choose x ∈E. Since V acts on E by translations,there exists some vψ ∈ V such that

ψ φ ψ−1(x) = ψ (vψ + ψ−1(x)).

This means that

ψ φ ψ−1(x)− x = ψ φ ψ−1(x)− ψ(ψ−1(x))

= ψ(vψ + ψ−1(x))− ψ(ψ−1(x))

= ψV (vψ)

which implies that ψ φ ψ−1(x) = x+ ψV (vψ), thus showing that ψ φ ψ−1 ∈ V . Thisis what we wanted to show.

Next, consider x ∈E. It can be shown that for ψ1, ψ2 ∈ Gal(G), we have (ψ1 ψ2)IG =ψ1,IGψ2,IG . To see this we recall that for any ψ ∈ Gal(G), the definition of ψIG gives

(ψ1 ψ2)IG(πG(x)) = πG(ψ1(ψ2(x))) = ψ1,IG(πG(ψ2(x)))

= ψ1,IG(ψ2,IG(πG(x))

which is what we want. Next, for φ ∈ Gal0(G), we have

(ψ φ ψ−1)IG = ψIG ψ−1IG

= idIG

This shows that ψ φ ψ−1 ∈ Gal0(G) thus showing that Gal0(G) is a normal subgroup.Finally, consider φ ∈ Galan(G). We need only show that (ψ φψ−1)V = idker(τ). However,this follows since (ψ1 ψ2)V = ψ1,V ψ2,V , ∀ψ1, ψ2 ∈ Gal(G) and the proof is complete.

4.10 Corollary: Let G = (E, V, g, τ) be a Galilean spacetime. The groups ker(τ) andGal0b(G) are normal subgroups of Gal(G).

Proof: We claim that ker(τ) = Gal0(G) ∩ V . To see this, let φ ∈ Gal0(G) ∩ V . This meansthat for every x ∈E, there exists some vφ ∈ V such that φ(x) = x+ vφ and φIG = idIG . Bydefinition, this means that

φIG(πG(x)) = πG(φ(x))

= πG(x) + τ(vφ) = πG(x)

which gives τ(vφ) = 0. The result now follows from the fact that the intersection of twonormal subgroups is itself normal.

32 A. Bhand

4.11 Remarks: (i) The subgroups defined in the above propositions are natural sub-groups of Gal(G) because they do not depend on any particular choice of points inE or vectors in V . By making these choices, additional subgroups of Gal(G) can bedefined. We shall, however, not deal with this situation.

(ii) Similar to the case of the isochronous canonical Galilean group Gal0(Gcan), the sub-group Gal0(G) is a normal subgroup of Gal(G).

Next, we look at the quotients of Gal(G) by these normal subgroups. We recall that thequotient G/H of a group G by its normal subgroup H is also a group.

4.12 Proposition: Let G = (E, V, g, τ) be a Galilean spacetime and let the subgroups Gal0(G),Galan(G)and Gal0b(G) be as before. The following statements hold.

(i) Gal(G)/Gal0(G) ' R.

(ii) Gal(G)/V = ψV : ψ ∈ Gal(G).(iii) Gal(G)/Gal0b(G) ' O(ker(τ))× R.

(iv) Gal(G)/Galan(G) ' O(ker(τ)).

Proof: (i) Let ψ ∈ Gal(G). Recall that there exists a unique tψ ∈ R such that ψIG(s) =s+ tψ. Now, the orbit of ψ is given by

orbGal0(G)(ψ) = ψ φ : φ ∈ Gal0(G).

Since Gal0(G) consists of “instant-preserving” Galilean maps, we know that for any ψ ∈orbGal0(G)(ψ), we have ψIG = ψIG . We therefore have the canonical projection

πGal0(G) : Gal(G) → Gal(G)/Gal0(G)

ψ 7→ tψ

This proves the first part.(ii) For ψ ∈ Gal(G) the orbit of ψ under V is

orbV (ψ) = ψ φ : φ ∈ V

Notice that for φ ∈ V ,

φV (w) = φ(w + x)− φ(x) = (w + x) + vφ − (x+ vφ) = w, ∀w ∈ V.

That is, φV = idV . For any ψ ∈ orbV (ψ) we therefore have ψV = ψV . Thus, we have thecanonical projection

πV : Gal(G) → Gal(G)/V

ψ 7→ ψV

which is what we wanted to show.(iii) In a similar manner, for ψ ∈ Gal(G), we form the orbit orbGal0b(G)(ψ) and observe

that for any ψ ∈ orbGal0b(G)(ψ), we have Rψ = Rψ and tψ = tψ. The canonical projectionis thus written as

πGal0b(G) : Gal(G) → Gal(G)/Gal0(G)

ψ 7→ (Rψ, tψ)


as desired.(v) It is easy to see that given ψ ∈ Gal(G), for any ψ ∈ orbGalan(G)(ψ), we have Rψ = Rψ

and therefore the projection map is given by

πGalan(G) : Gal(G) → Gal(G)/Galan(G)

ψ 7→ Rψ

This completes the proof.

4.3. Total velocities. Consider a rigid motion Ψ in a Galilean spacetime G = (E, V, g, τ).We first notice that Ψt ∈ TΨtGal(G). Therefore TΨtLΨ−1

t(Ψt) ∈ gal(G) and TΨtRΨ−1

t(Ψt) ∈

gal(G) (here LΨt and RΨt are as usual the left and right translation maps on Gal(G)). Weshall call these respectively, the total body velocity and the total spatial velocity of therigid motion Ψ. The following result provides the decomposition of total spatial and bodyvelocities in the presence of an observer.

4.13 Proposition: Let G = (E, V, g, τ) be a Galilean spacetime with O an observer and Ψa rigid motion. For s0 ∈ IG, let ιOs0 (Ψt) = (RΨ(t), rΨ,O(t), uΨ,O(t), t) Then, the followingstatements hold.

(i) The image of TΨtLΨ−1t

(Ψt) ∈ gal(G) under the isomorphism of the Lie algebras in-

duced by ιOs0 is

(ΩΨ(t), VΨ,O(t)−R−1Ψ (t)uΨ,O(t), R−1

Ψ (t)uΨ,O(t), 1) ∈ gal(Gcan).

(ii) The image of TΨt(RΨ−1t

Ψt) ∈ gal(G) under the isomorphism of Lie algebra induced

by ιOs0 is

(ωΨ(t), vΨ,O(t)− t(uΨ,O(t)+uΨ,O(t)×ωΨ(t)), uΨ,O(t)+uΨ,O(t)×ωΨ(t), 1) ∈ gal(Gcan).

where VΨ,O(t) = R−1Ψ (t)rΨ,O(t) and vΨ,O(t) = rΨ,O(t) + rΨ,O(t)× ωΨ(t).

Proof: We use the representation of Gal(Gcan) in the vector space W as described earlier.We then compute R u r

0 1 t0 0 1

−1

=

R−1 −R−1u R−1(tu− r)0 1 −t0 0 1

.With this expression both parts follow from direct computation.

We call the components in (i) as respectively the total body angular velocity, the total bodylinear velocity and the total body acceleration and denote them by Ωtotal

Ψ , V totalΨ,O and Atotal

Ψ,O

respectively. Similarly, we call the components in (ii) as respectively the total spatial angularvelocity, the total spatial linear velocity and the total spatial acceleration and represent themby ωtotal

Ψ , vtotalΨ,O and atotal

Ψ,O respectively . The next result shows how these velocities changewith the observer.

34 A. Bhand

4.14 Proposition: Let Ψ be a rigid motion in a Galilean spacetime G = (E, V, g, τ) and letO and O be two observers. For fixed s0 ∈ IG, let VΨ,O, VΨ,O, vΨ,O and vΨ,O be as defined inProposition 4.13. Then,

vΨ,O(t) = vΨ,O(t) + (Ot+s0 − Ot+s0)× ωΨ(t) + vO − vOVΨ,O(t) = VΨ,O(t) + (Os0 − Os0)× ΩΨ(t) +R−1

Ψ (t)(vO − vO)

vtotalΨ,O

(t) = vtotalΨ,O (t) + (Os0 − Os0)× ωΨ(t) + vO − vO

V totalΨ,O

(t) = V totalΨ,O (t) + (Os0 − Os0)× ΩΨ(t) + vO − vO

atotalΨ,O

(t) = atotalΨ,O (t) + (vO − vO)× ωΨ(t)

AtotalΨ,O

(t) = AtotalΨ,O (t) + (vO − vO)× ΩΨ(t)

Proof:

vΨ,O(t) = rΨ,O(t) + rΨ,O(t)× ωΨ(t)

= rΨ,O(t) + RΨ(t)(Os0 −Os0) + vO − vO + rΨ,O(t)× ωΨ(t)

+ ((RΨ(t)− idker(τ)(Os0 −Os0))× ωΨ(t) + t(vO − vO)× ωΨ(t)

= vΨ,O(t) + ωΨ(t) RΨ(t)(Os0 −Os0) + (RΨ(t)(Os0 −Os0))× ωΨ(t)

+ (Os0 − Os0)× ωΨ(t) + t(vO − vO)× ωΨ(t)

= vΨ,O(t) + (Ot+s0 − Ot+s0)× ωΨ(t) + vO − vONext,

VΨ,O(t) = R−1Ψ (t)(rΨ,O(t))

= R−1Ψ (t)(rΨ,O(t) + RΨ(t)(Os0 −Os0) + (vO − vO)

= VΨ,O(t) + ΩΨ(t)× (Os0 −Os0) +R−1(t)(vO − vO),

which is the required expression. Now,

vtotalΨ,O

(t) = vΨ,O(t)− t( ˙uΨ,O(t) + uΨ,O × ωΨ(t))

= vΨ,O(t)− t(uΨ,O(t) + uΨ,O(t)× ωΨ(t))

+ tRΨ(t)(vO − vO)− t(idker(τ) −RΨ(t))(vO − vO)× ωΨ(t)

= vtotalΨ,O (t) + (Ot+s0 − Ot+s0)× ωΨ(t) + vO − vO+

− t(vO − vO)× ωΨ(t)

= vtotalΨ,O (t) + (Os0 − Os0)× ωΨ(t) + vO − vO

which gives us what we wanted to show. Next,

V totalΨ,O

(t) = VΨ,O(t)−R−1Ψ (t)uΨ,O(t)

= VΨ,O(t)−R−1Ψ (t)

(uΨ,O(t) + (idker(τ) −RΨ(t))(vO − vO)

)= VΨ,O(t) + (Os0 − Os0)× ΩΨ(t) +R−1

Ψ (t)(vO − vO)−R−1Ψ (t)uΨ,O(t)

+ (idker(τ) −R−1Ψ (t))(vO − vO)

= V totalΨ,O (t) + (Os0 − Os0)× ΩΨ(t) + vO − vO


Finally, we compute the acceleration,

atotalΨ,O

(t) = uΨ,O(t) + uΨ,O(t)× ωΨ(t)

= uΨ(t)− RΨ(t)(vO − vO) + uΨ,O(t)× ωΨ(t)

(idker(τ) −RΨ(t))(vO − vO)× ωΨ(t)

= atotalΨ,O (t)− ωΨ(t)RΨ(t)(vO − vO) + (idker(τ) −RΨ(t))(vO − vO)× ωΨ(t)

= atotalΨ,O (t) + (vO − vO)× ωΨ(t)

and

AtotalΨ,O

(t) = R−1Ψ (t)( ˙uΨ,O(t))

= R−1Ψ (t)

(uΨ,O(t)− RΨ(t)(vO − vO)

)= Atotal

Ψ,O (t)− Ω(t)(vO − vO).

This finishes the proof.

5. Dynamics of rigid bodies in the presence of an observer

In Chapter 3, we formulated rigid body dynamics in an observer independent way. Inthis chapter, we shall explore the effect of introducing an observer to this formulation. InSection 5.1, we show that in the presence of an observer the body and spatial velocitiesdefined in Chapter 3, project to the corresponding total velocities. In the next section, weshow that the momenta also project to the well known quantities in the presence of theobserver. Finally, in Section 5.3, we illustrate how we recover the familiar Euler equationsfor a rigid body.

5.1. Angular and linear velocities. In this section we shall investigate the effect of anobserver on the angular and linear body and spatial velocities defined in Chapter 3.

Angular velocities. Recall that given a rigid motion Ψ in a Galilean spacetime G =(E, V, g, τ), the spatial angular velocity is defined as the map ω : R→ ker(τ) satisfying

ω(t) = RΨ(t)R−1Ψ (t)

and the body angular velocity is the map Ω : R→ ker(τ) satisfying

Ω(t) = R−1Ψ (t)RΨ(t)

It is clear that angular velocities remain the same even in the presence of an observer.

Linear velocities. Recall that given a motion Ψ in a Galilean spacetime, the body linearvelocity is the map V b

Ψ : E × R→ VG given by

V bΨ(x, t) = Ψ−1

t,V (d

dt(Ψt(x))

36 A. Bhand

and the spatial linear velocity is the map vsΨ : E × R→ VG given by

vsΨ(x, t) = −Ψt,V (d

dt(Ψ−1

t (x)))

Let’s see what these velocities look like in the presence of an observer. We look at bodylinear velocity first.

5.1 Proposition: Let Ψ be a motion in a Galilean spacetime G = (E, V, g, τ) and let O bean observer. Then,

PO(V bΨ(x, t)) = V total

Ψ,O (t)

Proof: We know that for each instant s0 ∈ IG, there exists an isomorphism ιOs0 such thatfor a motion Ψ, we have

ιOs0 (Ψt) = (RΨ(t), rΨ,O(t), uΨ,O(t), t).

Also,Ψt(x) = RΨ(t)(x−OπG(x)) + (πG(x)− s0)uΨ,O(t) + rΨ,O(t) +OπG(x)+t

Now, for x ∈ O, we haveΨt(x) = rΨ,O(t) +OπG(x)+t

so we haved

dt(Ψt(x)) = rΨ,O(t) + vO

Next,

Ψ−1t,V (

d

dt(Ψt(x))) = Ψ−1

t,V (rΨ,O(t) + vO)

= Ψ−1t (rΨ,O(t) + vO +Os0)−Ψ−1

t (Os0)

= R−1Ψ (t)(rΨ,O(t) + vO +Os0 −O1+s0)−R−1

Ψ (t)uΨ,O(t)

+R−1Ψ (t)(tuΨ,O(t)− rΨ,O(t)) + (1− t)vO +Os0

−R−1Ψ (t)(tuΨ,O(t)− rΨ,O(t)) + tvO −Os0

= R−1Ψ (t)rΨ,O(t)−R−1

Ψ (t)uΨ,O(t) + vO

= VΨ,O(t)−R−1Ψ (t)uΨ,O(t) + vO

From this, the result follows.

Let’s look at spatial linear velocity now.

5.2 Proposition: Let Ψ be a motion in a Galilean spacetime G = (E, V, g, τ) and let O bean observer. Then,

PO(vsΨ(x, t)) = vtotalΨ,O (t)

Proof: For x ∈ O, we compute,

Ψ−1t (x) = R−1

Ψ (t)(tuΨ,O(t)− rΨ,O(t))− tvO +Os0


Therefore,

d

dt(Ψ−1

t (x)) = R−1Ψ (t)(uΨ,O(t) + tuΨ,O(t)− rΨ,O(t))

+ (−R−1Ψ (t)RΨ(t)R−1

Ψ (t))(tuΨ,O(t)− rΨ,O(t)))− vO= R−1

Ψ (t)(uΨ,O(t) + tuΨ,O(t)− rΨ,O(t))

−R−1Ψ (t)ωΨ(t)(tuΨ,O(t)− rΨ,O(t))− vO

= R−1Ψ (t)uΨ,O(t) +R−1

Ψ (t)[t(uΨ,O(t) + uΨ(t)× ωΨ(t))]

−R−1Ψ (t)(rΨ,O(t) + rΨ,O(t)× ωΨ(t))− vO

= R−1Ψ (t)uΨ(t)−R−1

Ψ (t)(vΨ,O(t)− t(uΨ,O(t) + uΨ,O(t)× ωΨ(t)))− vO

Let’s call the last expression as Ψ−1t (x). Now, we compute

−Ψt,V (d

dt(Ψ−1

t (x)) = Ψt(−Ψ−1t (x) +Os0)−Ψt(Os0)

= RΨ(t)(−Ψ−1t (x) +Os0 −O1+s0) + uΨ,O(t) + rΨ,O(t) + tvO

+O1+s0 − rΨ,O(t)− tvO −Os0

= RΨ(t)(−Ψ−1t (x)− vO) + uΨ,O(t) + vO

= RΨ(t)(R−1

Ψ (t)(vΨ,O(t)− t(uΨ,O(t) + uΨ,O(t)× ωΨ(t)))

− uΨ,O(t) + uΨ,O(t) + vO

= vΨ,O(t)− t(uΨ,O(t) + uΨ,O(t)× ωΨ(t)) + vO

From this the result follows.

We notice that in the presence of an observer, the spatial linear and body linear velocitiesproject onto the total spatial linear and total body linear velocities respectively.

5.2. Spatial and body momenta. We let G = (E, V, g, τ) be a Galilean spacetime with Ψa rigid motion, O an observer and (B, µ) a rigid body with B ∈E(s0). We wish to see howour definitions of spatial and body momenta look like when we have an observer O. In sucha case, we have the following result.

5.3 Proposition: Let G = (E, V, g, τ) be a Galilean spacetime with Ψ a rigid motion, O anobserver and (B, µ) a rigid body with B ∈ E(s0)and let mΨ,B, `Ψ,B,MΨ,B and LΨ,B be asdefined in Section 3.4. For s0 ∈ IG, let ιOs0 (Ψt) = (RΨ(t), rΨ,O(t), uΨ,O(t), t). Then thefollowing statements hold.

(i) PO(mΨ,B(t)) = µ(B)rΨ,O(t)

(ii) PO(`Ψ,B(t)) = Ic(R−1Ψ (t)ωΨ(t))

(iii) PO(MΨ,B(t)) = µ(B)V totalΨ,O (t)

(iv) PO(LΨ,B(t)) = R−1Ψ (t)(Ic(R−1

Ψ (t)ωΨ(t))

38 A. Bhand

Proof: (i) We compute,

PO(mΨ,B(t)) = µ(B)PO(d

dt(Ψt(xc))

= µ(B)PO(rΨ,O(t) + vO)

= µ(B)rΨ,O(t)

where we have used the computations carried out in Proposition 5.1.(ii) and (iv) are just definitions. Since these quantities reside in ker(τ), the projection

PO is the identity map on ker(τ).To see (iii), we compute

PO(MΨ,B(t)) = µ(B)PO(Ψ−1t,V (

d

dt(Ψt(xc)))

= µ(B)PO(Ψ−1t,V (rΨ,O(t) + vO))

= µ(B)PO(R−1Ψ (t)rΨ,O(t)−R−1

Ψ (t)uΨ,O(t) + vO)

= µ(B)V totalΨ,O (t)

as desired.

5.3. Euler equations for a rigid body. Finally, we look at the Galilean Euler equations inthe presence of an observer. We shall see that the equations given in Section 3.4 reduce tothe familiar equations of motion for a rigid body.

5.4 Proposition: Let Ψ be a rigid motion in a Galilean spacetime G = (E, V, g, τ) and(B, µ) a rigid body with B ⊂ E(s0) and O be an observer. The Galilean–Euler equationsare given by

RΨ(t) = RΨ(t) ΩΨ(t)

rΨ,O(t) = 0

Ic(ΩΨ(t)) = Ic(ΩΨ(t))× ΩΨ(t)

Proof: The first equation is simply a definition. It is also independent of the observer soremains the same. Next, using the computations carried out in Proposition 5.1, we have

xc(t) = rΨ,O(t) + vO

from which the second equation follows. The third equation is the usual Galilean Eulerequation for the body angular momentum and remains unchanged in the presence of anobserver.

Discussion. The Galilean–Euler equations given in Proposition 5.4 coincide with the usualEuler equations for a rigid body, i.e. they generate the correct motion of the rigid body.It is interesting to note that in the presence of an observer, the expression for the spatialangular momentum computed using our definition differs from the classical one. However,the equations of motion for the body are the same in both cases. We note that the Galileanboosts take no part in the motion and can therefore take arbitrary values. Here too, we arenot able to recover the differential equation for the body linear momentum. We would liketo get a better understanding of why this is the case.


6. Conclusions and future scope

In this chapter we summarize the results of our investigation and outline the scope offuture work.

6.1. Conclusions. This project aimed at studying rigid body mechanics in abstract Galileanspacetimes. This is a considerable generalization of the classical or “canonical” setup ofstudying the motion of a rigid body in which the motion takes place in R3 and the rigid mo-tions essentially constitute SE(3). In such a treatment, it is not apparent to which spacesthe velocities and the momenta rightfully belong, since many identifications of spaces aremade. Our treatment sorts this problem out in a natural way. We defined rigid motions andthe fundamental quantities associated with it. We then formulated the dynamics of a rigidbody mechanics in an observer-independent manner. Our treatment is more general thanthe classical one on several counts. First of all, we have treated body and spatial momentaassociated with a rigid body in motion as fundamental quantities which can be definedwithout requiring additional structure. Equations of motion for a rigid body undergoingrigid motion are derived in this setting. Although we do not learn anything new by lookingat these Galilean-Euler equations in the abstract setting, we do come to an understandingof the role played by the Galilean structure when we study them in the presence of anobserver.

Next, the structure of the Galilean group is made clear in the presence of an observer.It is shown that an observes induces an isomorphism between the Galilean group and thecanonical Galilean group. Here, we have considered rigid motions with arbitrary Galileanboosts. We have also studied the notion of a rigid body carefully and discussed in detail, theproperties of the inertia tensor. The inertia tensor is an important object in the descriptionof the dynamics of a rigid body. It is then shown that we recover the familiar quantitiesand equations for the motion of a rigid body in a Galilean spacetime in the presence ofan observer. Although the observer plays no part in the motion of the rigid body itself, itfacilitates the study of the equations of motion.

We now summarize our results below.

(1) We have provided observer-independent definitions for spatial and body angular ve-locities (Section 3.2) and spatial and body linear velocities (Section 3.2)

(2) we have presented a definition of a rigid body in Section 3.3 and investigated theproperties of the inertia tensor of the rigid body.

(3) We have provided observer-independent definitions for spatial linear and angular mo-menta (Section 3.4) and body linear and angular momenta (Section 3.4).

(4) We have derived observer-independent equations of motion for a rigid body in anabstract Galilean spacetime (Section 3.4). These equations are considerably generalthat the Euler equations for a rigid body derived in the classical literature.

(5) We have shown that in the presence of an observer, the group Gal(G) decomposesinto rotations, translations, Galilean boosts and temporal origin shifts (Proposition4.5).

40 A. Bhand

(6) In Section 4.2, we have characterized the various subgroup of Gal(G) and shown howthey are related to the subgroups of Gal(Gcan). In Section 4.3, we have introducedtotal velocities associated with a rigid motion.

(7) In Section 5.1, we have shown that in the presence of an observer, the body and spatialvelocities project to the corresponding total velocities. The momenta also project ina similar manner.

(8) Finally, in Section 5.3, we show that Galilean-Euler equations derived in section 3.4reduce to the usual Euler equations for a rigid body.

6.2. Future scope. After bringing our work to this point, many questions still remainunanswered. In our treatment, we have not fully understood the role of velocity boosts inthe motion of the rigid body and the meaning of the fictitious accelerations that arise whenwe consider total velocities of Gal(G) is also not apparent. We would also like to come toa complete understanding of the physical implications of this formulation.

In Proposition 4.5, we have constructed an isomorphism ιOs0 between Gal(G) andGal(Gcan) induced by an observer O. We believe that this is the most general isomor-phism between these groups in the following sense. Given any isomorphism I betweenGal(G) and Gal(Gcan), there exists an observer O and an instant s0 such that I = ι

Os0. Our

efforts are currently directed towards showing that this is indeed true.It is interesting to note that although our expressions for body and spatial angular

momentum in the presence of an observer are different from those in the literature, theyproduce the same equations of motion. It would therefore be worthwhile to compare variousquantities in the usual setting with those in ours.

A more advanced extension of the problem is to consider relativistic mechanics in theframework of Minkowski spacetimes. A more general treatment of mechanics can be givenin this setting.

Acknowledgment

I am grateful to my supervisor Dr. Andrew Lewis for introducing me to the beautiful worldof Galilean spacetimes. I wish to thank him for providing constant support and motivationand most importantly, for being always accessible for discussion.

References

Arnol′d, V. I. [1978] Mathematical Methods of Classical Mechanics, translated by K. Vogt-mann and A. Weinstein, number 60 in Graduate Texts in Mathematics, Springer-Verlag:New York/Heidelberg/Berlin, isbn: 0-387-96890-3, New edition: [Arnol′d 1989].

— [1989] Mathematical Methods of Classical Mechanics, translated by K. Vogtmann and A.Weinstein, 2nd edition, number 60 in Graduate Texts in Mathematics, Springer-Verlag:New York/Heidelberg/Berlin, isbn: 978-0-387-96890-2, First edition: [Arnol′d 1978].

Artz, R. E. [1981] Classical mechanics in Galilean space-time, Foundations of Physics, AnInternational Journal Devoted to the Conceptual Bases and Fundamental Theories ofModern Physics, 11(9-10), pages 679–697, issn: 0015-9018, doi: 10.1007/BF00726944.

https://doi.org/10.1007/BF00726944


Berger, M. [1987] Geometry I, Universitext, Springer-Verlag: New York/Heidelberg/Berlin,isbn: 978-3-540-11658-5.

Levy-Leblond, J.-M. [1971] Galilei group and Galilean invariance, in Group Theory and itsApplications, volume 2, edited by E. M. Loebl, pages 221–299, Academic Press: NewYork, NY.

Souriau, J.-M. [1997] Structure of Dynamical Systems, A Symplectic View of Physics, trans-lated by C. H. Cushman-de Vries, number 149 in Progress in Mathematics, Birkhauser:Boston/Basel/Stuttgart, isbn: 978-0-8176-3695-1.

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