In a few previous tutorials, we briefly talked about physical properties, without really explaining the mathematics behind those concepts. This tutorial tries to remedy that situation. In what follows, space is three-dimensional and euclidean, and time is one-dimensional.

The goal of this tutorial is to define and investigate the structure of space-time and the corresponding group of transformations and to analyze the constraints imposed by Galileo's principle of relativity on Newton's equation of motion.


Galileo's Principle of Relativity

In the tutorial about Newton's three laws of motion, Galileo's principle of relativity was briefly mentioned when the concept of an inertial frame of reference was explained. Galileo stated that there exist coordinate systems, called inertial frames of reference, possessing the following two properties:

  • All the laws of nature are the same in all inertial coordinate systems, at all moments of time.

  • All coordinate systems in uniform rectilinear motion with respect to an inertial one, are themselves inertial.

In other words, as seen in the example in the previous tutorial, if a coordinate system attached to the earth is inertial, then it is impossible to detect the motion of the train, which is moving uniformly in a straight line, by experiments conducted entirely inside the cars of the train.


Newton's Principle of Determinacy

In this tutorial about kinematics it was mentioned, that Newton determined that the initial state of a mechanical system, i.e. knowing the initial positions and velocities of the objects in the system, uniquely determine the motion of the system.


In what follows, we try to bring those two concepts together. To do so, we must first define the space we live in.


A Space of Time

Let $\mathbb{R}^n$ denote the standard $n$-dimensional vector space, and $\mathbb{A}^n$ the $n$-dimensional affine space, distinguished from $\mathbb{R}^n$ by the fact that there is no fixed origin. As a group, $\mathbb{R}^n$ acts on $\mathbb{A}^n$ as the group of translations, i.e. as mappings of the form $\mathbb{R}^n \times \mathbb{A}^n \to \mathbb{A}^n$, $(v, a) \mapsto v+a$. Thus, the sum of two points in the affine space is not defined, but their difference is defined to be an element of $\mathbb{R}^n$. More precisely, for every points $a, b \in \mathbb{A}^n$, there exists a unique $v \in \mathbb{R}^n$, denoted $b-a$, such that $b=v+a$.

In other words, an affine space is a principal homogenous space, or a torsor, for the action of the additive group of the vector space as defined above, which means that the stabilizer subgroup of every point is trivial. Equivalenty, this means that the vector space acts freely and transitively on the affine space.

Further let the vector space be equipped with a positive definite symmetric bilinear form $b: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$, called a scalar product. The scalar product defines a so-called euclidean structure on the vector space $\mathbb{R}^n$ and enables the definition of the distance between two points $a,b \in \mathbb{A}^n$ in the corresponding affine space: \[d(a,b) := \lVert b-a \rVert := \sqrt{b(b-a,b-a)}.\]An affine space with such a distance function is called an Euclidean space.


Galilean Structure

To model physical phenomena, one often gives them a home in a Galilean space-time structure. This structure consists of three components.

The Universe

The actual universe is a four-dimensional affine space $\mathbb{A}^4$ - note that for many centuries, the universe was given by a linear space, the geocentric system of the universe. The points of $\mathbb{A}^n$ are called events. It is clear that the translation space of the universe is the vector space $\mathbb{R}^4$.

Time

Time is defined as a linear map $t: \mathbb{R}^4 \to \mathbb{R}$, from the translation space of the affine space to the so-called time-axis. The time interval between two events $a$ and $b$ is $t(b-a)$. If that number is zero, then the two events are said to happen simultaneously. The set of all events simultaneous to an event $a$ forms a three-dimensional affine subspace $S_a \subset \mathbb{A}^4$.

The time map.
Distance

The distance between simultaneous events $a$ and $b$ is defined as the norm of the vector joining the two events: \[d(a,b) = \lVert b-a \rVert = \sqrt{b(b-a, b-a)},\]where $b$ is a scalar product on $\mathbb{R}^3$. Thus the space of simultaneous events is a three-dimensionsal euclidean space.


An affine space $\mathbb{A}^4$, equipped with a Galilean space-time structure, is called a Galilean space. The Galilean group is now defined as the set of all structure-preserving mappings from the Galilean space into itself. Its elements are called Galilean transformations. It is clear that Galilean transformations are affine maps $\mathbb{A}^4 \to \mathbb{A}^4$, which preserve time intervals as well as the distance between simultaneous events, i.e. they must also preserve the euclidean structure, the so-called metric, and are thus also called isometries.


The standard Galilean coordinate space is the space $\mathbb{R} \times \mathbb{R}^3$, as the euclidean structure of $\mathbb{R}^3$ naturally induces a Galilean structure. Furthermore, every Galilean transformation of $\mathbb{R} \times \mathbb{R}^3$ is a composition of a rotation, a translation and a uniform motion.

A uniform motion with velocity $v$ is a map of the form: \[u: \mathbb{R} \times \mathbb{R}^3 \to \mathbb{R} \times \mathbb{R}^3, \, (t,x) \mapsto (t, x + vt).\]

Translations by a fixed vector $s$ obviously take the following form: \[\tau_{s}: \mathbb{R} \times \mathbb{R}^3 \to \mathbb{R} \times \mathbb{R}^3, \, (t, x) \mapsto (t + \lVert s \rVert, x + s).\]

A rotation of the coordinate axes looks like this: \[\rho: \mathbb{R} \times \mathbb{R}^3 \to \mathbb{R} \times \mathbb{R}^3, \, (t, x) \mapsto (t, Rx),\]where $R: \mathbb{R}^3 \to \mathbb{R}^3$ is an orthogonal transformation.


Since the standard Galilean coordinate space is easy to work with, it would be desirable for all other Galilean spaces to be isomorphic to $\mathbb{R} \times \mathbb{R}^3$, and indeed it is true that all Galilean spaces are isomorphic to each other and in particular, isomorphic to the standard Galilean coordinate space $\mathbb{R} \times \mathbb{R}^3$.


The Equation of Motion

A motion in an $n$-dimensional space is a differentiable map $x: I \subseteq \mathbb{R} \to \mathbb{R}^n$, also sometimes called a position map. The image of $x$ is called a trajectory or curve in $\mathbb{R}^n$.

The derivative \[\dot{x}(t_0) = \dfrac{dx}{dt}\bigg\rvert_{t=t_0}=\lim_{h \to 0}\dfrac{x(t_0+h)-x(t_0)}{h}\]is called the velocity vector at the point $t_0 \in I$.

The second derivative \[\ddot{x}(t_0) = \dfrac{d^2x}{dt}\bigg\rvert_{t=t_0}\]is called the acceleration vector at the point $t_0 \in I$.

In what follows, all functions are continuously differentiable as many times as necessary.


The movement of $N$ objects, seen as points, in a three-dimensional euclidean space, is given by $N$ position maps: $x_i: \mathbb{R} \to \mathbb{R}^3$, $i=1, 2,$ ... $, n$, and their graphs are called world lines. The direct product of $N$ copies of the $\mathbb{R}^3$ is called the configuration space of the system of $N$ points and the motion maps naturally define a new map $x: \mathbb{R} \to \mathbb{R}^{3N}$, of the time axis into the configuration space. This new map is called the motion of a system of $N$ points in the Galilean coordinate system on $\mathbb{R} \times \mathbb{R}^3$.


As seen before, according to Newton, all motions of a system are uniquely determined by their initial positions and velocities. In particular, the initial configuration of a system uniquely determines the acceleration, that is, there exists a function $f: \mathbb{R} \times \mathbb{R}^{3N} \times \mathbb{R}^{3N} \to \mathbb{R}^{3N}$ such, that \[\ddot{x}=f(t,x,\dot{x}).\]This second-order differential equation is called Newton's equation of motion. From a mathematical point of view, knowing $f$ means knowing the entire system of objects in motion.

Note that in the previous tutorials the motion of a single object, thus the case $N=1$, was studied.


Galilean Invariance

Since under Galileo's principle of relativity the transformation of the world lines of a given system define new world lines of the same system with new initial conditions, several conditions are imposed on Newton's equation of motion.

Time Invariance

One of the three possible Galilean Transformations was a time translation, or a uniform movement. Invariance under a time translation means that the laws of nature must be true at any moment in time, which means that if $x=\varphi(t)$ is a solution to the equation of motion, then for any $s \in \mathbb{R}$, so must be $x=\varphi(t+s)$.

It follows that, in an inertial frame of reference, or an inertial coordinate system, the function $f$ does not depend on the time, i.e. Newton's equation of motion can be rewritten as: \[\ddot{x} = f(x, \dot{x}).\]


Translation Invariance

A second possible Galilean Transformation was the simple translation. Invariance with respect to a translation means that the space must be homogeneous. More precisely, this means that if $x_i = \varphi_i(t)$, $i=1, 2$, ... ,$n$, is the motion of a system satisfying Newton's equation of motion, then so must be $\varphi_i(t)+\tau$, where $\tau \in \mathbb{R}^3$ is an arbitrary translation vector.

It follows that the right-hand side of the equation of motion in an inertial frame of reference does not depend on the choice of a coordinate system, i.e. it does not matter which point is taken as the considered origin of the affine space, i.e. it is possible to work with an affine basis. Further using the invariance with respect to time, Newton's equation of motion in an inertial frame of reference can be rewritten as: \[\ddot{x_i} = f_i((x_j - x_k)_{j,k}, (\dot{x}_j - \dot{x}_k)_{j,k}),\]with $i,j,k = 1, 2$, ..., $n$.


Rotational Invariance

The last Galilean transformation to consider is the rotation in three-dimensional space. Invariance with respect to such rotations means that the space is isotropic, which means "uniform in all orientations". The word isotropy is derived from the two Greek words isos (ἴσος), meaning "equal", and tropos (τρόπος), meaning "way", thus, basically speaking, rotational invariance means that "all directions are the same". If $\varphi_i$ is the motion of a system of points satisfying Newton's equation of motion, and $G: \mathbb{R}^3 \to \mathbb{R}^3$ is an orthogonal map, then so must the motion $G \circ \varphi_i$ satisfy Newton's equation of motion.

In other words, invariance with respect to an orthogonal map means just that, invariance under an orthogonal map: \[f(Gx, G\dot{x})=Gf(x,\dot{x}).\]


Law of Inertia

In a mechanical system consisting of only one point, the acceleration in an inertial coordinate system is zero.

As there is only one point in motion, the equation of motion can be written as \[\ddot{x} = f(t,x,\dot{x}),\]with $f: \mathbb{R} \times \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$. Now, as in an inertial frame of reference the equation must be invariant with respect to time, we get $\ddot{x} = f(x, \dot{x}).$ It follows from the invariance with respect to translations, that $\ddot{x}$ does not depend on the position and velocity: $\ddot{x} = f(0,0)$. Thus, with the invariance with respect to orthogonal maps, it follows from $Gf(0,0) = f(G(0),G(0)) = f(0,0)$, that $a = \ddot{x} = 0$.


References

  • Differential Equations, by Prof. Dr. K.-F. Siburg
  • Dynamical Systems, by Prof. Dr. K.-F. Siburg
  • Geogebra
  • Wikipedia