The Principles of Relativity and Determinacy

A series of experimental facts is at the basis of classical mechanics. These facts are summarized in the following principles:

Space and Time Principle

Our space is three dimensional and Euclidean, and time is one-dimensional.

Galileo's Principle of Relativity

There exist coordinate systems (called inertial) possessing the following two properties:

1. All laws of nature at all moments of time are the same in all inertial coordinate systems.
2. All coordinate systems in uniform rectilinear motion with respect to an inertial system are themselves inertial systems.

e.g. If a coordinate system attached to the earth is inertial, then an experimenter on a train which is moving uniformly in a straight line with respect to the earth cannot detect the motion of the train by experiments conducted entirely inside his car.

Remark

In reality, the coordinate system associated with the earth is only approximately inertial. Coordinate systems associated with the sun, the stars, etc. are more nearly inertial.

Newton's Principle of Determinacy

The initial state of a mechanical system (the totality of positions and velocities of its points at some moment of time) uniquely determines all of its motion.

The Galilean Group

Affine Space

An $n$-dimensional affine space is a set $A$ equipped with a $n$-dimensional vector space $V$ and a transitive and free action of the additive group of $V$ on $A$. The elements of the affine space $A$ are called points, the group action is called a parallel displacement.

Remark

The sum of two points in $A$ is not defined, but their difference can be uniquely identified as a vector in $V$.

Galilean Space

A Galilean spacetime is a 4-dimensional affine space $A^4$ equipped with a linear mapping $t:{\mathbb{R}}^4\to \mathbb{R}$ called the time. The points of  are called events or world points. The time interval from event $a\in A$  and $b\in A$  is the number $t(b-a)$ , if $t(b-a)=0$ , then the events  $a$ and $b$  are called simultaneous. The distance between simultaneous events $\rho (a,b)=\Vert a-b\Vert =\sqrt{(a-b,a-b)},a,b\in A^3$

Galilean Group

The Galilean group is the group of all transformations of a Galilean space which preserve its structure (i.e., affine transformations that preserves intervals of time and the distance between simultaneous events). The elements of this group is called Galilean transformations.

Theorem

The Galilean group is generated by time translations, space translations, spatial rotations, and constant‑velocity boosts. In other words, in the coordinate form, any Galilean transformation looks like $(x,t)\to (Rx+vt+a,t+s)$ where $R\in S{O}_3$  is a rotation, $v\in {\mathbb{R}}^3$ is a constant velocity boost, $a\in {\mathbb{R}}^4$ is a constant spatial translation, and $s\in \mathbb{R}$ is a time translation.

Newton’s Equation

According to Newton’s principal of determinacy all motions of a system are uniquely determined by their initial positions and initial velocities.

Principal

There is a function $F:{\mathbb{R}}^N\times {\mathbb{R}}^N\times \mathbb{R}\to {\mathbb{R}}^N$ such that $\ddot{x}=F(x,\dot{x},t)$ If we subject the world lines of all the points of any mechanical system to one and the same galilean transformation, we obtain world lines of the same system. So Newton’s equation must be invariant with respect to the group of galilean transformations.

Remark

Invariance with respect to time translations means that “the laws of nature remain constant”, i.e. if $x=\phi (t)$ is a solution to Newton’s Equation, then for any $s\in \mathbb{R}$, $x=\phi (t+s)$ is also a solution. Invariance with respect to translation in 3-dimension space means that the space is homogeneous or “has the same properties at all of its points”. It follows that $F(x,\dot{x},t)$ in the inertial coordinate system can depend only on the relative coordinates. Invariance with respect to rotations means the space is isotropic, i.e. there are no preferred directions.