The Postulates of Classical Mechanics
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:
- All laws of nature at all moments of time are the same in all inertial coordinate systems.
- 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 Spacetime
Affine Space
An
-dimensional affine space is a set equipped with an -dimensional (real) vector space and a transitive and free action of the additive group of on . The elements of the affine space are called points, the group action is called a parallel displacement. We shall denote such an affine space as . An affine map between affine spaces and is a function together with a linear map such that
Remark
The sum of two points in
is not defined, but their difference can be uniquely identified as a vector in . Thus the above condition for an affine map can be rewritten as
Galilean Space
A Galilean spacetime is a 4-dimensional affine space
equipped with a linear mapping called the time. The points of are called events or world points. The time interval from event and is the number , if , then the events and are called simultaneous.
Proposition
The set of events simultaneous with a given event forms a 3-dimensional affine subspace, it is called the simultaneous events.
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.
Lemma
All Galilean spacetimes are isomorphic to each other, and, in particular, isomorphic to the coordinate space
.
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
where is a rotation, is a constant velocity boost, is a constant spatial translation, and is a time translation.