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 Group and Newton’s Equations
Affine Space
An
-dimensional affine space is a set equipped with a -dimensional 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.
Remark
The sum of two points in
is not defined, but their difference can be uniquely identified as a vector in .