The following builds on the book *Mathematical Aspects of Classical and Celestial Mechanics, 3rd edition,* by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt (Springer 2006).

Basic Principles of Classical Mechanics (cf. Chapter 1)

Space and Time

The space where the motion takes place is three-dimensional and Euclidean with a fixed orientation. We shall denote it by

E^{3}. We fix some pointo ∈ E^{3}called the “origin of reference”. Then the position of every pointsinE^{3}is uniquely determined by its position vector=osr(whose initial point isoand end point iss). The set of all position vectors forms the three-dimensional vector space ℝ^{3}, which is equipped with the scalar product 〈 , 〉.

The time in which motion takes place has the same structure as the abstract space above. The combined vector space is ℝ^{3} × ℝ^{3}. The abstractions for space and time are unconnected unless there is defined a fixed relationship between them. Examples of such a fixed relationship include a default rate of motion or a maximum rate of motion. Let us begin without such a relationship.

Position in space is called *location* and in time is called *chronation*. The Euclidean metric for space is called *length* and for time is called *duration* (or *time*).

A *frame of reference* (“frame”) is a method to assign every *particle* a unique position in a coordinate system of points in ℝ^{3}. Such assignment is known continually and universally, without signals, from the universal extent of the frame. The coordinate system is commonly Cartesian.

A *system of reference *(“reference system”) is a method to assign every *event* a unique position in a coordinate system of points in ℝ^{3} × ℝ^{3}. A reference system is composed of dual frames of reference, one called the *space frame* and the other called the *time frame*, such that the time frame is in standard uniform motion relative to the space frame. This requires that given the magnitudes *s*_{1} and *s*_{2} of any two intervals of the curve of motion in the space frame, then the corresponding intervals of the time frame, *t*_{1} and *t*_{2}, relative to the space frame satisfy the proportion: *s*_{1}:*s*_{2} :: *t*_{1}:*t*_{2}.