Abstract classical mechanics

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 E3. We fix some point o ∈ E3 called the “origin of reference”. Then the position of every point s in E3 is uniquely determined by its position vector os = r (whose initial point is o and end point is s). The set of all position vectors forms the three-dimensional vector space ℝ3, 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 ℝ3 × ℝ3. 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.

Spatial position is called location and temporal position 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 ℝ3. 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 ℝ3 × ℝ3. 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 s1 and s2 of any two intervals of the curve of motion in the space frame, then the corresponding intervals of the time frame, t1 and t2, relative to the space frame satisfy the proportion: s1:s2 :: t1:t2.

First Law of Dynamics: There exists an elementary reference system such that a body continues in its state of motion unless compelled or constrained otherwise.

By convention either the space frame or the time frame is primary; the dual frame is secondary. The position and motion of the secondary frame is relative to the primary frame.

The secondary frame moves linearly relative to the primary frame, so the curve of the secondary frame relative to the primary frame is a line, i.e., a single dimension. If the space frame is primary, the system of reference is spatio-temporal, the space frame is called space, and the time frame is one dimension of time, called time. If the time frame is primary, the system of reference is temporo-spatial, the time frame is called time or chron, and the space frame is one dimension of space, called stance. By convention the space frame is primary in a space-time reference system and the time frame is primary in the temporo-spatial reference system.

If primary and secondary frames are given, then rates of motion may be defined. If the space frame is the secondary frame, then length is the independent variable. If the time frame is the secondary frame, then duration is the independent variable.

A space rate of motion is the duration of motion per unit of independent length. A time rate of motion is the length of motion per unit of independent duration. The independent duration is called the elapsed time or simply time. The independent length is called here the stance (a stance interval is a distance).

An independent variable is either bound or free. A bound variable is specified, as for example, a race with a length specified or a sport with a time specified. A free variable is unspecified and appears to continue at a constant rate forever.

“Ever while time flows on and on and on, / That narrow noiseless river” ‒ Christina Rossetti, A Life’s Parallels

Stance as an independent variable is like an odometer connected to a vehicle that travels at a constant rate, rather like the spacecraft Voyager 1 and 2 that continue to travel beyond the solar system at a constant rate with respect to the Sun (see here).

Representation of the physical universe can be either as three-dimensional space with independent time or as three-dimensional time with independent stance.

Abstract Motion

The trajectory of a body in motion is one-dimensional; it is denoted by t. The set R = {t} is called the trajectory axis.

A motion (or path) of the point s is a smooth map Δ → E3, where Δ is an interval of the trajectory axis. We say that the motion is defined on the interval Δ. If the origin (point o) is fixed, then every motion is uniquely determined by a smooth vector-function r: Δ → R3.

The image of the interval Δ under the map tr(t) is called the trajectory or orbit of the point s.

The rate of motion v of the point s at a point of t ∈ Δ is by definition the derivative dr/dt = r˙(t) ∈ R3. Clearly the rate of motion is independent of the choice of the origin.

The rate of the rate (rate squared) of motion of the point is by definition the vector a = v˙ = r¨ ∈ R3. The rate of motion and rate squared of motion are usually depicted as vectors with initial point at the point s.