This post is based on Mathematical Aspects of Classical and Celestial Mechanics, Third Edition by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt (Springer 2006). Here it is generalized to (3 + 3) dimensions.
Motion takes place in two spaces that are three-dimensional and Euclidean with a fixed orientation. Denote them by E3 and E3. One space is the length space and the other is the duration space, which are determined by the frame of reference system described elsewhere.
Fix a point of the length space o1 ∈ E3 and a point of the duration space o2 ∈ E3 called the “origin of reference” of each space. Then the position of every point x in length space is uniquely determined by its position vectors o1r (whose initial point is o1 and end point is r). The position of every point z in duration space is uniquely determined by its position vectors o2q (whose initial point is o2 and end point is q).
The set of all position vectors in each space forms a three-dimensional vector space R3, which is equipped with the scalar product 〈 , 〉. Length space has an associated one-dimensional time, an independent duration, which is denoted by t throughout. The set R = {t} is called the time axis. Duration space has an associated one-dimensional distance, an independent length, which is denoted by s throughout. The set Q = {s} is called the distance axis.
A length space motion (or path) of the point r is a smooth map Δt → E3, where Δt is an interval of the time axis. We say the length space motion is defined on the interval Δt.
A duration space motion (or path) of the point q is a smooth map Δs → E3, where Δs is an interval of the distance axis. We say the motion is defined on the interval Δs.
If the length space origin point is fixed, then every motion is uniquely determined by a smooth vector-function r: Δt → R3. The image of the interval Δt under the map t → r(t) is called the trajectory or orbit of the point r.
If the duration space origin point is fixed, then every motion is uniquely determined by a smooth vector-function q: Δs → R3. The image of the interval Δs under the map s → q(s) is called the trajectory or orbit of the point q.
The velocity v of the length space point r at the time point t ∈ Δt is defined as the derivative dr/dt ∈ R3. The lenticity w of the duration space point q at the distance point s ∈ Δs is defined as the derivative dq/ds ∈ R3. Clearly the velocity and lenticity are independent of the choice of the origins.
The acceleration of the length point r is the vector a = v′ = r″ ∈ R3. The velocity and acceleration are usually depicted as vectors with initial point at the point r. The relentment of the duration point q is the vector b = w′ = q″ ∈ R3. The velocity, acceleration, lenticity, and relentment are usually depicted as vectors with initial point at the point q.
The set E3 is also called the configuration space of the point r or q. Each pair (r, v) or (q, w) is called the state of the point, and each set E3 × R3{v}, the phase (or state) space.