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 *E*^{3} and *E*^{3}. 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 *o*_{1} ∈ *E*^{3} and a point of the duration space *o*_{2} ∈ *E*^{3} 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 *o*_{1}*r* (whose initial point is *o*_{1} and end point is *r*). The position of every point *z* in duration space is uniquely determined by its position vectors *o*_{2}*q* (whose initial point is *o*_{2} and end point is *q*).

The set of all position vectors in each space forms a three-dimensional vector space *R*^{3}, 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 Δ_{t} → *E*^{3}, where Δ_{t} is an interval of the time axis. We say the length space motion is defined on the interval Δ_{t}.

A duration space motion (or path) of the point *q* is a smooth map Δ_{s} → *E*^{3}, where Δ_{s} is an interval of the distance axis. We say the motion is defined on the interval Δ_{s}.

If the length space origin point is fixed, then every motion is uniquely determined by a smooth vector-function **r**: Δ_{t} → *R*^{3}. The image of the interval Δ_{t} 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**: Δ_{s} → *R*^{3}. The image of the interval Δ_{s} 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* ∈ Δ_{t} is defined as the derivative d**r**/d*t* ∈ *R*^{3}. The lenticity **w** of the duration space point *q* at the distance point *s* ∈ Δ_{s} is defined as the derivative d**q**/d*s* ∈ *R*^{3}. 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**″ ∈ *R*^{3}. 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**″ ∈ *R*^{3}. The velocity, acceleration, lenticity, and relentment are usually depicted as vectors with initial point at the point *q*.

The set *E*^{3} 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 *E*^{3} × *R*^{3}{*v*}, the phase (or state) space.