Kinematics, the geometry of motion, studies the location of geometric objects parameterized by time. This is a 3D length space with functions representing the path or trajectory as the locus of places occupied by points. It has a dual mathematics of 3D duration space with functions representing the course of motion as the locus of times occupied by events. Below is an introduction to both, following the exposition in Principles of Engineering Mechanics: Kinematics by Millard Beatty Jr. (Springer)
1.3 Motion and Particle Path
To locate an object spatially, we need a reference system. The only reference we have is other objects. Therefore, the physical nature of what we shall call a reference frame is an assigned set of objects whose mutual distances do not change with time – at least not very much. …
We define a three-dimensional Euclidean reference frame φ as a set consisting of a point O of space, called the origin of the reference frame, and a vector basis {ei} ≡ {e1, e2, e3}. That is, φ = {O; ei}. We shall require for convenience that the basis is an orthonormal basis, i.e., a triple of mutually perpendicular unit vectors.
The location of a particle P in the frame φ at time t is given by the location vector xφ of P from O. Of course, as time goes on the place occupied by P generally will vary. The time sequence of positions of P in φ is called a motion of P relative to φ, and it is defined by the equation
x = xφ(P, t), (1.1)
in which x is the place in frame φ occupied at time t by the particle P in its motion relative to φ. The locus L of places occupied by P in the motion is called the path or trajectory of P. The place x0 that P occupies along its path at some instant t = t0, say, which we may sometimes consider as the initial instant t0 = 0, is determined by the motion (1.1):
x0 = xφ(P, t0). (1.2)
When the particular choice of φ is clear, as it is when only one frame is being considered, for the sake of simplicity we shall discard the subscript φ and write (1.1) and (1.2) as
x = xφ(P, t), (1.3a)
x0 = xφ(P, t0). (1.3b)
Frequently we shall identify {em} as the familiar Cartesian basis {im} = {i1, i2, i3} with i1 = i, i2, = j and i3 = k, as usual.
Dual of above
Motion and Eventicle Path
To temporally locate, i.e., chronate, a subject, we need a reference system. The only reference we have is other subjects. Therefore, the physical nature of what we shall call a reference timeframe is an assigned set of subjects whose mutual distimes do not change with distance – at least not very much. …
We define a three-dimensional Euclidean reference timeframe ψ as a set consisting of a point O of duration, called the origin of the reference timeframe, and a vector basis {ei} ≡ {e1, e2, e3}. That is, ψ = {O; ei}. We shall require for convenience that the basis is an orthonormal basis, i.e., a triple of mutually perpendicular unit vectors.
The position of an eventicle Q in the timeframe ψ at distance s is given by the chronation (time position) vector xψ of Q from O. Of course, as distance goes on the place occupied by Q generally will vary. The distance sequence of chronations of Q in ψ is called a motion of Q relative to ψ, and it is defined by the equation
w = wψ(Q, s), (1.1)
in which w is the place with timeframe ψ occupied at distance s by the eventicle Q in its motion relative to ψ. The locus L of places in duration space occupied by Q in the motion is called the duration path or trajectory of Q. The place in duration space w0 that Q occupies along its duration path at some point s = s0, say, which we may sometimes consider as the initial point s0 = 0, is determined by the motion (1.1):
w0 = wψ(Q, s0). (1.2)
When the particular choice of ψ is clear, as it is when only one timeframe is being considered, for the sake of simplicity we shall discard the subscript ψ and write (1.1) and (1.2) as
w = wψ(Q, s), (1.3a)
w0 = wψ(Q, s0). (1.3b)
Frequently we shall identify {em} as the familiar Cartesian basis {im} = {i1, i2, i3} with i1 = i, i2, = j and i3 = k, as usual.