Temporo-spatial polar coordinates

This post follows the material on polar coordinates from MIT Open Courseware, here. Instead of the space position vector r, we’ll use the time position vector w, and replace (arc) time with arc length, s.

In polar coordinates, the time position of a tempicle A is determined by the value of the radial duration to the origin, w, and the angle that the radial line makes with an arbitrary fixed line, such as the tx axis (x axis with time metric). Thus the trajectory of a tempicle will be determined if we know w and θ as a function of s, i.e., w(s) and θ(s). The directions of increasing w and θ are defined by the orthogonal unit vectors ew and eθ.

The time position vector of a tempicle has a magnitude equal to the radial duration, and a direction determined by ew. Thus

w = w ew.

Since the vectors ew and eθ are clearly different from point to point, their variation will have to be considered when calculating the lenticity and relentation. Over an infinitesimal interval of arc length ds, the coordinates of time point A will change from (w, θ) to (w + dw, θ + ).

We note that the vectors ew and eθ do not change when the coordinate w changes. Thus dew/dw and eθ/dw = 0. On the other hand, when θ changes to (θ + ), the vectors ew and eθ are rotated by an angle .

A mathematical approach to obtaining the derivatives of the unit vectors is to express ew and eθ in terms of their Cartesian components along i and j. We have that

ew =  cos θ i + sin θ j

eθ = – sin θ i + cos θ j.

Therefore, when we differentiate we obtain

dew/dw = 0,   dew/ = – sin θ i + cos θ j = eθ

deθ/dw = 0,   deθ/ = – cos θ i – sin θ j = –ew.

Lenticity vector

We can now differentiate w = w ew with respect to arc length and write

u = w′ = wew + w ew′,

where the prime indicates differentiation with respect to arc length. Or, using the above expression for eθ, we have

u = w′ = wew + eθ.

Here, uw = w′ is the radial lenticity component, and uθ = ′ is the circumferential lenticity component. We also have that  The radial component is the rate at which w changes magnitude, or stretches, and the circumferential component is the rate at which w changes direction, or swings.

Differentiating again with respect to arc length, we obtain the relentation

b = u′ = wew + wew′ + w′θ′eθ + wθ″eθ + wθ′eθ′.

Using the previous relations, we obtain

b = u′ = (w″wθ′²) ew + (wθ″ + 2w′θ′) eθ,

where bw = (w″wθ′²) is the radial relentation component, and bθ = (wθ″ + 2w′θ′) is the circumferential relentation component. Also, we have that

Change of basis

In many practical situations, it will be necessary to transform the vectors expressed in polar coordinates to Cartesian coordinates and vice versa. Since we are dealing with free vectors, we can translate the polar reference frame for a given point (w, θ) to the origin, and apply a standard change of basis procedure. This will give, for a generic vector A,

Equations of motion

In two dimensional polar coordinates, the release and relentation vectors are Γw = Γwew + Γθeθ and b = bwew + bθeθ. Thus, in component form, with vass n, we have

Γw = nbw = n (w″wθ′²)

Γθ = nbθ = n (wθ″ + 2w′θ′).