Displacement
//Bidang// (detail), Iban people. Upper Rajang river, Kapit Division of Sarawak, 20th century, 110 x 61 cm. Ikat technique. From the Teo Family collection, Kuching. Photograph by D Dunlop.
Bidang (detail), Iban people. Upper Rajang river, Kapit Division of Sarawak, 20th century, 110 x 61 cm. Ikat technique. From the Teo Family collection, Kuching. Photograph by D Dunlop.

Let particle P be described by an ordered chain of events

$\Psi ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \ \ldots \ \sf{P}_{\it{k}} \ \ldots \ \right)$

that is repetitive so that $\Psi$ may also be written as

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1} , \sf{\Omega}_{2} , \sf{\Omega}_{3} \ \ldots \ \right)$

where each orbital cycle $\sf{\Omega}$ is composed of $N$ sub-orbital events

$\sf{\Omega} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \ \ldots \ \sf{P}_{ \it{N}} \right)$

Let P be described by $\delta _{\hat{m}}$, $\delta _{\hat{e}}$ and $\delta _{z}$ which specify its spatial orientation, $R$ its orbital radius and $\lambda$ its wavelength. We use these characteristics to define the following numbers for describing sub-orbital events

$\begin{align} &dx \equiv \delta _{\hat{m}} \frac{ R \sin{\! 2 \theta } }{N } \\ \ \\ &dy \equiv \delta _{\hat{e}} \frac{ R \cos{\! 2 \theta } }{N } \\ \ \\ &dz \equiv \delta _{z} \frac{ \lambda }{\,N \,} \end{align}$

where $\theta$ is the phase angle. Then using the spatial axes $\hat{x}$, $\hat{y}$ and $\hat{z}$ the displacement vector is defined as

$d\bar{r} \equiv dx \, \hat{x} + dy \, \hat{y} + dz \, \hat{z}$

If we switch to implicitly using Cartesian basis vectors, we can express the displacement as an ordered set

$d\bar{r} = \left( dx, dy, dz \right)$

WikiMechanics uses a finite categorical scheme of binary distinctions to describe sensation. So the number of sub-orbital events may be large but not infinite. In principle $N$ is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we may assume that $N$ is large enough to make an approximation to spatial continuity, then allowing the use of calculus.

Right.png Next step: position.
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