Position
//Bidang 129//, Iban people. Upper Rajang river, Kapit Division of Sarawak, 20th century, 118 x 46 cm. Pilih technique. From the Teo Family collection, Kuching. Photograph by D Dunlop.
Bidang 129, Iban people. Upper Rajang river, Kapit Division of Sarawak, 20th century, 118 x 46 cm. Pilih technique. From the Teo Family collection, Kuching. Photograph by D Dunlop.

Let particle P be described by a chain of events

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

where each event is characterized by $d\bar{r}$ its displacement. Definition: The position of event $\sf{P}_{\it{k}}$ is

$\begin{align} \bar{r}_{k} \equiv \bar{r}_{0} + \sum_{j=1}^{k} d \bar{r}_{j} \end{align}$

where $\bar{r}_{0}$ is arbitrary. Please notice that this algebraic vector has been defined entirely through a systematic description of sensation. So our ideas about position are based on an empirical approach that is scientific and consistent with the premise of WikiMechanics. If all events are assigned a position, $\Psi$ can be expressed as an ordered set of position vectors

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

Consider an ordered pair of events from $\Psi$

$\left( \sf{P}_{\it{i}}, \sf{P}_{\it{f}} \right)$

Definition: the separation vector between these events is

$\Delta \bar{r} \equiv \bar{r}_{f} - \bar{r}_{i}$

Definition: the norm of the separation is called the distance between events

$\Delta r \equiv \left\| \, \Delta \bar{r} \vphantom{\sum^{2}} \, \right\|$

These definitions imply that position, separation and distance are quantized. Their variation is discontinuous because WikiMechanics is based on a finite categorical scheme of binary distinctions. Quantization comes from the logical structure of the descriptive method, even for a continuous sensorium. In principle, motion is always some sort of quantum leaping or jumping from event to event. Phenomena like this have certainly been observed in twentieth-century physics and can, for example, be used to understand zener diodesXlink.png and the Stern-GerlachXlink.png experiment. For WikiMechanics, smoothly continuous motion is therefore presumed to be a macroscopic approximation. We are cautious about using calculus because the logical foundations of both differential and integral calculus are proven using assumptions about continuity. So WikiMechanics does not require calculus; instead calculations are designed to be implemented on digital computers, in a finite number of discrete steps.
Right.png Next step: Cartesian coordinates.
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