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First Law of Motion

Consider a particle P described by a repetitive chain  \Psi of historically ordered space-time events  \mathsf{\Omega}. Let these events be characterized by their position  \bar{r} and time of occurrence  t. We represent this trajectory of P with the expression

\Psi \! \left( \bar{r}, t \right)^{\mathsf{P}} = \left( \mathsf{\Omega}_{1}, \, \mathsf{\Omega}_{2} \, \ldots  \;  \mathsf{\Omega}_{\it{i}} \; \ldots \; \mathsf{\Omega}_{\it{f}} \; \ldots \, \right)

These characteristics may be used to establish the momentum  \overline{p} for each event \mathsf{\Omega}. By the second law of motion, any change in the momentum of P is related to the action of some force  \overline{F} that is described by

\overline{F} \equiv  \dfrac{\Delta \overline{p}}{\Delta t}

So if there are no forces acting on P, then there are also no changes in P’s momentum, and vice versa

\overline{F} = \left(0, 0, 0 \right) \; \; \Longleftrightarrow \; \; \Delta \overline{p} = \left(0, 0, 0 \right)

The forgoing statement is just a special case of the second law of motion. Yet Newton included this null relationship as part of his first law of motion. It may seem redundant, but the first law is more than simply a special case of the second law because it also establishes exactly what we mean by a straight line segment or a straight rod. The first law is also known as the law of inertia, it has been translated1Isaac Newton, Mathematical Principles of Natural Philosophy, page 416. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999. into modern English as

An icon indicating a quotation.Every body perseveres in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by forces impressed.

β€” Sir Isaac Newton

For EthnoPhysics this first law is uniquely important because by our premise we prefer to avoid mysteriously received knowledge about length and lines. So this aspect of Newton’s first law is formally restated in the following explicit definition: If P has the same momentum for all events in its trajectory, then  \Psi describes uniform linear motion and we say that P is moving in a straight line. This sort of force-free motion is obtained if the frame of reference is inertial and P is isolated It is only well-defined for particles that are at least as big as atoms.

References
1Isaac Newton, Mathematical Principles of Natural Philosophy, page 416. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999.