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Second 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}. Then let P interact with some particle called  \mathsf{X} between initial and final events so that there is a change in P’s motion described by

\Delta  \overline{p} = \overline{p}_{f} - \overline{p}_{i}

\Delta t = t_{f} - t_{i}

According to the usual narrative of Newtonian mechanics, the particle  \mathsf{X} impresses a force like a push or a pull that causes the change in P’s motion. Sir Isaac Newton says that1Isaac Newton, Mathematical Principles of Natural Philosophy, page 416. Translated by I. Bernard Cohen and Anne Whitman. University of California Press 1999.

An icon indicating a quotation.A change in motion is proportional to the motive force impressed and takes place along the straight line in which that force is impressed.

β€” Sir Isaac Newton

This relationship is called Newton’s second law of motion. It can be mathematically expressed by defining an algebraic vector called the force as

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

If P is a Newtonian particle in dynamic equilibrium, then its momentum is related to its velocity by \overline{p} = m \overline{\mathsf{v}} where  m is its mass. Also for many sorts of interactions the mass of a Newtonian particle can be considered constant, then \Delta \overline{p} = m  \Delta\overline{\mathsf{v}} and the force can be written in terms of the acceleration  \overline{a} as

\overline{F} = m \dfrac{\Delta \overline{\mathsf{v}}}{\Delta t} = m \overline{a}

Momentum is conserved. So all particles that have momentum can cause changes in the momentum of another particle if they are absorbed or emitted. Then X, the particle that is absorbed or emitted, is often called a force-carrying particle. Force-carrying particles that are material or charged are usually easy to detect in the immediate vicinity of an interaction. So the forces they impart are called contact forces. Phenomena like automobile collisions and gunshot wounds can be understood using contact forces. X is much less conspicuous if it is ethereal and neutral. Such particles can be difficult to detect, and the forces that they carry may seem to come from far away. So they are often referred to as exchange particles to suggest a remote origin, and their effects are called action-at-a-distance forces. If X is imaginary then its force may seem like some random background fluctuation coming from nowhere specific. Many of these force-carrying particles are difficult to characterize and distinguish as individuals, so it is often more convenient to group lots of them together and refer to them collectively as force fields. For example we may vaguely refer to a set of photons as an electromagnetic field, or a collection of gravitons as a gravitational field.

Bidang, Iban people. Sarawak 20th century, 50 x 101 cm. Lintah motif. From the Teo Family collection, Kuching. Photograph by D Dunlop.
References
1Isaac Newton, Mathematical Principles of Natural Philosophy, page 416. Translated by I. Bernard Cohen and Anne Whitman. University of California Press 1999.