Newtonian Mechanics – EthnoPhysics

Outline

The theory of mechanics presented by Sir Isaac Newton in 1687 is the foundation of classical mechanics and modern physics. This article relates to his work by first considering velocity and acceleration. For EthnoPhysics, these dynamic concepts are logically defined from sensations, and quantified by measurement. Then we develop the notion of force based on Newton’s second law of motion. His first law is used to define straight lines and linear motion. And finally, Newton’s third law is interpreted as a scientific reiteration of venerable knowledge that can be traced back as far as the Vedic literature . These widespread ancient ideas about cause and effect are often referred to as karma .

Velocity

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 $\overline{r}$ and time of occurrence $t .$ Then we write

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

The separation between some arbitrary initial and final pair of events is given by $\Delta \overline{r} = \overline{r}_{f} - \overline{r}_{i} \, .$ And the elapsed time between these events is $\Delta t = t_{f} - t_{i} \, .$ Then a velocity vector is defined by the ordered set of three numbers

$\overline{\mathsf{v}} \equiv \dfrac{\Delta \overline{r}}{\Delta t}$

The speed of P is defined by the norm of the velocity

$\mathsf{v} \equiv \left\| \, \overline{\mathsf{v}} \, \right\|$

Consider measuring this quantity based on observations of length and time. Recall that length is defined only for particles that are at least as big as atoms. And we also presume that P is compared to a frame of reference that includes a calibrated rod and a calibrated clock $\mathbf{\Theta} .$ To determine the velocity first measure the elapsed time between events $\mathsf{\Omega}_{\it{i}}$ and $\mathsf{\Omega}_{ \it{f}}$ as

$\Delta t = \left( k-j \right) \widehat{\tau}^{\, \mathbf{\Theta}} = \left( f-i \right) \widehat{\tau}^{\, \mathsf{P}} = \dfrac{f-i}{\nu}$

where $\left( k-j \right)$ is the number of clock cycles between initial and final events, $\hat{\tau}$ is the period and $\nu$ is the frequency of P. Then make three length measurements along the spatial axes that are noted by $\ell_{x} ,$ $\ell_{y}$ and $\ell_{ z} .$ Combine these measurements to obtain the observed separation vector $\Delta \overline{r} = \left( \ell_{x}, \, \ell_{y}, \, \ell_{z} \right)$ between events which may be written as

$\Delta \overline{r} = \Delta \overline{r} \! \left(\mathsf{\Omega}_{i} \right) + \Delta \overline{r} \! \left( \mathsf{\Omega}_{i+1} \right) + \; \ldots \; + \Delta \overline{r} \! \left( \mathsf{\Omega}_{f} \right)$

where $\Delta \overline{r} \! \left(\mathsf{\Omega} \right)$ characterizes the motion of P during one complete orbital cycle $\mathsf{\Omega}$ . As discussed earlier atomic cycles are separated from each other by one wavelength $\lambda .$ And if measurements are not too disruptive so that $\lambda$ is constant, then the distance between initial and final events is

$\Delta r \equiv \left\| \Delta \overline{r} \right\| = \left\| \left( \ell_{x}, \, \ell_{y}, \, \ell_{z} \right) \right\| = \left( f-i \right) \lambda$

Combining these observations gives the measured speed of P as

$\mathsf{v} = \left\| \dfrac{\Delta \overline{r}}{\Delta t} \right\| = \dfrac{\Delta r}{\Delta t} = \dfrac{\left(f-i \right) \lambda}{\left(f-i \right) \widehat{\tau}} = \dfrac{\lambda}{\widehat{\tau}} = \nu \lambda$

If P contains many quarks we can use Planck’s postulate to substitute the mechanical energy $E$ for the frequency to obtain

$\mathsf{v} = \nu \lambda = \left( \dfrac{E}{h} \rule{0px}{14px} \right) \lambda$

And if the frame of reference is inertial then de Broglie’s postulate can be used to replace the wavelength with the momentum $p$ to obtain

$\mathsf{v} = \left( \dfrac{E}{h} \rule{0px}{14px} \right) \! \left( \dfrac{h}{p} \rule{0px}{14px} \right) =\dfrac{E}{p}$

The Speed of a Photon

If the particle under consideration is a graviton or a photon $\boldsymbol{\gamma}$ then its mass $m$ is zero, and its energy is given by $E=cp \, .$ Substituting this into the expression for speed developed above gives

$\mathsf{v} \! \left( \boldsymbol{\gamma} \right) = \dfrac{E}{p} = \dfrac{cp}{p} = c$

And so the constant number $c$ is usually called the speed of light. For photons $\lambda \nu = c.$ Thus the wavenumber $\kappa$ and other photon characteristics are all related as

$\kappa = \dfrac{2 \pi}{\lambda} = \dfrac{2\pi}{h} p = \dfrac{2\pi}{hc} E$

The Speed of a Newtonian Particle

If P is Newtonian then $m \ne 0$ and the particle is presumably in dynamic equilibrium with its environment. For this case, the mechanical energy is given by $E = 2K$ where $K$ is the kinetic energy. So for Newtonian particles

$\mathsf{v} = \dfrac{E}{p} = \dfrac{2K}{p}$

But recall that kinetic energy is defined by $K \equiv p^{ 2} / \, 2m \, .$ So for Newtonian particles

$\mathsf{v} = \dfrac{2K}{p} = \dfrac{2}{p} \left( \dfrac{p^{2}}{2m} \right) = \dfrac{p}{m}$

This equation can be rearranged to obtain the conventional statement

Momentum is identified by Sir Isaac Newton, pictured here, as a central idea in mechanics. — Sir Isaac Newton. Painted by G Kneller 1689.

$p = m \mathsf{v}$

The term momentum is the modern English word used for translating the phrase; quantity of motion.¹Isaac Newton, Mathematical Principles of Natural Philosophy, page 639. Translated by Andrew Motte and Florian Cajori. University of California Press, 1934. So the foregoing relationship was articulated by Sir Isaac Newton when he wrote²Isaac Newton, Mathematical Principles of Natural Philosophy, page 404. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999.

Quantity of motion is a measure of motion that arises from the velocity and the quantity of matter jointly.

— Sir Isaac Newton

This direct proportionality between speed and momentum is traditional and simple. It can be used to eliminate the momentum in some previously defined quantities such as the Lorentz factor $\gamma$ which can now be expressed as

$\gamma \equiv \dfrac{1}{\sqrt{\; 1 - \left(p/mc \right)^{2} \; }} = \dfrac{1}{\sqrt{\; 1 - \left( \mathsf{v}/c \right)^{2} \; }}$

And for Newtonian particles the kinetic energy can be written as

$K \equiv \dfrac{\; p^{2}}{2m} = \frac{1}{2} m \mathsf{v}^{2}$

Momentum

Old discussion of Momentum

Momentum is the modern English word used for translating the phrase “quantity of motion” that Sir Isaac Newton uses on the very first page of his great book, the Principia³Isaac Newton, Mathematical Principles of Natural Philosophy, page 639. Translated by A Motte and F Cajori. University of California Press, 1934.. So to understand motion EthnoPhysics starts by using sensation to define the momentum as follows.

Consider some particle P characterized by its wavevector $\overline{ \kappa }$ and the total number of quarks it contains $N_{\mathsf{q}} \, .$ Report on any changes relative to a frame of reference F which is characterized using $\widetilde{ \kappa }$ the average wavevector of the quarks in F. We define the momentum of particle P, in reference frame F, as the ordered set of three numbers

$\overline{p}^{\,\mathsf{P}} \equiv \dfrac{h}{2\pi} \left( \overline{\kappa}^{\,\mathsf{P}} \! - N^{\mathsf{P}}_{\mathsf{q}} \, \widetilde{\kappa}^{\,\mathsf{F}} \right)$

where $h$ and $\pi$ are constants.

The norm of the momentum is marked without an overline as $p \equiv \left\| \, \overline{p} \, \right\| .$

Momentum is traditionally understood as a product of the mass with a velocity. But the premise of EthnoPhysics questions how we accept spatial ideas like velocity. So instead we start with this sensation-based definition of momentum. Then later, after untangling some entwined concepts, we show how $p \! = \! m \mathsf{v}$ for many particles and conditions.

Sensory Interpretation: The momentum is defined by a difference between the wavevector of P and a scaled-down version of the frame’s wavevector. Recall that the wavevector has previously been interpreted as a mathematical representation of lateral and visual sensation. So momentum is like the audio-visual contrast between a particle and its reference frame. This juxtaposition attracts and holds our attention because it is necessary for situational awareness and survival. The EthnoPhysics definition of momentum works by expressing the relevance of reference sensations. Seeing the Sun, seeing blood and seeing gold are vividly pertinent for understanding motivation, movement and motion.

Momentum is Conserved

Recall that quarks are conserved. So if some particles $\mathbb{X}$ , $\mathbb{Y}$ and $\mathbb{Z}$ interact like $\mathbb{X} + \mathbb{Y} \leftrightarrow \mathbb{Z}$ but are otherwise isolated, then $N_{\mathsf{q}}$ the total number of quarks, is constrained as $N_{\mathsf{q}}^{\mathbb{X}} + N_{\mathsf{q}}^{\mathbb{Y}} = N_{\mathsf{q}}^{\mathbb{Z}} .$ Also, as shown earlier, wavevectors are combined as $\overline{\kappa}^{\mathbb{X}} + \overline{\kappa}^{\mathbb{Y}} = \overline{\kappa}^{\mathbb{Z}}$ if these particles are free. Then by substitution into the definition of momentum

$\begin{align*} \overline{p}^{\,\mathbb{Z}} &\equiv \dfrac{h}{2\pi} \left[ \overline{\kappa}^{\,\mathbb{Z}} \! - N^{\mathbb{Z}}_{\mathsf{q}} \, \widetilde{\kappa}^{\,\mathsf{F}} \right] \\ &= \rule{0px}{15px}\dfrac{h}{2\pi} \left[ \left( \overline{\kappa}^{\mathbb{X}} + \overline{\kappa}^{\mathbb{Y}} \right) - \left( N_{\mathsf{q}}^{\mathbb{X}} + N_{\mathsf{q}}^{\mathbb{Y}} \right) \widetilde{\kappa}^{\,\mathsf{F}} \rule{0px}{15px} \right] \\ &= \dfrac{h}{2\pi} \left[ \left( \overline{\kappa}^{\,\mathbb{X}} \! - N^{\mathbb{X}}_{\mathsf{q}} \, \widetilde{\kappa}^{\,\mathsf{F}} \right) + \left( \overline{\kappa}^{\,\mathbb{Y}} \! - N^{\mathbb{Y}}_{\mathsf{q}} \, \widetilde{\kappa}^{\,\mathsf{F}} \right) \rule{0px}{15px} \right] \\ &= \rule{0px}{15px} \dfrac{h}{2\pi} \left( \overline{\kappa}^{\,\mathbb{X}} \! - N^{\mathbb{X}}_{\mathsf{q}} \, \widetilde{\kappa}^{\,\mathsf{F}} \right) + \dfrac{h}{2\pi} \left( \overline{\kappa}^{\,\mathbb{Y}} \! - N^{\mathbb{Y}}_{\mathsf{q}} \, \widetilde{\kappa}^{\,\mathsf{F}} \right) \\ &= \rule{0px}{15px} \; \overline{p}^{\mathbb{X}} + \overline{p}^{\mathbb{Y}} \end{align*}$

Thus we say that momentum is conserved when compound quarks are formed or decomposed. Newtonian mechanics is built on this relationship. It is important but not unique. Recall that we also have conservation laws for seeds, quarks, charge, lepton number, baryon number and enthalpy. All of these conservation rules follow from the logical requirements of our descriptive method. EthnoPhysics depends on mathematics. Therefore we are constrained by the law of noncontradiction and the associative properties of addition. So any characteristic defined by simple sums of quark coefficients will always be conserved.

Definition of Momentum

$\overline{p} \equiv p \widehat{p}$

where $\widehat{p}$ is called the direction of motion of P when presented in a Cartesian frame of reference. It is identified with the central-axis of P, written as $\widehat{z}$ in quark-space. And it is completely arbitrary except for being a unit-vector.

De Broglie’s Postulate

In a perfectly inertial frame of reference $\widetilde{ \kappa } ^{ \mathsf{F}}= \left( 0, 0, 0 \right)$ . Then the momentum of P is given by

$\overline{p} = \dfrac{h \overline{\kappa}}{2\pi}$

And recall that for particles in motion, the wavelength is $\lambda = 2 \pi / \kappa$ . So taking the norm of the momentum and eliminating the wavenumber obtains Louis de Broglie’s statement about the inverse relationship between momentum and wavelength

$p = h / \lambda$

Thus de Broglie’s postulate notes a conditional proportionality between $\overline{p}$ and $\overline{ \kappa }$ that is just built-in to the EthnoPhysics definitions of these characteristics. Moreover these definitions apply to all sorts of particles, photons as well as Newtonian particles. So De Broglie’s postulate is often used to determine the momentum of a photon since photons are usually described by their wavelength.

Momentum of a Graviton

Gravitons have been defined by the union of a photon and its associated anti-photon. This is written as $\mathsf{\Gamma} \equiv \left\{ \, \boldsymbol{\gamma}, \, \overline{\boldsymbol{\gamma}} \, \rule{0px}{9px} \right\}.$ Also, recall that the wavevector $\overline{\kappa}$ is defined from sums of quark coefficients, and that quarks are conserved. So the wavevector of a graviton is the sum of the wavevectors of its component photons. But the wavevector of any particle is symmetrically opposed to the wavevector of its matching anti-particle. Thus

$\begin{align*} \overline{\kappa} \! \left( \mathsf{\Gamma} \rule{0px}{10px} \right) &= \overline{\kappa} \! \left( \boldsymbol{\gamma} \rule{0px}{10px} \right) + \overline{\kappa} \! \left( \overline{\boldsymbol{\gamma}} \rule{0px}{10px} \right) \\ &= \overline{\kappa} \! \left( \boldsymbol{\gamma} \rule{0px}{10px} \right) - \overline{\kappa} \! \left( \boldsymbol{\gamma} \rule{0px}{10px} \right) \rule{0px}{15px} \\ &= (0, 0, 0)\rule{0px}{14px} \end{align*}$

Then, in a frame of reference noted by F, the momentum of a graviton is given by

$\overline{p}^{\, \mathsf{\Gamma}} \! = - \dfrac{h \widetilde{\kappa}^{\,\mathsf{F}} }{2\pi} \, N_{\mathsf{q}}^{\mathsf{\Gamma}}$

where $N_{\mathsf{q}}^{\mathsf{\Gamma}}$ is the total number of quarks in $\mathsf{\Gamma},$ including all types. This expression shows that all of the gravitons in a description have their momenta pointed in the same direction, and this direction is determined by attributes of the reference frame.

Acceleration

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

Use these characteristics to calculate a velocity $\overline{\mathsf{v}}$ for each event $\mathsf{\Omega}.$ Then changes between some arbitrary initial and final events are noted by

$\Delta \overline{\mathsf{v}} = \overline{\mathsf{v}}_{f} - \overline{\mathsf{v}}_{i}$

and

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

The acceleration vector is defined by the ordered set of three numbers

$\overline{a} \equiv \dfrac{\Delta \overline{\mathsf{v}}}{\Delta t}$

This acceleration is used to describe changes in the trajectory of P. It can be experimentally determined by measuring lengths and elapsed times. The norm of the acceleration is written without an overline

$a \equiv \left\| \, \overline{a} \, \right\|$

This Malaysian weaving highlights a warp and weft structure somewhat like Cartesian depictions of particle dynamics. — Bidang, Iban people. Sarawak 20th century, 54 x 121 cm. Pilih technique. From the Teo Family collection, Kuching. Photograph by D Dunlop.

Force and Newton’s Second Law

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

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

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

and

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

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 that⁴Isaac Newton, Mathematical Principles of Natural Philosophy, page 416. Translated by I. Bernard Cohen and Anne Whitman. University of California Press 1999.

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

— 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 exchange particle.

Exchange 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.

If X is imaginary then its force may seem like some random background fluctuation coming from nowhere specific. Many of these exchange 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.

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Construction Zone

Types of Forces

Preparations for completing this discussion can be seen here: fields and forces

Centripetal Force

$\mathscr{F}_{centripetal} \equiv - \dfrac{ \partial \mathcal{U}_{binding} }{ \partial \rho_{z} }$

For circular motion (as established by a choice for the spatial displacement vector) the magnitude of the centripetal force is a constant.

$\mathscr{F}_{centripetal} = - k_{\mathsf{F}}$

From definitions of velocity and acceleration for circular motion

$\mathscr{F}_{centripetal} = \dfrac{\, m {\mathsf{v}}^{2} }{ R }$

Magnetic Force

$\mathscr{F}_{m} = - \dfrac{ \partial \mathcal{U}_{m} }{ \partial \rho_{m} } = - \delta_{\widehat{m}} \, k_{\mathsf{F}} \sqrt{k_{mm}}$

Observed as force on a current carrying wire.

Electric Force

$\mathscr{F}_{e} = - \dfrac{ \partial \mathcal{U}_{e} }{ \partial \rho_{e} } = -\delta_{\widehat{e}} \, k_{\mathsf{F}} \sqrt{k_{ee}}$

Observed as force in uniform field between capacitor plates.

$\mathscr{F}_{e} = q \dfrac{ \Delta \! \mathsf{V} }{ d }$

Coulomb Force

$\mathscr{F}_{Coulomb} \equiv - \dfrac{ \partial \mathcal{U}_{Coulomb} }{ \partial R }$

Exchange particle: Coulomb force due to an interaction with a Coulomb photon formed by some field quanta from the ground-state of hydrogen which include electronic and muonic quarks .

Observed as: Coulomb’s law

$\mathscr{F}_{Coulomb} = \dfrac{1}{4\pi \varepsilon_{\textsf{o}}} \dfrac{ \; q \, q^{\mathsf{F}} }{ R^{2} }$

Gravitational Force

$\mathscr{F}_{centripetal} \equiv - \dfrac{ \partial \mathcal{U}_{binding} }{ \partial \rho_{z} }$

Exchange particle: Let $\partial \rho_{z}$ be due to an interaction with a graviton $\Gamma$ only.

Observed as 1) Classroom field

$\mathscr{U}_{g} = - k^{'}_{g} m \rho_{z}$

$\mathscr{F}_{g} = k^{'}_{g} m$

Observed as 2) Planetary field

$\mathscr{U}_{g} = - 2k_{g} \dfrac{\; mm^{\mathsf{F}} }{ R }$

$\mathscr{F}_{g} = k_{g} \dfrac{\; mm^{\mathsf{F}} }{ R^{2} }$

Strong Force

Exchange particle: strong force quanta

Observed as: transitions between energy levels of hydrogen

The entwined images and three-fold phases in this Iban weaving suggest a karmic knot. — Bidang, Iban people. Sarawak 20th century, 50 x 101 cm. Lintah motif. From the Teo Family collection, Kuching. Photograph by D Dunlop.

First Law of Motion

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

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 is meant by a straight line segment or a straight rod. The first law is also known as the law of inertia, it has been translated⁵Isaac 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

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.

— 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.

Collisions and Explosions

Here is an archetypal vignette from Newtonian mechanics. Two atoms called $\mathbf{A}$ and $\mathbf{B}$ have an interaction with each other by swapping another particle, $\mathsf{X}$ , which is called the exchange particle. The interaction is caused when $\mathbf{A}$ emits $\mathsf{X}$ at event $\mathbf{A}_{i}$ which is called the initial event of the interaction. This is written as

$\mathbf{A}_{\, i-1} \to \mathbf{A}_{i} + \mathsf{X}_{i}$

Particle $\mathsf{X}$ then has an effect on $\mathbf{B}$ by being absorbed at event $\mathbf{B}_{f}$ which is called the final event of the interaction. We express this by writing

$\mathbf{B}_{f} + \mathsf{X}_{f} \to \mathbf{B}_{f+1}$

For EthnoPhysics, the interaction is described using three repetitive chains of historically ordered events written as

$\Psi \! \left( \overline{r}, t \right)^{\mathbf{A}} = \left( \mathbf{A}_{1}, \, \mathbf{A}_{2} \, \ldots \, \mathbf{A}_{i} \, \ldots \, \mathbf{A}_{f} \, \ldots \right)$

$\Psi^{\mathsf{X}} = \left( \mathsf{X}_{i} \, \ldots \, \mathsf{X}_{f} \right)$

$\Psi \! \left( \overline{r}, t \right)^{\mathbf{B}} = \left( \mathbf{B}_{1}, \, \mathbf{B}_{2} \, \ldots \, \mathbf{B}_{i} \, \ldots \, \mathbf{B}_{f} \, \ldots \right)$

Since $\mathbf{A}$ and $\mathbf{B}$ are composed from atoms, we assume that they can be described by space-time events with a position $\overline{r}$ and time of occurrence $t$ . We do not assume that $\mathsf{X}$ is an atom, rather we often take it to be a photon or a graviton. So we cannot always describe $\mathsf{X}$ using a trajectory. And the position of $\mathsf{X}$ is well-defined only for the initial and final events where it is included as part of an atom. Overall, the interaction is characterized by the following quantities.

Momentum Change by Emission

The interaction is caused when $\mathbf{A}$ emits $\mathsf{X}$ at an event $\mathbf{A}_{\, i}$ which is called the initial event of the interaction. We write $\mathbf{A}_{\, i-1} \to \mathbf{A}_{i} + \mathsf{X}_{i} \, .$ But momentum is conserved so a total over all momenta $\overline{p}$ are the same before and after the interaction

$\overline{p}_{i-1}^{\mathbf{A}} = \overline{p}_{i}^{\mathbf{A}} + \overline{p}_{i}^{\mathsf{X}}$

Then the change in $\mathbf{A}$ ‘s momentum due to the emission of $\mathsf{X}$ is given by

$\Delta \overline{p}^{\mathbf{A}} \equiv \overline{p}_{i}^{\mathbf{A}} - \overline{p}_{i-1}^{\mathbf{A}} = - \, \overline{p}^{\mathsf{X}}$

Momentum Change by Absorption

Particle $\mathsf{X}$ has an effect on $\mathbf{B}$ by being absorbed at event $\mathbf{B}_{f}$ which is called the final event of the interaction. We write $\mathbf{B}_{f} + \mathsf{X}_{f} \to \mathbf{B}_{f+1} \, .$ But momentum is conserved so a total over all momenta $\overline{p}$ are the same before and after the interaction

$\overline{p}_{f}^{\mathbf{B}} + \overline{p}^{\mathsf{X}}_{f} = \overline{p}_{f+1}^{\mathbf{B}}$

Then the change in $\mathbf{B}$ ‘s momentum due to the emission of $\mathsf{X}$ is given by

$\Delta \overline{p}^{\mathbf{B}} \equiv \overline{p}_{f+1}^{\mathbf{B}} - \overline{p}_{f}^{\mathbf{B}} = \overline{p}^{\mathsf{X}}$

Karma and Newton’s Third Law

The third law of motion from Sir Isaac Newton is about a balance between the forces of cause and effect. It has been translated⁶Isaac Newton, Mathematical Principles of Natural Philosophy, page 417. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999. as

To any action there is always an opposite and equal reaction; in other words, the actions of two bodies upon each other are always equal and always opposite in direction.

— Sir Isaac Newton

This has something in common with the ancient Indian idea of karma a key concept in Hindu, Jain, Buddhist, Tao, Shinto and Sikh philosophies. It is also similar to Western aphorisms like, ‘what goes around comes around’. And we suppose that Newton himself was well aware of passages in the Jewish and Christian texts about reaping and sowing.

But Newton’s third law is much more than just a vague claim of cosmic balance. It has a scientifically precise expression in terms of two atoms called $\mathbf{A}$ and $\mathbf{B}$ that have an interaction with each other by exchanging a third particle called $\mathsf{X} \, .$

The interaction begins when $\mathbf{A}$ emits $\mathsf{X} \, .$ As discussed earlier, the momentum change due to this event is given by $\Delta \overline{p}^{\mathbf{A}} = - \, \overline{p}^{\mathsf{X}} .$ The emission causes the effect of $\mathsf{X}$ being absorbed into $\mathbf{B}$ . The momentum change due to absorption is $\Delta \overline{p}^{\mathbf{B}} = \overline{p}^{\mathsf{X}} .$ The elapsed time between emission and absorption is noted by $\Delta t$ . The forces acting on $\mathbf{A}$ and $\mathbf{B}$ are found by substituting their momentum changes into the definition of force to obtain

$\overline{F}^{\mathbf{A}} \equiv \dfrac{\; \Delta \overline{p}^{\mathbf{A}}}{\Delta t} = \dfrac{- \overline{p}^{\mathsf{X}}}{\Delta t}$

and

$\overline{F}^{\mathbf{B}} \equiv \dfrac{\; \Delta \overline{p}^{\mathbf{B}}}{\Delta t} = \dfrac{\; \overline{p}^{\mathsf{X}}}{\Delta t}$

so the force of the cause is of equal size and in the opposite direction to the force of the effect.

$\overline{F}^{\mathbf{A}} = - \, \overline{F}^{\mathbf{B}}$

Thus Newton succeeded in articulating a wisp of widespread karmic wisdom. His approach leads to the enormous benefit of being susceptible of scientific investigation. Its range and accuracy can be tested in our laboratories. This process of measuring and checking is what makes Newtonian mechanics so trustworthy.

The bilateral symmetries in this bead panel suggest a balance between cause and effect. — Baby Carrier Panel, Bahau people. Borneo 20th century, 35 x 28 cm. From the Teo Family collection, Kuching. Photograph by D Dunlop.

Tests and Predictions

Experimental tests of Ethnophysics are discussed. Magnetic susceptibility and magnetic moments are defined and compared with observations.

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References
1	Isaac Newton, Mathematical Principles of Natural Philosophy, page 639. Translated by Andrew Motte and Florian Cajori. University of California Press, 1934.
2	Isaac Newton, Mathematical Principles of Natural Philosophy, page 404. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999.
3	Isaac Newton, Mathematical Principles of Natural Philosophy, page 639. Translated by A Motte and F Cajori. University of California Press, 1934.
4, 5	Isaac Newton, Mathematical Principles of Natural Philosophy, page 416. Translated by I. Bernard Cohen and Anne Whitman. University of California Press 1999.
6	Isaac Newton, Mathematical Principles of Natural Philosophy, page 417. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999.