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Newtonian Mechanics

Outline

The theory of mechanics presented by Sir Isaac Newton in 1687 is the foundation of classical mechanics and modern physics. This article reviews his work by first considering velocity and acceleration. For EthnoPhysics, these dynamic concepts are defined by sensation, and established 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. This separation is due to a sum of displacements that 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) notes the displacement 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 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 utterly conventional relationship

 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 stated 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.

An icon indicating a quotation.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}

Acceleration

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 . We express this trajectory of P by writing

\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\|

Acceleration due to emission is suggested by the pilih formed imagery in this bidang from Sarawak.
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

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 . We represent this trajectory of P with the expression

\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 that3Isaac 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.

— 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.

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  \overline{r} and time of occurrence  t . We express this trajectory of P by writing

\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 translated4Isaac 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.

— 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 translated5Isaac Newton, Mathematical Principles of Natural Philosophy, page 417. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999. as

An icon indicating a quotation.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}}

Understanding these forces depends on what we mean by mass, length and time because these concepts are thoroughly entwined in all foregoing definitions. And for EthnoPhysics, these terms have been explicitly defined by sensation. In principle, these Newtonian forces can be observed and measured. Thus the range of validity, and the accuracy of Newton’s assertion may be scientifically tested in our laboratories. So next we discuss some EthnoPhysics faviconTests and Predictions.

A mechanics of cause and effect is suggested by this highly symmetric bead panel from Borneo.
Baby Carrier Panel, Bahau people. Borneo 20th century, 35 x 28 cm. From the Teo Family collection, Kuching. Photograph by D Dunlop.
References
1Isaac Newton, Mathematical Principles of Natural Philosophy, page 639. Translated by Andrew Motte and Florian Cajori. University of California Press, 1934.
2Isaac Newton, Mathematical Principles of Natural Philosophy, page 404. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999.
3, 4Isaac Newton, Mathematical Principles of Natural Philosophy, page 416. Translated by I. Bernard Cohen and Anne Whitman. University of California Press 1999.
5Isaac Newton, Mathematical Principles of Natural Philosophy, page 417. Translated by I. Bernard Cohen and Anne Whitman. University of California Press, 1999.