Author Archives: vinaire

I am originally from India. I am settled in United States since 1969. I turned 65 in June 2011. I am working as Production Systems Analyst. I love mathematics, philosophy and clarity in thinking.

Motion, Force & Inertia

ReferenceA Logical Approach to Theoretical Physics

With the discovery of quanta, we have a better understanding of particle and void. As we break down the particle of matter it ultimately reduces to quanta of radiation. The void is filled with quanta, and it ultimately reduces to emptiness.

Per Newton’s Definition II:

DEFINITION II: The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjunctly.

The motion of the whole is the sum of the motions of all the parts; and therefore in a body double in quantity, with equal velocity, the motion is double; with twice the velocity, it is quadruple.

A body of matter, such as, the heavenly bodies, moves in space at a uniform velocity. This velocity is shared by all particles that make up that body. The total motion of the body is the sum of the motion of all its particles. The measure of the velocities of different bodies is consistent only when they are relative to a standard reference body.

Velocities are absolutely consistent when measured relative to a reference-body at absolute rest.

Newton used the background of fixed stars as the reference-body at rest. Einstein, on the other hand, used the fixed velocity of light as a reference body of infinite velocity, which works in reverse.

Velocities may also be absolutely consistent when measured relative to a reference-body moving at infinite velocity.

Newton, however, was not sure if that background of fixed stars was at absolute rest. As far as the velocity of light goes, we know that it is very large but not infinite.

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Per Newton’s Definition III:

DEFINITION III: The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly forward in a right line.

This force is ever proportional to the body whose force it is; and differs nothing from the inactivity of the mass, but in our manner of conceiving it. A body, from the inactivity of matter, is not without difficulty put out of its state of rest or motion. Upon which account, this vis insita, may, by a most significant name, be called vis inertia, or force of inactivity. But a body exerts this force only, when another force, impressed upon it, endeavours to change its condition; and the exercise of this force may be considered both as resistance and impulse; it is resistance, in so far as the body, for maintaining its present state, withstands the force impressed; it is impulse, in so far as the body, by not easily giving way to the impressed force of another, endeavours to change the state of that other. Resistance is usually ascribed to bodies at rest, and impulse to those in motion; but motion and rest, as commonly conceived, are only relatively distinguished ; nor are those bodies always truly at rest, which commonly are taken to be so.

The body, when pushed, changes in velocity; but this change is inversely proportional to the mass. The velocity of a body of large mass may only be changed with difficulty. Newton viewed it as a resistance put up by the body and called it the “force of inertia”. Inertia keeps the body moving at a uniform velocity in a straight line, when there are no external forces. In other words,

Inertia smooths out the deviations, if any, from the uniform velocity of the body.

But deviations from uniform velocity can occur only when the body is being pushed around randomly. This means that mass of the body is a factor in smoothing out these deviations. The uniform velocity, which the body naturally acquires, will then depend upon its mass.

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Per Newton’s Definition IV:

DEFINITION IV: An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of moving uniformly forward in a right line.

This force consists in the action only; and remains no longer in the body, when the action is over. For a body maintains every new state it acquires, by its vis inertia only. Impressed forces are of different origins as from percussion, from pressure, from centripetal force.

The impressed force, or push, will definitely influence the uniform velocity of the body, but that velocity shall be restored back by inertia soon after the push is over. This restoration shall occur as argued above under the previous definition. But, according to Newton, the velocity increased by the momentary push is now maintained by inertia. This could only mean that the momentary push has somehow overcome part of the inertia permanently.

In essence, Newton is implying that the uniform velocity of a body is arbitrary and independent of its mass.

But this contradicts the logic that a body of infinite mass shall have no velocity because it cannot be moved. Other bodies of lesser mass shall have finite velocities relative to it. In general, a body of lesser mass shall revolve around a body of greater mass, and not the other way around. Light that has no mass, has a velocity many degrees of magnitude greater than a body with mass. Therefore,

Evidence suggests that the uniform velocity of a body is higher when its mass is lower. So, there is a relationship between a body’s uniform velocity and its mass.

This conclusion is supported by Faraday’s principle of Conservation of Force, because both uniform velocity and mass are manifestations of force, and the total force must be conserved.

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Per Newton’s Definition V:

DEFINITION V: A centripetal force is that by which bodies are drawn or impelled, or any way tend, towards a point as to a centre.

Of this sort is gravity, by which bodies tend to the centre of the earth; magnetism, by which iron tends to the loadstone; and that force, whatever it is, by which the planets are perpetually drawn aside from the rectilinear motions, which otherwise they would pursue, and made to revolve in curvilinear orbits… It is necessary, that the force be of a just quantity, and it belongs to the mathematicians to find the force, that may serve exactly to retain a body in a given orbit, with a given velocity; and vice versa, to determine the curvilinear way, into which a body projected from a given place, with a given velocity, may be made to deviate from its natural rectilinear way, by means of a given force…

A centripetal force requires a fixed location. A body of infinite mass shall have a completely fixed location. A body of lesser mass shall have a less fixed location. There is a certain degree of uncertainty in its location, which is expressed as a less fixed body revolving around a more fixed body. In case of void there are quanta instead of mass particles and, therefore, there is no fixity in the void. It will have an infinite radius of revolution that would make quanta to move in straight lines.

The geometry of the real world has mass integral to it.

Newton assumed space (aether) to be immovable. For this to be true, it must be fixed everywhere, and this requires aether to have infinite mass. But that is not possible. Newton’s assumption of “immovable space” seems to arise from his observation of the background of fixed stars.

Absolutely fixed points are only those that are infinite in mass.

Other points in space are less fixed and have uncertainties associated with them in the form of uniform velocities. Less fixed means less mass and greater velocity.

The uniform velocity is a measure of the lack of fixity of a point. The higher is the velocity of a point, the lower is the mass there.

Note that the location of a body of lot of mass particles will be determined by the center of mass.

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Particle, Void & Emptiness

ReferenceA Logical Approach to Theoretical Physics

Per Newton’s Definition I:

DEFINITION I: The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

Thus air of a double density, in a double space, is quadruple in quantity; in a triple space, sextuple in quantity. The same thing is to be understood of snow, and fine dust or powders, that are condensed by compression or liquefaction and of all bodies that are by any causes whatever differently condensed. I have no regard in this place to a medium, if any such there is, that freely pervades the interstices between the parts of bodies. It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight, as I have found by experiments on pendulums, very accurately made, which shall be shewn hereafter.

The quantity of matter depends on how densely its particles are packed in a volume of space. The particles of matter are separated from each other by a continuous void. The actual sense of void is “void of matter”. Void may still contain other things, such as, radiation and force fields.

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Matter and Particle

Today we may see the actual particle as the nucleus of the atom where the matter is concentrated; and it is surrounded by void. On a different scale, particle may represent a heavenly body surrounded by vast distances. We think of a particle as a ball-like body of solid matter. According to Wikipedia:

“A particle is a small localized object to which can be ascribed several physical or chemical properties such as volume, density or mass.”

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The Void and Aether

The void between the nuclei of a molecule is filled with non-material electromagnetic field. The distance between nuclei gives us an estimate of the radius of an atom. Beyond the atoms and molecules we have the larger void of space filled with radiation, such as, light.

Void may be actually be described by the “electromagnetic” spectrum of the non-atomic radiation. It is much denser when close to the mass of nucleus. It becomes lighter as it spreads out from the nucleus. It is not homogenous. This fact was recognized for the first time in his 1905 paper on Light quanta. Einstein says,

According to Maxwell’s theory, the energy must be considered to be a continuous function in space for all purely electromagnetic phenomena, thus also for light, while according to the present-day ideas of physicists the energy of a ponderable body can be written as a sum over the atoms and electrons. The energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts…

Maxwell looked at the radiative phenomenon to be continuous in terms of energy, which distributed continuously over an ever increasing volume. According to Newton energy resides in the motion of particles. Maxwell was, therefore, assuming aether-like substance distributed evenly throughout space. Since the atoms and electrons were looked upon as discrete points of energy in space, Einstein focused on what happens when light is emitted and transformed from atoms.

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Einstein and Light Quanta

In his paper, Einstein sets up a model of blackbody radiation similar to the model in kinetic theory of gases. He is then able to show mathematically that that the entropy of monochromatic radiation follows the same rules as the entropy of a perfect gas. This is an ingenious way of proving that the energy distribution of radiation becomes particle-like at higher frequencies. Einstein thus showed that Planck’s postulate of energy quanta was more than a mathematical device. Einstein, therefore, concludes:

According to the assumption considered here, when a light ray starting from a point is propagated, the energy is not continuously distributed over an ever increasing volume, but it consists of a finite number of energy quanta, localised in space, which move without being divided and which can be absorbed or emitted only as a whole.

In other words, there is no aether-like substance distributed evenly throughout space. The substance of void is not homogenous. Its “density” depends on its frequency. It becomes denser with higher frequency and less dense with lower frequency.

Einstein makes the assumption: “the energy of a ponderable body cannot be split into arbitrarily many, arbitrarily small parts”. In other words, Einstein assumes that the solidity of a particle is fixed: it can only be cut into smaller particles, while its solidity remains fixed. This is where quantum mechanics stops theoretically.

The truth is that the solidity of a particle can become less solid without limit.

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Quanta and Substantiality

The idea of ponderability, or solidity, of a particle is tied with substantiality of substance. The particle of matter is considered to have the same substantiality while it becomes smaller. Of course, there is a limit to how small a particle may become. But the dimension of lessening substantiality is not considered here.

We may say that matter is a substance of very high substantiality called mass. On the other hand, void represents a substance of very low substantiality called quanta.

Thus, the substantiality of matter is not fixed. The substantiality of quanta lessens as its frequency decreases.

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The “Quantum Particle”

Mass and quanta are two different aspect of substance. A “quantum particle” does not refer to solid ball-like mass particle. It is much less solid and stretches out in all directions. With increasing frequency these “stretched-out particles” compress to become thicker and less stretched until they become solid and ball-like. In other words, the increasing frequency of “quantum particle” of radiation ultimately compresses it into a matter particle. This follows Bohr’s correspondence principle that demands that classical physics and quantum physics give the same answer when the systems become large.

A “quantum particle” stretches out in all directions like Faraday’s lines of force.

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Substance and Force

Substance is perceived by its force and resistance. Energy is the dynamic aspect of force. Thus, mass, quanta, and energy account for the force of substance. This definition of force was used by Faraday when he insisted on the principle of Conservation of Force. This definition is much broader than the mathematical definition of “force” given by Newton.

Faraday’s conservation of force means the conservation of mass, quanta and energy.

Substance occupies space and endures in time. This applies to both mass of matter and quanta of void.

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Universe and Emptiness

The universe may be perceived as a huge molecule that is occupying space and persisting in time. The interstices of this universe are filled by void. The void may extend beyond the universe until it loses any and all substantiality. With no substantiality there is no substance to occupy space and endure in time. We then have an “emptiness” characterized by absence of substance, space and time.

Beyond this universe we have emptiness characterized by absence of substance, space and time.

This concept of “emptiness” is difficult to grasp. It may be new to science, but this concept has existed in the East since ancient times.

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The Background of Modern Physics

ReferenceA Logical Approach to Theoretical Physics

Copernicus (1473 – 1543)
Tycho Brahe (1546 – 1601)
Galileo Galilei (1564 – 1642)
Johannes Kepler (1571 – 1630)
René Descartes (1596–1650)
Isaac Newton (1642 – 1726)

We start learning physics through mathematics that describes space and time in a material world. The mathematics uses the Cartesian coordinates, which assign continuity and uniformity to space and time. One learns to plot graphs of the relationships between space and time. These graphs describe the paths traced by moving particles of matter.

Matter appears in the form of particles because it is not continuous. Particle ends where void starts. The “void” is the empty space. Particles occupy space by displacing the void.  Particles move in the void. They have certain uniform velocities. These particles accelerate when pushed and their velocities change. When the force of the push goes away the velocity becomes uniform again.

We study the motion of particles in straight lines and in circles and spirals. This requires the consideration of more than one dimension of space. We consider motion to be relative because we can’t image a particle being at absolute rest. We find this subject of mechanics thoroughly explored by Newton, and so we study Newton’s Laws of Motion.

The background of physics starts with simple concepts of particle, void, motion and force.

These concepts seem to model the big picture of the cosmos adequately, where the laws of motion apply to the satellites, planets and stars in the huge void. With a little modification, these concepts also appear to model the phenomenon of heat. This brings us to the laws of thermodynamics. Furthermore, these concepts are employed to explain the phenomena of sound, electricity, magnetism, and light. We see them used even in the explanation of atomic, nuclear and quantum phenomenon. In the succeeding chapters we shall review each of these basic concepts and see how they have evolved over time.

Physics is characterized by the scientific method, which started with Galileo. This method establishes consistency between theory and reality. It uses experiments and mathematics. The scientific method starts with certain postulates and assumptions. New discoveries are made when we examine past assumptions and improve upon them. This is where logic comes in.

This work examines that logic.

Principia 1687: Laws of Motion

ReferenceA Logical Approach to Theoretical Physics

This paper presents the chapter on LAWS OF MOTION from the English translation of NEWTON’S PRINCIPIA, American edition, 1846.

The paragraphs of original material are accompanied by brief comments in color based on present understanding. The heading below links to the original materials.

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Laws of Motion

LAW I:  Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.

Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve then motions both progressive and circular for a much longer time.

The uniform motion of a body is characterized by constant velocity. This velocity changes only when the body is being pushed around by forces. The greater is the mass of a body, the lesser are the fluctuations in its velocity. This is seen as the uniform motion of the body being maintained by the inertia of matter (see Definition III).

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LAW II: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.

If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impressed altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force), if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joined, when they are oblique, so as to produce a new motion compounded from the determination of both.

The greater is the push the bigger is the change in the resulting velocity of a body. The magnitude of the push is determined by its force and duration. It is noted that for a push to continue, the body pushing must attempt to move faster than the body being pushed. The overall alteration of motion is proportional to the push.

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LAW III: To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tied to a rope, the horse (if I may so say) will be equally drawn back towards the stone: for the distended rope, by the same endeavour to relax or unbend itself, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinges upon another and by its force change the motion of the other, that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities but in the motions of bodies; that is to say, if the bodies are not hindered by any other impediments. For, because the motions are equally changed, the changes of the velocities made towards contrary parts are reciprocally proportional to the bodies. This law takes place also in attractions, as will be proved in the next scholium.

A push (impressed force) requires a contact. Here actions are considered to be reciprocal because they are balanced for the duration of the contact, for example, in the case of a person standing on the floor, or in an elastic collision. But, when the push is continuous over a long duration, it is balanced by the mass (inertia) coming into play through acceleration.

CRITICAL COMMENT

These three laws of Newton, when examined closely, lead to the conclusion that the natural uniform velocity of a body in space shall ultimately depend on its mass or inertia. The higher is the mass the lesser is the velocity and vice versa. You cannot keep increasing the velocity of a body without decreasing its mass (inertia). This is not the same consideration as given by relativity.

This explains why the velocity of light (that has no mass) is many degrees of magnitude higher than the velocity of matter (that has mass).

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COROLLARY I: A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it would describe the sides, by those forces apart.

When a body is pushed in two different directions simultaneously, it is equivalent to a single push along the diagonal of the parallelogram formed as above by the two pushes M and N. This geometry is based on the uniform velocity due to inertia.

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COROLLARY II: And hence is explained the composition of any one direct force AD, out of any two oblique forces AC and CD ; and, on the contrary, the resolution of any one direct force AD into two oblique forces AC and CD : which composition and resolution are abundantly confirmed from mechanics.

Thus, two forces AC and CD may be shown as equivalent to a single force AD. Similarly, a single force AD may be resolved in any two directions AC and CD.

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COROLLARY III: The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.

This is the principle of conservation of momentum of a system. In a collision, motion may transfer from one body to another as velocity. This change in mass is so small that it is ignored. If there is a change in the mass of a body it is in the inertia (quantization) of each particle, and not in the number of particles.

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COROLLARY IV: The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line.

This geometry is based on the rigidity of matter, and on the fact that the effect of a force is inversely proportional to the distance from the point of effect.

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COROLLARY V: The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.

Newton is assuming that the same relative velocities shall be maintained by the objects of a system if the same absolute velocity is added to all of them. This may be so if the velocity added is small. But when that velocity added is high the mass starts to reduce significantly (see the CRITICAL COMMENT under Law III).  Objects of different masses may respond differently, and the relative speeds may not be maintained.

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COROLLARY VI: If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces.

Again, the objects of different masses may behave differently when the push is very large and/or applied for a long duration. The relative velocities among objects may not be maintained.

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SCHOLIUM

The experiments described by Newton in this scholium deal with low velocities only. Hence deviations from these conclusions at high velocities shall not be detectable.

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Walking Meditation

Reference: A Scientific Approach to Meditation

Walking meditation is done while walking in an open and pleasant environment, such as, in a farm, park or a garden. One meditates on the body and the physical environment while walking. Like in any meditation, the guiding principle is “being there and seeing things as they are.” Walking meditation on the body and on the physical environment is best done separately.

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The Body

In walking meditation on the body, one allows stresses in the body to unwind, so the body regains its relaxed and natural form.

Once you start walking you, simply start observing the body without interfering with it.  Become aware of the natural pattern of your breathing. Notice the disposition of the various parts of the body at different times: in walking, turning, bending, stretching, stopping, etc. Get the feel of the clothes on the body, their weight, temperature, etc. After a little while, start putting you attention on different parts of the body and feel any sensations, aches or pains present.  You let the stresses in the body unwind on their own, so the body starts to become increasingly relaxed.

All this while, you use your breathing as the stabilizing factor. In other words, whenever your attention strays you bring it back to your breathing and start observing the body again.

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The Physical Environment

In walking meditation on the physical environment, one observes one’s physical perceptions until they become clear and sharp.

Once you start walking you, simply start observing the physical environment without interfering with your perceptions.  Use breathing as your stabilizing factor, as before. At first you focus on the physical perception of sight. Notice the size, shape and color of the things in the environment and their overall visual pattern. Look as far as you can see. Then start putting attention on the perception of hearing also. Notice the quality, tone and loudness of sounds. Do this until your perceptions of sight and hearing start to become sharper.

At the end of your walk you may go to a coffee or tea place. There you practice the perceptions of touch, taste and smell as above.

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Summary

The whole idea of walking meditation is to perceive things as they are. One perceives the sensations, pains and aches in the body as they are. Similarly, one perceives things in the environment just as they are.

This exercise may be done again and again until one’s perceptions of the body and the physical environment improve.

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