KNOWABLE THOUGHTS By: Richard j.Kosciejew

KNOWABLE THOUGHTS



RICHARD J.KOSCIEJEW





The title has a dramatic quality that does not rest exclusively on the theory of relativity or quantum mechanics. Perhaps, the most startling and potentially revolutionary of implications in human terms is a new perspective on the relationship between mind and the world that is utterly different from that sanctioned by classical physics. René Descartes, for reasons of which was among the first to realize that mind or consciousness in the mechanistic world-view of classical physics appeared to exist in a realm separate and distinct from nature. The prospect was that the realm of the mental is a self-contained and self-referential island universe with no real or necessary connection with the universe itself.

It also tends the belief . . . that all men dance to the tune of an invisible piper. Yet, this may not be so, as whenever a system is really complicated, indeterminacy comes in, not necessarily because of ‘h’ ( Planck constant ) but because to make a prediction so we must know many things that the stray consequences of studying them will disturb the status quo, due to which formidable comminations can never therefore answer -history is not and cannot be determined. The supposed causes may only produce the consequences we expect. This has rarely been more true of those whose thought and action in science and life became interrelated in a way no dramatist would dare to conceive, this itself has some extraordinary qualities if determinacy, which in physics is so reluctant to accept.

A presence awaiting to the future has framed its proposed new understanding of the relationship between mind and world within the larger context of the history of mathematical physics, the origin and extensions of the classical view of the fundamentals of scientific knowledge, and the various ways that physicists have attempted to prevent previous challenges to the efficacy of classical epistemology. There is no basis in contemporary physics or biology for believing in the stark Cartesian division between mind and world that some have moderately described as ‘the disease of the Western mind’. The dialectic orchestrations will serve as background for understanding a new relationship between parts and wholes in physics, with a similar view of that relationship that has emerged in the co-called ‘new biology’ and in recent studies of the evolution of a scientific understanding to a more conceptualized representation of ideas, and includes its allied ‘content’.

Descartes, the founder of modern philosophy quickly realized that there appears of nothing in viewing nature that shows possibilities of reconciliation between a full-fledged comparison, as between Plotinus and Whitehead view for which posits of itself outside the scope of concerns, in that the comparability is with the existent idea of ‘God’, especially. However, that ‘the primordial nature of God’, whom in which is eternal, a consequent of nature, which is in flux, as far as, this difference of thought remains but comprises no bearing on the relationship or either with the quantum theory, as it addresses the actual notion that authenticates the representation of actual entities as processes of self-creation.

Nonetheless, it seems a strong possibility that Plotonic and Whitehead connect upon the issue of the creation of the sensible world may by looking at actual entities as aspects of nature’s contemplation. The contemplation of nature is obviously an immensely intricate affair, involving a myriad of possibilities, therefore one can look at actual entities as, in some sense, the basic elements of a vast and expansive process.

We could derive a scientific understanding of these ideas with the aid of precise deduction, as Descartes continued his claim that we could lay the contours of physical reality out in three-dimensional co-ordinates. Following the publication of Isaac Newton’s “Principia Mathematica” in 1687, reductionism and mathematical modeling became the most powerful tools of modern science. The dream that we could know and master the entire physical world through the extension and refinement of mathematical theory became the central feature and principals of scientific knowledge.

The radical separation between mind and nature formalized by Descartes served over time to allow scientists to concentrate on developing mathematical descriptions of matter as pure mechanism without any concern about its spiritual dimensions or ontological foundations. Meanwhile, attempts to rationalize, reconcile or eliminate Descartes’s merging division between mind and matter became the most central feature of Western intellectual life.

Philosophers like John Locke, Thomas Hobbes, and David Hume tried to articulate some basis for linking the mathematical describable motions of matter with linguistic representations of external reality in the subjective space of mind. Descartes’ compatriot Jean-Jacques Rousseau reified nature as the ground of human consciousness in a state of innocence and proclaimed that “Liberty, Equality, Fraternities” are the guiding principles of this consciousness. Rousseau also fabricated the idea of the ‘general will’ of the people to achieve these goals and declared that those who do not conform to this will were social deviants.

The Enlightenment idea of ‘deism’, which imaged the universe as a clockwork and God as the clockmaker, provided grounds for believing in a divine agency, from which the time of moment the formidable creations also imply, in of which, the exhaustion of all the creative forces of the universe at origins ends, and that the physical substrates of mind were subject to the same natural laws as matter. In that the only means of mediating the gap between mind and matter was pure reason, causally by the traditional Judeo-Christian theism, which had previously been based on both reason and revelation, responded to the challenge of deism by debasing tradionality as a test of faith and embracing the idea that we can know the truths of spiritual reality only through divine revelation. This engendered a conflict between reason and revelation that persists to this day. And laid the foundation for the fierce completion between the mega-narratives of science and religion as frame tales for mediating the relation between mind and matter and the manner in which they should ultimately define the special character of each.

The nineteenth-century Romantics in Germany, England and the United States revived Rousseau’s attempt to posit a ground for human consciousness by reifying nature in a different form. Goethe and Friedrich Schelling proposed a natural philosophy premised on ontological Monism ( the idea that adhering manifestations that govern toward evolutionary principles have grounded inside an inseparable spiritual Oneness ) and argued God, man, and nature for the reconciliation of mind and matter with an appeal to sentiment, mystical awareness, and quasi-scientific attempts, as he afforded the efforts of mind and matter, nature became a mindful agency that ‘loves illusion’, as it shrouds man in mist, presses him or her heart and punishes those who fail to see the light. Schelling, in his version of cosmic unity, argued that scientific facts were at best partial truths and that the mindful creative spirit that unities mind and matter is progressively moving toward self-realization and ‘undivided wholeness’.

The British version of Romanticism, articulated by figures like William Wordsworth and Samuel Taylor Coleridge, placed more emphasis on the primary of the imagination and the importance of rebellion and heroic vision as the grounds for freedom. As Wordsworth put it, communion with the “incommunicable powers” of the “immortal sea” empowers the mind to release itself from all the material constraints of the laws of nature. The founders of American transcendentalism, Ralph Waldo Emerson and Henry David Theoreau, articulated a version of Romanticism that commensurate with the ideals of American democracy.

The American envisioned a unified spiritual reality that manifested itself as a personal ethos that sanctioned radical individualism and bred aversion to the emergent materialism of the Jacksonian era. They were also more inclined than their European counterpart, as the examples of Thoreau and Whitman attest, to embrace scientific descriptions of nature. However, the Americans also dissolved the distinction between mind and natter with an appeal to ontological monism and alleged that mind could free itself from all the constraint of assuming that by some sorted limitation of matter, in which such states have of them, some mystical awareness.

Since scientists, during the nineteenth century were engrossed with uncovering the workings of external reality and seemingly knew of themselves that these virtually overflowing burdens of nothing, in that were about the physical substrates of human consciousness, the business of examining the distributive contribution in dynamic functionality and structural foundation of mind became the province of social scientists and humanists. Adolphe Quételet proposed a ‘social physics’ that could serve as the basis for a new discipline called sociology, and his contemporary Auguste Comte concluded that a true scientific understanding of the social reality was quite inevitable. Mind, in the view of these figures, was a separate and distinct mechanism subject to the lawful workings of a mechanical social reality.

More formal European philosophers, such as Immanuel Kant, sought to reconcile representations of external reality in mind with the motions of matter-based on the dictates of pure reason. This impulse was also apparent in the utilitarian ethics of Jerry Bentham and John Stuart Mill, in the historical materialism of Karl Marx and Friedrich Engels, and in the pragmatism of Charles Smith, William James and John Dewey. These thinkers were painfully aware, however, of the inability of reason to posit a self-consistent basis for bridging the gap between mind and matter, and each remains obliged to conclude that the realm of the mental exists only in the subjective reality of the individual.

The fatal flaw of pure reason is, of course, the absence of emotion, and purely explanations of the division between subjective reality and external reality, of which had limited appeal outside the community of intellectuals. The figure most responsible for infusing our understanding of the Cartesian dualism with contextual representation of our understanding with emotional content was the death of God theologian Friedrich Nietzsche 1844-1900. After declaring that God and ‘divine will’, did not exist, Nietzsche reified the ‘existence’ of consciousness in the domain of subjectivity as the ground for individual ‘will’ and summarily reducing all previous philosophical attempts to articulate the ‘will to truth’. The dilemma, forth in, had seemed to mean, by the validation, . . . as accredited for doing of science, in that the claim that Nietzsche’s earlier versions to the ‘will to truth’, disguises the fact that all alleged truths were arbitrarily created in the subjective reality of the individual and are expressed or manifesting the individualism of ‘will’.

In Nietzsche’s view, the separation between mind and matter is more absolute and total than previously been imagined. Based on the assumption that there is no really necessary correspondence between linguistic constructions of reality in human subjectivity and external reality, he deuced that we are all locked in ‘a prison house of language’. The prison as he concluded it, was also a ‘space’ where the philosopher can examine the ‘innermost desires of his nature’ and articulate a new message of individual existence founded on ‘will’.

Those who fail to enact their existence in this space, Nietzsche says, are enticed into sacrificing their individuality on the nonexistent altars of religious beliefs and democratic or socialists’ ideals and become, therefore, members of the anonymous and docile crowd. Nietzsche also invalidated the knowledge claims of science in the examination of human subjectivity. Science, he said. Is not exclusive to natural phenomenons and favors reductionistic examination of phenomena at the expense of mind? It also seeks to reduce the separateness and uniqueness of mind with mechanistic descriptions that disallow and basis for the free exercise of individual will.

Nietzsche’s emotionally charged defense of intellectual freedom and radial empowerment of mind as the maker and transformer of the collective fictions that shape human reality in a soulless mechanistic universe proved terribly influential on twentieth-century thought. Furthermore, Nietzsche sought to reinforce his view of the subjective character of scientific knowledge by appealing to an epistemological crisis over the foundations of logic and arithmetic that arose during the last three decades of the nineteenth century. Through a curious course of events, attempted by Edmund Husserl 1859-1938, a German mathematician and a principal founder of phenomenology, wherefor to resolve this crisis resulted in a view of the character of consciousness that closely resembled that of Nietzsche.

The best-known disciple of Husserl was Martin Heidegger, and the work of both figures greatly influenced that of the French atheistic existentialist Jean-Paul Sartre. The work of Husserl, Heidegger, and Sartre became foundational to that of the principal architects of philosophical postmodernism, and deconstructionist Jacques Lacan, Roland Barthes, Michel Foucault and Jacques Derrida. It obvious attribution of a direct linkage between the nineteenth-century crisis about the epistemological foundations of mathematical physics and the origin of philosophical postmodernism served to perpetuate the Cartesian two-world dilemma in an even more oppressive form. It also allows us better to understand the origins of cultural ambience and the ways in which they could resolve that conflict.

The mechanistic paradigm of the late n nineteenth century was the one Einstein came to know when he studied physics. Most physicists believed that it represented an eternal truth, but Einstein was open to fresh ideas. Inspired by Mach’s critical mind, he demolished the Newtonian ideas of space and time and replaced them with new, “relativistic” notions.

Two theories unveiled and unfolding as their phenomenal yield held by Albert Einstein, attributively appreciated that the special theory of relativity (1905) and, also the tangling and calculably arranging affordance, as drawn upon the gratifying nature whom by encouraging the finding resolutions upon which the realms of its secreted reservoir in continuous phenomenons, in additional the continuatives as afforded by the efforts by the imagination were made discretely available to any the unsurmountable achievements, as remain obtainably afforded through the excavations underlying the artifactual circumstances that govern all principle ‘forms’ or ‘types’ in the involving evolutionary principles of the general theory of relativity (1915). Where the special theory gives a unified account of the laws of mechanics and of electromagnetism, including optics. Before 1905 the purely relative nature of uniform motion had in part been recognized in mechanics, although Newton had considered time to be absolute and postulated absolute space. In electromagnetism the ether was supposed to give an absolute bases respect to which motion could be determined. The Galilean transformation equations represent the set of equations:

χʹ = χ ‒ vt

yʹ = y

zʹ = z

tʹ = t

They are used for transforming the parameters of position and motion from an observer at the point ‘O’ with co-ordinates (z, y, z) to an observer at Oʹ with co-ordinates (χʹ, yʹ z). The axis is chosen to pass through O and Oʹ. The times of an event at ‘t’ and tʹ in the frames of reference of observers at O and Oʹ coincided. ‘V’ is the relative velocity of separation of O and Oʹ. The equation conforms to Newtonian mechanics as compared with Lorentz transformation equations, it represents a set of equations for transforming the position-motion parameters from an observer at a point O(χ, y, z) to an observer at Oʹ(χʹ, yʹ, z ), moving compared with one another. The equation replaces the Galilean transformation equation of Newtonian mechanics in reactivity problems. If the x-axes are chosen to pass through Oʹ and the time of an event are t and tʹ in the frame of reference of the observers at O and Oʹ respectively, where the zeros of their time scales were the instants that O and Oʹ supported the equations are:

χʹ = β( χ ‒ vt )

yʹ = y

zʹ =z

tʹ = β( t ‒ vχ / c2 ),

Where ‘v’ is the relative velocity of separation of O, Oʹ, c is the speed of light, and β is the function

(1 ‒ v2 / c2 )-½.

Newton’s laws of motion in his “Principia,” Newton (1687) stated the three fundamental laws of motion, which are the basis of Newtonian mechanics.

The First Law of acknowledgement concerns that all bodies persevere in its state of rest, or uniform motion in a straight line, but in as far as it is compelled, to change that state by forces impressed on it. This may be regarded as a definition of force.

The Second Law to acknowledge is, that the rate of change of linear momentum is propositional to the force applied, and takes place in the straight line in which that force acts. This definition can be regarded as formulating a suitable way by which forces may be measured, that is, by the acceleration they produce,

F = d( mv ) / dt

i.e., F = ma = v( dm / dt ),

Where F = force, m = masses, v = velocity, t = time, and ‘a’ = acceleration, from which case, the proceeding majority of quality values were of non-relativistic cases of, dm / dt = 0, i.e., the mass remains constant, and then

F = ma.

The Third Law acknowledges, that forces are caused by the interaction of pairs of bodies. The forces exerted by ‘A’ upon ‘B’ and the force exerted by ‘B’ upon ‘A’ are simultaneous, equal in magnitude, opposite in direction and in the same straight line, caused by the same mechanism.

Appreciating the popular statement of this law in terms of significant “action and reaction” leads too much misunderstanding. In particular, any two forces that happen to be equal and opposite if they act on the same body, one force, arbitrarily called “reaction,” are supposed to be a consequence of the other and to happen subsequently, as two forces are supposed to oppose each other, causing equilibrium, certain forces such as forces exerted by support or propellants are conventionally called “reaction,” causing considerable confusion.

The third law may be illustrated by the following examples. He gravitational force exerted by a body on the earth is equal and opposite to the gravitational force exerted by the earth on the body. The intermolecular repulsive force exerted on the ground by a body resting on it, or hitting it, is equal and opposite to the intermolecular repulsive force exerted on the body by the ground. More general system of mechanics has been given by Einstein in his theory of relativity. This reduces to Newtonian mechanics when all velocities relative to the observer are small compared with those of light.

Einstein rejected the concept of absolute space and time, and made two postulates (i ) The laws of nature are the same for all observers in uniform relative motion, and (ii) The speed of light in the same for all such observers, independently of the relative motions of sources and detectors. He showed that these postulates were equivalent to the requirement that co-ordinates of space and time used by different observers should be related by Lorentz transformation equations. The theory has several important consequences.

The transformation of time implies that two events that are simultaneous according to one observer will not necessarily be so according to another in uniform relative motion. This does not affect the construct of its sequence of related events so does not violate any conceptual causation. It will appear to two observers in uniform relative motion that each other’s clock runs slowly. This is the phenomenon of ‘time dilation’, for example, an observer moving with respect to a radioactive source finds a longer decay time than found by an observer at rest with respect to it, according to:

Tv = T0 / ( 1 ‒ v2 / c2 ) ½

Where Tv is the mean life measurement by an observer at relative speed ‘v’, and T(v) is the mean life maturement by an observer at rest, and ‘c’ is the speed of light.

This formula has been verified in innumerable experiments. One consequence is that no body can be accelerated from a speed below ‘c’ with respect to any observer to one above ‘c’, since this would require infinite energy. Einstein educed that the transfer of energy δE by any process entailed the transfer of mass δm where δE = δmc2, hence he concluded that the total energy ‘E’ of any system of mass ‘m’ would be given by:

E = mc2

The principle of conservation of mass states that in any system is constant. Although conservation of mass was verified in many experiments, the evidence for this was limited. In contrast the great success of theories assuming the conservation of energy established this principle, and Einstein assumed it as an axiom in his theory of relativity. According to this theory the transfer of energy ‘E’ by any process entails the transfer of mass m = E/c2./ hence the conservation of energy ensures the conservation of mass.

In Einstein’s theory inertial and gravitational masses are assumed to be identical and energy is the total energy of a system. Some confusion often arises because of idiosyncratic terminologies in which the words mass and energies are given different meanings. For example, some particle physicists use “mass” to mean the rest-energy of a particle and “energy” to mean ‘energy other than rest-energy’. This leads to alternate statements of the principle, in which terminology is not generally consistent. Whereas, the law of equivalence of mass and energy such that mass ‘m’ and energy ‘E’ are related by the equation E = mc2, where ‘c’ is the speed of light in a vacuum. Thus, a quantity of energy ‘E’ has a mass ‘m’ and a mass ‘m’ has intrinsic energy ‘E’. The kinetic energy of a particle as determined by an observer with relative speed ‘v’ is thus ( m ‒ m0 )c2, which tends to the classical value ½mv2 if ≪ C.

Attempts to express quantum theory in terms consistent with the requirements of relativity were begun by Sommerfeld (1915), eventually. Dirac (1928) gave a relativistic formulation of the wave mechanics of conserved particles (fermions). This explained the concept of spin and the associated magnetic moment, which had been postulated to account for certain details of spectra. The theory led to results of extremely great importance for the theory of standard or elementary particles. The Klein-Gordon equation is the relativistic wave equation for ‘bosons’. It is applicable to bosons of zero spin, such as the ‘pion’. In which case, for example the Klein-Gordon Lagrangian describes a single spin-0, scalar field:

L = ½[∂t∂t‒ ∂y∂y‒ ∂z∂z] ‒ ½(2πmc / h)22

In this case

∂L/∂(∂) = ∂μ

Leading to the equation

∂L/∂ = (2πmc/h)22+

And hence the Lagrange equation requires that

∂μ∂μ + (2πmc / h)2 2 = 0.

Which is the Klein-Gordon equation describing the evolution in space and time of field ‘’? Individual ‘’ excitation of the normal modes of represents particles of spin -0, and mass ‘m’.

A mathematical formulation of the special theory of relativity was given by Minkowski. It is based on the idea that an event is specified by there being a four-dimensional co-ordinates, three of which are spatial co-ordinates and one in a dimensional frame in a time co-ordinates. These continuously of dimensional co-ordinate give to define a four-dimensional space and the motion of a particle can be described by a curve in this space, which is called “Minkowski space-time.” In certain formulations of the theory, use is made of a four-dimensional do-ordinate system in which three dimensions represent the spatial co-ordinates χ, y, z and the fourth dimension are ‘ict’, where ‘t’ is time, ‘c’ is the speed of light and ‘I’ is √ - 1, points in this space are called events. The equivalent to the distance between two points is the interval (δs) between two events given by the Pythagoras law in a space-time as:

(δs)2 = ij ηij δ χi χj

Where:

χ = χ1, y = χ2, z = χ3 . . . , t = χ4 and η11 (χ) η33 (χ) = 1? η44 (χ) = 1

is component of the Minkowski metric tensor. The distances between two points are variant under the ‘Lorentz transformation’, because the measurements of the positions of the points that are simultaneous according to one observer in uniform motion with respect to the first. By contrast, the interval between two events is invariant.

The equivalents to a vector in the four-dimensional space are consumed by a ‘four vector’, in which has three space components and one of time component. For example, the four-vector momentum has a time component proportional to the energy of a particle, the four-vector potential has the space co-ordinates of the magnetic vector potential, while the time co-ordinates corresponds to the electric potential.

between nonaccelerated frames of reference. The general theory reals with general relative motion between accelerated frames of reference. In accelerated systems of reference, certain fictitious forces are observed, such as the centrifugal and Coriolis forces found in rotating systems. These are known as fictitious forces because they disappear when the observer transforms to a nonaccelerated system. For example, to an observer in a car rounding a bend at constant velocity, objects in the car appear to suffer a force acting outward. To an observer outside the car, this is simply their tendency to continue moving in a straight line. The inertia of the objects is seen to cause a fictitious force and the observer can distinguish between non-inertial (accelerated) and inertial (Nonaccelerated) frames of reference.

A further point is that, to the observer in the car, all the objects are given the same acceleration irrespective of their mass. This implies a connection between the fictitious forces arising from accelerated systems and forces due to gravity, where the acceleration produced is independent of the mass. Near the surface of the earth the acceleration of free fall, ‘g’, is measured with respect to a nearby point on the surface. Because of the axial rotation the reference point is accelerated to the centre of the circle of its latitude, hence ‘g’ is not quite in magnitude or direction to the acceleration toward the centre of the earth given by the theory of ‘gravitation’ in 1687 Newton presented his law of universal gravitation, according to which every particle evokes every other particle with the force, ‘F’ given by:

F = Gm1 m2 / χ2,

Where m1, m2 is the masses of two particles a distance ‘χ’ apart, and ‘G’ is the gravitational constant, which, according to modern measurements, has a value

6.672 59 x 10-11 m3 kg -1 s -2.

For extended bodies the forces are found by integrations. Newton showed that the external effect of a spherical symmetric body is the same as if the whole mass were concentrated at the centre. Astronomical bodies are roughly spherically symmetrical so can be treated as point particles to a very good approximation. On this assumption Newton showed that his law was consistent with Kepler’s Laws. Until recently, all experiments have confirmed the accuracy of the inverse square law and the independence of the law upon the nature of the substances, but in the past few years evidence has been found against both.

The size of a gravitational field at any point is given by the force exerted on unit mass at that point. The field intensity at a distance ‘χ’ from a point mass ‘m’ is therefore Gm/χ2, and acts toward ‘m’ Gravitational field strength is measured in the newton per kilogram. The gravitational potential ‘V’ at that point is the work done in moving a unit mass from infinity to the point against the field, due to a point mass. Importantly, ( a ) Potential at a point distance ‘χ’ from the centre of a hollow homogeneous spherical shell of mass ‘m’ and outside the shell:

V = ‒ Gm/χ

The potential is the same as if the mass of the shell is assumed concentrated at the centre, ( b ) At any point inside the spherical shell the potential is equal to its value at the surface:

V = ‒ Gm/r

Where ‘r’ is the radius of the shell, thus there is no resultant force acting at any point inside the shell and since no potential difference acts between any two points. (c) potential at a point distance ‘χ’ from the centre of a homogeneous solid sphere and outside the sphere is the same as that for a shell;

V = ‒ Gm/χ

(d) At a point inside the sphere, of radius ‘r’:

V = ‒ Gm( 3r2 ‒ χ2 ) /2r3The essential property of gravitation is that it causes a change vin motion, in particular the acceleration of free fall (g) in the earth’s gravitational field. According to the general theory of relativity, gravitational fields change the geometry of spacetime, causing it to become curved. It is this curvature of spacetime, produced by the presence of matter, that controls the natural motions of matter, that controls the natural motions of bodies. General relativity may thus be considered as a theory of gravitation, differences between it and Newtonian gravitation only appearing when the gravitational fields become very strong, as with ‘black holes’ and ‘neutron stars’, or when very accurate measurements can be made.

Accelerated systems and forces due to gravity, where the acceleration produced are independent of the mass, for example, a person in a sealed container could not easily determine whether he was being driven toward the floor by gravity or if the container were in space and being accelerated upward by a rocket. Observations extended in space and time could distinguish between these alternates, but otherwise they are indistinguishable. His leads to the ‘principle of equivalence’, from which it follows that the inertial mass is the same as the gravitational mass. A further principle used in the general theory is that the laws of mechanics are the same in inertial and non-inertial frames of reference.

Still, the equivalence between a gravitational field and the fictitious forces in non-inertial systems can be expressed by using Riemannian space-time, which differs from Minkowski Space-time of the special theory. In special relativity the motion of a particle that is not acted on by any force is represented by a straight line in Minkowski Space-time. In general relativity, using Riemannian Space-time, the motion is represented by a line that is no longer straight, in the Euclidean sense but is the line giving the shortest distance. Such a line is called geodesic. Thus, a space-time is said to be curved. The extent of this curvature is given by the ‘metric tensor’ for spacetime, the components of which are solutions to Einstein’s ‘field equations’. The fact that gravitational effects occur near masses is introduced by the postulate that the presence of matter produces this curvature of the space-time. This curvature of space-time controls the natural motions of bodies.