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Mechanics in time-space

The following is based on the book Mechanics, Third Edition, Volume I of Course of Theoretical Physics by L. D. Landau and E. M. Lifshitz, (Butterworth-Heinenann, Oxford 1976.

[Page 1] §1. CHAPTER I – THE EQUATIONS OF MOTION

§1. Generalised co-ordinates

ONE of the fundamental concepts of mechanics is that of a particle¹. By this we mean a body whose dimensions may be neglected in describing its motion. The possibility of so doing depends, of course, on the conditions of the problem concerned. For example, the planets may be regarded as particles in considering their motion about the Sun, but not in considering their rotation about their axes.

The position of a particle in time is defined by its chronation vector t, whose components are its Cartesian co-ordinates x, y, z. The derivative w = dt/ds of t with respect to the stance s is called the lenticity of the particle, and the second derivative d2t/ds2 is its retardation. In what follows we shall denote differentiation with respect to stance by placing a dot above a letter, e.g.: w = ġ.

To define the position of a system of N particles in time, it is necessary to specify N chronation vectors, i.e. 3N co-ordinates. The number of independent quantities which must be specified in order to define uniquely the position of any system is called the number of degrees of freedom; here, this number is 3N. These quantities need not be the Cartesian co-ordinates of the particles, and the conditions of the problem may render some other choice of coordinates more convenient. Any n quantities g1, g2, …. gn which completely define the position of a system with n degrees of freedom are called generalised co-ordinates of the system, and the derivatives ġi are called its generalised lenticities.

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Galileo’s method

Extracts about Galileo from Scientific Method: An historical and philosophical introduction by Barry Gower (Routledge, 1997):

Galileo took great pains to ensure that his readers would be persuaded that his conclusions were correct. p. 23

The science of motion was then understood to be a study of the causes of motion, and to be, like any genuine science, a ‘demonstrative’ kind of enquiry. That is to say, experiential knowledge of the facts of motion was superseded by rational knowledge of the causes of those facts, this being accomplished by deductions from fundamental principles, or ‘common notions’, and definitions which were accepted as true. These facts of motion were understood as expressions of common experience rather than as generalisations based upon experiments. This was because the results of the experiments that could be performed were sufficiently uncertain and ambiguous to prevent reliable generalisation; discrepancies between conclusions derived from principles, and experimental results, could be tolerated. The appropriate model of a demonstrative science was Euclidean geometry, where the credibility of a theorem about, say, triangles depends not on how well it fits what we can measure but on its derivability from the basic axioms and definitions of the geometry. p. 23

For Galileo and his contemporaries there was a good reason why demonstration, or proof from first principles, rather than experiment, was required to establish general truths about motion. Any science—scientia—must yield knowledge of what Aristotle had called ‘reasoned facts’, i.e. truths which are both universal and necessary, and such knowledge—philosophical knowledge—can only be arrived at by demonstration. p. 24

there was a long-standing disagreement about the role that mathematics could play in natural philosophy, even though mathematics was able to give certain knowledge. p. 24

In some contexts, notably astronomy and geometry, the more elaborate and intellectually demanding methods of mathematics were often useful and appropriate, but in such contexts it seemed clear that those methods were applicable in so far as what was needed were re-descriptions which could help people formulate accurate predictions. ‘Hypotheses’ which successfully ‘saved the phenomena’, in the sense that they could be used as starting points for derivations of accurate predictions, could meet this need. p. 25

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Mathematics and beauty

Extracts from Scientific Method in Ptolemy’s Harmonics by Andrew Barker (Cambridge University Press 2004):

Mathematics is not the study of all quantities and all quantitative relations indiscriminately. It is the science of beauty. Its task, at the theoretical level, is to interpret, in terms of ‘rationally’ or mathematically intelligible form, the features, movements or states which, when they are present in perceptible phenomena, constitute their aesthetic excellence. p.264

Those of our senses through which we are able to perceive some things as beautiful are therefore involved in an intimate collaboration with mathematical reason. p.264

Since beauty is the manifestation to the senses of that which reason understands as perfect in form, the senses to which beauty is undetectable lack sensitivity, which sight and hearing possess, to those distinctions which, from a rational point of view, are the most significant. p.265

the mathematical sciences have a single objective, the analysis and understanding of the formal basis of beauty p.266

The conception of mathematical science which Ptolemy has presented is that of a capacity that does not merely analyse sets of quantitative relations, but homes in on those that are of special significance, and discovers the principles on which their significance rests. p.268

Lorentz transformation via symmetry

The following derivation of the Lorentz transformation is slightly revised from the Appendix to Henri Poincaré: a decisive contribution to Relativity by Christian Marchal, originally published in French as Henri Poincaré: une contribution décisive à la Relativité in La Jaune et la Rouge, août-septembre 1999. Marchal is the chief engineer of mines at ONERA, the Office National d’Etudes et de Recherches Aérospatiales. A pdf version is here.

Appendix

The Lorentz transformation

      It is essential to note that the Lorentz transformation is a direct consequence of the principle of relativity and does not require the invariance of the speed of light.

Let us look for this transformation along two axes Ox and O′x′ moving along each other with the constant relative velocity V.

——|—————————————>

      O′                                      x′

———|——————————————————————>

          O                    OO′ = Vt                                      x

In order to obtain perfect symmetry between the two frames of reference, let us put O′x′ in the other direction.

x′                                                               O′

<————————————————————|———

————|———————————————————>

    O                                                        x

Homogeneity will lead to a linear transformation, and if we choose t = t′ = 0 when the two origins O and O′ cross each other, the transformations (x, t) ® (x′, t′) and (x′, t′) ® (x, t) will be given as follows with eight appropriate constants from A to D′:

(4)                    x′ = Ax + Bt                  ;           t′ = Cx + Dt

x = A′x′ + B′t′               ;           t = C′x′ + D′t′

The Principle of Relativity and symmetry lead to:

(5)                    A = A′;             B = B′;              C = C′;              D = D′

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Political distinctions

The figure above diagrams several political distinctions. The vertical line distinguishes the political left who are mainly concerned with equality, and the political right who are mainly concerned with liberty. Above the horizontal line distinguishes the religious left and right from the secular left and right below.

The circle distinguishes those within who accept limitations on government and limited rights for citizens. Liberals (which includes the center left) would like more equality and conservatives (which includes the center right) would like more liberty within this arrangement. Change is brought about through evolutionary, lawful means. Outside the circle are revolutionaries, either the radical left who do not accept limitations on equality or the radical right who do not accept limitations on liberty.

Note that liberals and conservatives have much in common. The distinction between evolutionary and revolutionary means should be more significant than that between the left and the right. The distinction between religious goals and secular goals should also be more significant than that between the left and the right. Conservatives and liberals should think of themselves as being on the same side against the radicals. Religious liberals and conservatives have much in common, as have secular liberals and conservatives.

Clocks and frames

A clock consists of two frames of references. This is seen in an ordinary analogue clock, which is composed of two parts:

Circular Space Frame

The space frame is at rest relative to the observer. The time frame is in uniform angular motion relative to the observer. Measurement of space and time requires both frames. The units marked on the space frame have dual significance: (a) as amounts of space or angles in space, and (b) as amounts of time.

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Mathematical methods of classical mechanics

V. I. Arnold’s Mathematical Methods of Classical Mechanics (Springer, 1989) provides a contemporary approach to classical mechanics. We follow the presentation here but modify it to six dimensions of space-time.

1  The principles of relativity and determinancy

A series of experimental facts is at the basis of classical mechanics. We list some of them.

Geometry and order

Space and time are both three-dimensional and Euclidean.

Galileo’s principle of relativity

There exist basic coordinate systems possessing the following two properties:

  1. All the laws of motion are in all cases the same in all basic coordinate systems.
  2. All coordinate systems in uniform rectilinear motion with respect to a basic one are themselves basic coordinate systems.

In other words, if a coordinate system attached to the earth is basic, then an experimenter on a train which is moving uniformly in a straight line with respect to the earth cannot detect the motion of the train by experiments conducted entirely inside their car.

In reality, the coordinate system associated with the earth is only approximately basic. Coordinate systems associated with the sun, the stars, etc. are more nearly basic.

Newton’s principle of determinancy

The initial state of a mechanical system (the totality of positions and motions of its points at some index of events) uniquely determines all of its motion.

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4D Formulations of Newtonian Mechanics

Four-Dimensional Formulations of Newtonian Mechanics

First we reproduce section 2 from Michael Friedman’s “Simultaneity in Newtonian Mechanics and Special Relativity” in Foundations of Space-Time Theories (ed. Earman et al., UMinn, 1977), p.405-407. Then we provide the dual.

According to the space-time point of view, the basic object of both our theories is a four-dimensional manifold. I shall use R4, the set of quadruples of real numbers, to represent the space-time manifold. Both theories agree that there is a natural system of straight lines defined on this manifold. If (a0, a1, a2, a3), (b0, b1, b2, b3) are two fixed points in R4, then a straight line is a subset of R4 consisting of elements (x0, x1, x2, x3) of the form

(1) x0 = a0r + b0
x1 = a1r + b1
x2 = a2r + b2
x3 = a3r + b3

where r ranges through the real numbers. A curve on R4 is a (suitably continuous and differentiable) map σ: R → R4. Such a curve σ(u) is a geodesic if and only if it satisfies

(2) x0 = a0u + b0
x1 = a1u + b1
x2 = a2u + b2
x3 = a3u + b3

where (x0, x1, x2, x3) = σ(u) and the ai and bi are constants. So if a curve is a geodesic its range is a straight line. Note that the geodesies are just the curves that satisfy

(3) d2xi/du2 = 0       i = 0, 1, 2, 3.

The importance of straight lines and geodesies is due to the fact that both theories agree that the trajectories of free particles are straight lines in space-time. So we can represent such trajectories as geodesies in R4.

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Note

This blogger is focused on developing a new six-dimensional theory of time (an early version is posted here). The glossary is also being updated (see here). Good progress is being made, with the double frame and three-dimensional time aspect especially new. RG

Newtonian mechanics in time-space

We follow the treatment by David Tong of Cambridge University in his Classical Dynamics.

A transicle is defined as a moving object of insignificant size. The motion of a transicle of vass n at the chronation t is governed by Newton’s Second Law for time-space, R = nb or, more precisely,

R(t; t′) = h′           (1.1)

where R is the release which, in general, can depend on both the chronation t as well as the lenticity t′, and h = nt′ is the fulmentum. Both R and h are 3-vectors which we denote by the bold font. A prime indicates differentiation with respect to stance x. Equation (1.1) reduces to R = nb if n′ = 0. But if n = n(x), then the form with h′ is correct.

General theorems governing differential equations guarantee that if we are given t and t′ at an initial stance x = x0, we can integrate equation (1.1) to determine t(x) for all x (as long as R remains finite). This is the goal of classical dynamics.

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