The following is based on A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres (Cambridge UP, 2004) starting with Example 2.29 on page 54 and modifying it for time-space.

The Galilean group. To find the set of transformations of space and time that preserve the laws of Newtonian mechanics we follow the lead of special relativity and define an event to be a point of R⁴ characterized by four coordinates (t₁, t₂, t₃, s). Define Galilean time G⁴ to be the time of events with a structure consisting of three elements:

1. Distance intervals Δs = s₂ − s₁.
2. The duration intervals Δt = |q₂ − q₁| between any pair of simulstanceous events (events having the same stance coordinate, s₁ = s₂).
3. Motions of facilial (free) particles, otherwise known as rectilinear motions,
   q(s) = ws + q₀,                 (2.19)
   where w and q₀ are arbitrary constant vectors.

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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, …. g_n which completely define the position of a system with n degrees of freedom are called generalised co-ordinates of the system, and the derivatives ġ_i are called its generalised lenticities.

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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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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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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.

A  Geometry and order

Space and time are both three-dimensional and Euclidean.

B  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.

C  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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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 a_i and b_i 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) d²x_i/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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