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Ballistic table based on launching from a height and angle with coasting ascent and descent (no drag, no thrust). Note the handy trigonometry identity for range: 2 sin θ cos θ = sin 2θ. This table is in pdf form here.



Initial space angle = θ Initial time angle = φ
Initial height distance = y0 Initial height distime = b0
Elapsed time interval = t Elapsed stance interval = s
Distance downrange or horizontal location = x Distime downrange or horizontal chronation = a
Altitude distance or vertical location = y Altitude distime or vertical chronation = b
Gravitational acceleration = g Levitational rapidation = h
Initial velocity = v₀ Initial legerity = w₀
Initial horizontal velocity = v0x = v0 cos θ Initial horizontal legerity = w0a = w0 cos φ
Initial vertical velocity = v0y = v0 sin θ Initial vertical legerity = w0b = w0 sin φ
Horizontal velocity = vx = v0x Horizontal legerity = wa = w0a
Vertical velocity = vy = v0y – gt Vertical legerity = wb = w0b – hs
Velocity at apex point: vy = 0 Legerity at apex instant: wb = 0
Horizontal location x = v0x t Horizontal chronation a = w0a s
Vertical location y = v0yt – ½ gt2 Vertical chronation b = w0bs – ½ hs2
Vertical location at impact point: y = 0 Vertical chronation at impact instant: b = 0
Time of flight to apex tapex = v0y/g Stance of flight to apex sapex = w0b/h
Total time of flight ttotal = 2tapex = 2v0y/g Total stance of flight stotal = 2sapex = 2w0b/h
Distance range to apex xapex = vox voy/g Distime range to apex aapex = woa wob/h
Total distance range xtotal = 2vox voy/g Total distime range atotal = 2woa wob/h
Max altitude distance yapex = ½ v0y2/g Max altitude distime bapex = ½ w0b2/h
Trajectory formula: y = y0 + x tan θ − ½ gx²/v0x2 Trajectory formula: b = b0 + a tan φ − ½ ha²/w0a2

Abstract classical mechanics

The following builds on the book Mathematical Aspects of Classical and Celestial Mechanics, 3rd edition, by Vladimir I. Arnold, Valery V. Kozlov, and Anatoly I. Neishtadt (Springer 2006).

Basic Principles of Classical Mechanics (cf. Chapter 1)

Space and Time

The space where the motion takes place is three-dimensional and Euclidean with a fixed orientation. We shall denote it by E3. We fix some point o ∈ E3 called the “origin of reference”. Then the position of every point s in E3 is uniquely determined by its position vector os = r (whose initial point is o and end point is s). The set of all position vectors forms the three-dimensional vector space ℝ3, which is equipped with the scalar product 〈 , 〉.

The time in which motion takes place has the same structure as the abstract space above. The combined vector space is ℝ3 × ℝ3. The abstractions for space and time are unconnected unless there is defined a fixed relationship between them. Examples of such a fixed relationship include a default rate of motion or a maximum rate of motion. Let us begin without such a relationship.

Position in space is called location and in time is called chronation. The Euclidean metric for space is called length and for time is called duration (or time).

A frame of reference (“frame”) is a method to assign every particle a unique position in a coordinate system of points in ℝ3. Such assignment is known continually and universally, without signals, from the universal extent of the frame. The coordinate system is commonly Cartesian.

A system of reference (“reference system”) is a method to assign every event a unique position in a coordinate system of points in ℝ3 × ℝ3. A reference system is composed of dual frames of reference, one called the space frame and the other called the time frame, such that the time frame is in standard uniform motion relative to the space frame. This requires that given the magnitudes s1 and s2 of any two intervals of the curve of motion in the space frame, then the corresponding intervals of the time frame, t1 and t2, relative to the space frame satisfy the proportion: s1:s2 :: t1:t2.

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Rest in time-space

Speed is the travel distance per unit of duration (or time interval). Rest in space is a speed of zero. That is, there is no change in location per unit of time. A body does not change location (relative to an inertial observer) while time continues.

But rest in time seems different. It cannot be zero pace because that would mean it takes no time to go a positive distance, right? No, that is not what zero pace means.

Pace is the travel time per unit of distance (or stance interval). Time is the dependent variable and distance is the independent variable.

Consider a race that is about to begin. The runners are in place waiting for the signal to start. The official timer is set to begin. In terms of motion, the runners are at rest with speed of zero. They are not making any distance, but time continues as usual.

What is the pace of the runners in that case? There is no change on the official timer. But the stance continues as usual. For example, if stance is related to the distance from the Sun of a Voyager spacecraft (see here), it continues to increase as usual.

A map with a time scale shows a point for a pace of zero. Despite the distance made by an odologe, a body with a pace of zero remains in the same place in time. It is at rest in time.

Runner about to start

What about an infinite value for pace in time? The Galilean transformation implicitly has an infinite speed of information in space, which makes information spatially ubiquitous since it travels an infinite distance in a finite time. The symmetric Galilean transformation implicitly has an infinite pace of information in time, which makes information temporally ubiquitous since it takes an infinite time to travel a finite distance.

The Galilean group in time-space

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 = s2s1.
  2. The distime (temporal distance) Δt = |q2q1| between any pair of simulstanteous 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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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.


§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 legerity 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 legerities.

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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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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.


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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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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