This derivation of the Galilean transformations is similar to that of the Lorentz transformations here.

Since space and time are assumed to be homogeneous, the transformations must be linear. The most general linear relationship is obtained with four constant coefficients: A, B, C, and D:

x′ = Ax − Bt
t′ = Ct − Dx

Without loss of generality, if t = 0, let A = 1, and if x = 0, let C = 1. Then

x′ = x − Bt
t′ = t − Dx

The inverse transformation for the position from frame R′ to frame R should have the same form as the original but with its motion in the opposite direction, as is confirmed by algebra:

x = x′ + Bt′
t = x′ + Dt′

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Independent variables are measured first, independent of other variables. They may be either set to a fixed value or allowed to change at a fixed rate. An example of the former is a race in which the distance is the independent variable set for the race, and of the latter is a time variable, which increases with each tick of a clock.

Dependent variables are functionally dependent on an independent variable. A dependent variable may or may not be causally dependent on the independent variable. Dependent variables are measured relative to the independent variable. For example, given a time of four minutes (the independent variable), how far can someone run (the dependent variable)? Dependent variables are known by many names, including target variable.

The extent of motion

The extent of a motion is measured by its time intervals (“times”) and its space intervals (“spaces”). Let T represent an interval of time, and let S represent an interval of space. Uniform motion may be stated as a proportion in either of two ways:

(1) Given the ratio of two elapsed times, the corresponding ratio of two traversed spaces are in the proportion:

S₁ : S₂ :: T₁ : T₂

(2) Given the ratio of two elapsed spaces, the corresponding ratio of two traversed times are in the proportion:

T₁ : T₂ :: S₁ : S₂

In case (1) time is the independent variable and space is the dependent variable (the time speed). In case (2) space is the independent variable and time is the dependent variable (the space speed). Let us adopt the convention that the variable on the right side of the proportion is independent, and the variable on the left side is the dependent variable.

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The extent of a motion is measured in two ways: by its time (duration) and by its space (length). The relation between these two measures is the subject here.

Although a definition of uniform motion was given by Archimedes, Galileo was the first to give a complete definition:

Equal or uniform motion I understand to be that of which the parts run through by the moveable in any equal times whatever are equal to one another. (Galileo, Two New Sciences, “On Equable Motion,” tr. by Stillman Drake, p. 148)

Archimedes first stated a proportion for uniform motion:

If a point move at a uniform rate along any line, and two lengths be taken on it, they will be proportional to the times of describing them. (Archimedes, The Works of Archimedes, ed. by T. L. Heath, Dover, p.155.)

In other words, given a uniform motion and on it any two lengths, L and M, then the lengths and the corresponding times of motion, T and U, will satisfy the proportion L : M :: T : U.

Galileo gave the converse proportion:

If a moveable equably carried [latum] with the same speed passes through two spaces, the times of motion will be to one another as the spaces passed through. (Galileo, Two New Sciences, “On Equable Motion,” tr. by Stillman Drake, p. 149)

In other words, given a uniform motion and on it any two spaces, S1 and S2, then the times of motion, T1 and T2, for the corresponding lengths will satisfy the proportion T1 : T2 :: S1 : S2.

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The following presents the space-time and time-space versions of Newton’s laws based on the book Classical Dynamics of Particles and Systems by Thornton and Marion (Fifth Edition, 2008).

Start with page 49, section 2.2:

2.2 Newton’s Laws [for space-time]

We begin by simply stating in conventional form Newton’s laws of mechanics:

I. A body remains at rest [in space] or in uniform motion unless acted upon by a force.

II. A body acted upon by a force moves in such a manner that the time rate of change of momentum equals the force.

III. If two bodies exert forces on each other, these forces are equal in magnitude and opposite in direction.

…

To demonstrate the significance of Newton’s Third Law, let us paraphrase it in the following way, which incorporates the appropriate definition of mass:

III′. If two bodies constitute an ideal, isolated system [in space], then the accelerations of these bodies are always in opposite directions, and the ratio of the magnitudes of the accelerations is constant. This constant ratio is the inverse ratio of the masses of the bodies.

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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.

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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