space & time

Explorations of multidimensional space and time with linear and angular motion.

Equivalence principle

Inertia is the property of a body that resists changes in its motion. Inertial mass of a body is the ratio of the applied force divided by the body’s acceleration. Gravitational mass is the mass of a body as measured by its gravitational attraction to other bodies. The Equivalence Principle takes several forms. The Newtonian version …

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Galilean transformations derived

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 …

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Interchangeability of space and time

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 …

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Classical dynamics

The following presents the spatio-temporal and temporo-spatial 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 [Spatio-temporal] Laws We begin by simply stating in conventional form Newton’s laws of mechanics: I. A body remains at …

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Ballistics

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. Spatio-temporal Temporo-spatial Initial space angle = θ Initial time angle = φ Initial height …

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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 …

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Temporo-spatial rest

Speed is the length of travel per unit of duration (or time interval). Spatial rest 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 temporal rest seems different. It cannot be zero …

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Temporo-spatial Galilean group

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 a temporo-spatial context. The Galilean group. To find the set of transformations of space and time that preserve the laws of …

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Temporo-spatial mechanics

The following is a temporo-spatial modification of 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. …

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