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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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 …

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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 …

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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 …

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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 …

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