rag

Vector inverse and mean

This post is based on research papers by Anderson and Trapp, Berlinet, and the post on Harmonic algebra. The vector inverse x−1 is defined as with positive norm. For a non-zero scalar k, The harmonic (or parallel) sum is usually symbolized by a colon (:), but I prefer a circle plus to maintain its relation with …

Vector inverse and mean Read More »

With and between independent variables

This post continues the previous post here on independent and dependent variables. The selection of a physical independent variable (or variables) applies to a context such as an experiment. Within that context all other variables are, at least potentially, dependent on the independent variable(s) selected. Functions with the physical independent variable as a functional independent …

With and between independent variables Read More »

Distance as an independent variable

A previous post here gives “Motion as a Function of Distance” in which distance is a functional but not a physical independent variable. So distance is an independent variable of the inverse of the function of motion as a function of time. But this functional independence does not change the original independent variable of time. In …

Distance as an independent variable Read More »

Independent and dependent variables

There are two kinds of independent variables: (1) functional independent variables, and (2) physical independent variables. To avoid confusion an independent variable it is standard that a variable be of both kinds, since being of one kind does not imply being of the other kind. A physical independent in an experiment remains the independent variable …

Independent and dependent variables Read More »

Traffic flow in time and space

The following is based on Wilhelm Leutzbach’s Introduction to the Theory of Traffic Flow (Springer, 1988), which is an extended and totally revised English language version of the German original, 1972, starting with page 3 (with a few minor changes): I.1 Kinematics of a Single Vehicle I.1.1 Time-dependent Description I.1.1.1 Motion as a Function of …

Traffic flow in time and space Read More »

Dual Galilean transformation

The Galilean transformation is based on the definition of velocity: v = dx/dt, which for constant velocity leads to x = ∫ v dt = x0 + vt So for two observers at constant velocity in relation to each other we have x′ = x + vt with their time coordinates unchanged: t′ = t …

Dual Galilean transformation Read More »

Changing coordinates for the wave equation

The following is based on section 3.3.2 of Electricity and Magnetism for Mathematicians by Thomas A. Garrity (Cambridge UP, 2015). See also blog post Relative Motion and Waves by Conrad Schiff. The classical wave equation is consistent with the Galilean transformation. Reflected electromagnetic waves are also consistent with classical physics using the dual Galilean transformation, in …

Changing coordinates for the wave equation Read More »

Michelson-Morley experiment

This post relates to a previous post here. The Michelson-Morley experiment is a famous “null” result that has been understood as leading to the Lorentz transformation. However, an elementary error has persisted so that the null result is fully consistent with classical physics. Let us look at it in detail: The Michelson-Morley paper of 1887 …

Michelson-Morley experiment Read More »

Temporo-spatial light clock

This post builds on the post about the Michelson-Morley experiment here. One “Derivation of time dilation” (e.g., here) uses a light clock, pictured below: The illustration on the left shows a light clock at rest, with a light beam reflecting back and forth between two mirrors. The distance of travel is set at the beginning …

Temporo-spatial light clock Read More »

Michelson-Morley re-examined

There are many expositions of the famous Michelson-Morley experiment (for example here) but they all assume the independent variable is time, which is not the case. As we shall see, distance is the independent variable, and so the experiment is temporo-spatial (1+3). Let us examine the original experiment as it should have been done: The configuration …

Michelson-Morley re-examined Read More »