## Defining multi-dimensional time

The revised SI metric system is based on seven defined constants and seven base units. One defined constant is c, the (round-trip mean) speed of light in a vacuum, is defined as exactly 299 792 458 metres per second (before metres and seconds are defined). The unit of time, the second, is defined as (2019): …

## Newtonian light wave front

Consider the standard relativity configuration. Let a spherical light wave be emitted from the coincident coordinate origins at t=0 and t′=0. For the rest frame, the spherical wave front is given by x² + y² + z² = c²t². For the frame moving at velocity v parallel to the x-axis, two-way light must be considered. …

## Time-space transformations

This post is based on David Tong’s Newtonian Mechanics, 1.2.1 Galilean Relativity. Given one facilial frame system, S, in which a tempicle has coordinates t(x), we can always construct another facilial frame system, S′, in which a tempical has coordinates t′(x) by any combination of the following transformations: Translations: t′ = t + a, for …

## Lagrange’s equations in time-space

This post is based on the article Deriving Lagrange’s equations using elementary calculus by Josef Hanc, Edwin F. Taylow, and Slavomir Tuleja (AJP 72(4) 2004), which provides a derivation of Lagrange’s equations from the principle of least action using elementary calculus. A tempicle moves along the t axis with potential lethargy W(t), which is location-independent. …

## Definitions of mass and vass

The conservation of momentum states (see here): For a system of objects, a component of the momentum (p = mv, the mass times the velocity) along a chosen direction is constant, if no net outside force with a component in this chosen direction acts on the system. The corresponding principle for levamentum states: For a system …

## Polar plot in space and time

This post builds on others such as this. It’s not unusual to see a map of travel time from a central location, for example, this map of Washington, DC, USA (click to enlarge): Two-dimensions of travel time are represented as in space but could be represented as in time. Here is an example, first of …

## Relative velocity and lenticity

Consider a particle P in uniform motion. Suppose two inertial observers observe its motion. Observer K is stationary relative to the ground, and observer L is in uniform motion in the same direction as P but at a different rate. (A) Say the spatial position of P relative to K is x(P, K), the spatial …

## Mathematical methods of classical mechanics, part 2

Part 1 is here. C Measures of motion A motion in RN is a differentiable mapping x: I → RN, where I is an interval on the real axis. The derivative is called the velocity vector at the point r0 ∈ I. The second derivative is called the acceleration vector at the point r0. We …

## Ratios and quotients

The traditional ratio, x : y, represents both x / y and y / x. In order to represent a ratio as quotients, both forms are required. Let us define a ratio as an ordered pair of quotients: For vectors this means One can either exclude zero or include infinity as follows; A rate of …

## Opposite velocities and lenticities

Two opposite velocities — or lenticities — are invariant over time and space. The standard Galileian transformation in the space-time domain is Velocity u transforms as Velocity is not invariant relative to a single inertial observation, but it is relative to observations with opposite relative velocities: That is Harmonic velocities are opposites and so are …