### Tag: Relativity

Classical free particles (i.e. Klein-Gordon)

## Classical free particles (i.e. Klein-Gordon)

TL;DR – The extended Hamiltonian for a classical free particle is the classical version of the Klein-Gordon equation. After having seen the classical version of antiparticles we will see the classical version of the Klein-Gordon equation. This is the essentially the extended Hamiltonian for the free particle. Let’s look at the details. 1. Extended Hamiltonian for …

Classical antiparticles

## Classical antiparticles

TL;DR – Antiparticle states are those for which time is a decreasing function $t(s)$ of the trajectory parametrization instead of an increasing function (i.e. time and trajectory parametrization are anti-aligned). In a previous post we introduced the extended Hamiltonian equations. In the next few posts we will see that they allow us to create parallels …

Speed of light and sugary water

## Speed of light and sugary water

TL;DR – The speed of light tells us how many different positions can fit in an interval of time. I was having a run with a buddy of mine. “The speed of light is fascinating” he panted. “Could we ever go faster?” “If we go faster, do we travel back in time or something?” I …

Hamiltonian mechanics on the extended phase space

## Hamiltonian mechanics on the extended phase space

TL;DR – The most complete version of Hamiltonian mechanics lives on the extended phase space that includes position, momentum, time and energy. In this post we look at the most complete version of Hamiltonian mechanics. It is kind of a shame that this is not the version taught because it is the one that works …

How Hamiltonians vary under coordinate changes

## How Hamiltonians vary under coordinate changes

TL;DR – Hamiltonians are not invariant: they change as the time component of a covector (i.e. covariant component) in phase space. In the previous post we saw how momentum varies as covariant components and that keeps the Hamiltonian equations unchanged under coordinate transformations. We have also seen, though, that under coordinate transformations that mix time …