### Author: Gabriele Carcassi

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 …

Relativity and the mathropologist

## Relativity and the mathropologist

TL;DR – The laws of physics are the same for all observers because nature couldn’t care less what coordinate system we use. I recently met someone who had some slight misconceptions about the principle of relativity. “Is it true that time and space are relative concepts?” “That contemporary events for you may not be contemporary …

Galilean transformations are not compatible with Hamiltonian mechanics

## Galilean transformations are not compatible with Hamiltonian mechanics

TL;DR Galilean boosts do not leave Hamilton’s equations and phase space volumes unchanged. They give good approximations only for small changes in velocity. When studying physics, you learn that relativistic mechanics is more “correct” and non-relativistic mechanics is just an approximation. But you may also get a sense that non-relativistic mechanics is self-consistent: it just …