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 …
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 …
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 …
TL;DR – Classical Hamiltonian mechanics already includes an uncertainty relationship that is similar to Heisenberg’s uncertainty principle of quantum mechanics. In previous posts we have looked at information entropy, the number of yes/no questions you need to identify an element within a distribution, and the fact that Hamiltonian dynamics conserves that. Here we will show …
TL;DR – Hamiltonian systems are those that conserve information entropy during time evolution. The idea is the following: suppose you have a distribution over position and momentum $\rho(x, p)$. Suppose you evolve it in time and get a final distribution $\hat{\rho}(x, p)$ using Hamiltonian evolution. That is: you take each little area in phase space …
TL;DR – Inertial mass tells us how many states are there within a given range of velocity. More massive bodies are harder to accelerate because they have to go through more states to reach the same change in velocity. In this post I want to look a bit more closely at the concept of mass. …