TL;DR – Hamiltonian systems for which two states have the same trajectory $q(t)$ are not Lagrangian. Lagrangian systems are Hamiltonian systems where the velocity is a function $v=v(x,p)$ strictly increasing with respect to momentum (i.e. an increase in momentum yields an increase in velocity). In a previous post we saw how the photon, as a …
TL;DR – Only convex Lagrangians (i.e. positive second derivative in all directions) can give unique solutions when the action is minimized. This is the kind of post I do mainly for my future self. Sometimes it takes far longer than I would have expected to nail down a detail and I want a record of …
TL;DR – The laws of physics are not the laws of nature. They can only describe what is experimentally accessible. I am often cautious about sharing my interest in fundamental physics to strangers. “Ah, fundamental physics!” They may get excited and step too close to me. “What can you tell me about the nature of …