TL;DR – We cannot always validate all answers to a scientific question with the same process. For some yes/no questions, only one side can be actually experimentally verified. Some people write me things like “Is it true that absent of evidence is not evidence of absence?” “Is it true that there my be a China …
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 …
TL;DR – Time dependent Hamiltonian mechanics is for conservative forces in non inertial frames, not for non-conservative forces. In a previous post we saw how Hamiltonian and Newtonian mechanics are different as the first cannot describe dissipative systems. Yet, if we allow the Hamiltonian to be time dependent, the system would appear to change its …
TL;DR – Particle under friction is Newtonian, but neither Lagrangian or Hamiltonian. Photon as a particle is Hamiltonian, but neither Lagrangian or Newtonian. After taking a course in advanced mechanics, one is often left with the impression that Netwonian, Lagrangian and Hamiltonian mechanics are all equivalent. Unfortunately, while this is often what a book would …