TL;DR – Classical mechanics describes a divergence-free flow of states. The Hamiltonian is the time component of its vector potential. The Lagrangian is the scalar product between the flow and the vector potential. For the longest time I didn’t know what the Lagrangian was and why its integral is minimized along trajectories. A couple of …
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 – 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 …