Month: May 2017

Lagrangian mechanics is a subset of Hamiltonian mechanics

Lagrangian mechanics is a subset of Hamiltonian mechanics

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 …

Lagrangians must be convex

Lagrangians must be convex

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 …