What is mass?

## What is mass?

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. …

Classical mechanics in one post

## Classical mechanics in one post

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 …

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 …

Can Hamiltonian mechanics describe dissipative systems?

## Can Hamiltonian mechanics describe dissipative systems?

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 …