Background: Quantum Theory of Electromagnetism: Hamiltonian Dynamics
Since we’re now being good physicists in this series by working to understand what’s happening in various electromagnetic systems, in the last post we looked at one very important type of measurement we can make on such systems. For many purposes, that basic measurement might provide all the framing for a question that we need; from there, we can compute all of the specific quantities of interest. That measurement, the Lagrangian, is formulated in terms of the system’s kinetic and potential energy: 
From there, we’re going to look at another measurement that can be derived from the Lagrangian, called the Hamiltonian. With the Lagrangian, we’re tracking the difference in kinetic and potential energy of the system. With the Hamiltonian, we’re tracking the total energy of the system, which is the sum of its kinetic and potential energies. All this means in the simplest version of the math is that we’ve gone from a ‘-‘ sign to a ‘+’ sign: 
Knowing a quantity representing the total energy of a system is already pretty useful, by itself. But computing the Hamiltonian is made even more useful by the fact that you can harness various mathematical relationships it has with other quantities of interest in a system to find out those other quantities. You can also use the Lagrangian to find these, but with the Hamiltonian formulation the calculations are often a bit easier. There are several ways to get from the Lagrangian to the Hamiltonian that are more rigorous and enlightening than just substituting a sign. Maybe the canonical way is called via the Legendre transform, which I’ll cover in a separate post that does a whole bunch of math in full rigor and detail—which I realize will be of interest to even fewer people than my informal math/physics posts.;-D
The Hamiltonian is also interesting for historical reasons, because it was used to reformulate Newtonian physics (roughly, the physics of our everyday world based on Newton’s laws of motion and gravitation). This, in turn, helped make a conceptual step that enabled some of the bold ideas of quantum mechanics, which is why I’m including background about it in this series. In broad terms, from the total energy of a system you can work our the set of all possible outcomes of a measurement made at the quantum scale.
At this point, we’ve spent a lot of time motivating ideas and laying out in English how they’re connected to each other. In order to actually do and understand what these mean for particular systems, we need to work through how they play out in specific examples of physical systems and crunch on the math that describes them. Once we start to get a feel for how these mathematical tools work, we’ll grok what’s going on at a much deeper level. So, in the next post, I’ll start us off in that effort by looking at the nuts and bolts of the Lagrangian, Hamiltonian, and the Legendre transform. You’ll be able to skip that installment in this series and stick to the words- and ideas-heavy posts if you’re not into that sort of thing!
References
- https://scholar.harvard.edu/files/david-morin/files/cmchap15.pdf
- http://www.srl.caltech.edu/phys106/p106b01/topic1.pdf