The World is Gonna Know Your Name

This week's post involves William Rowen Hamilton, so I thought I'd piggyback on another Hamilton's popularity! This week I saw a news article about improving neural networks, like the one I talked about a few weeks ago, by adding information from the system's Hamiltonian, which characterizes the total energy available.

Like the Newtonian and Lagrangian techniques, the Hamiltonian is yet another completely equivalent way to determine system dynamics. In most cases, the Hamiltonian is defined as the sum of kinetic and potential energy

where the quantities are defined in terms of some coordinates q, and corresponding momenta p. Then the equations of motion are given by

As with the Lagrangian, this can often be a simpler way to define a system where energy is conserved.

The paper that the group wrote concerns chaotic systems, where small changes to conditions can cause wildly different results. Often a chaotic system will have a subset of simple results – I actually wrote about an example long ago, but since I hadn't yet taken any nonlinear dynamics, I wasn't sure if it qualified. If you're trying to train a neural network to predict a system's dynamics, this transition between simple and chaotic motion can cause problems.

One of the example systems the paper studies is a circular billiards table, with a peg in the center:

Figure 8

A simple solution could be a ball bouncing between the peg and the wall on a radial line. If we gave the ball a random direction though, chances are it would bounce around the circle in a complex pattern. Those dynamics would be entirely deterministic, but impossible to perfectly predict in a real-world situation.

The group's idea was to change the way the neural network is trained, incorporating the Hamiltonian to give it awareness of the chaotic nature of the system. They compared the results from a standard network, a Hamiltonian network, and from solving the differential equations:

Figure 9

From top to bottom, the cases are non-chaotic hitting the outer circle only, non-chaotic hitting both boundaries, and chaotic. While the Hamiltonian network is far from an exact match to the true trajectory, it does a much better job characterizing the behavior of the ball with the same amount of training. By giving the network a rule to conserve energy, we can teach it a little bit of physics to better train it (potentially to become our robot overlords)!

Dentist Time

[Title from my grandfather's favorite joke: When is the best time to go to the dentist? 2:30! (Tooth-hurty)]

Earlier I said I would describe what I'm doing here in Florida, and this week I'm going to talk about my work on LIGO. I'm part of the engineering department here, so rather than the data analysis that I usually do, I'm more involved in the mechanics of the detectors. I described in my first post on LIGO how gravitational waves stretch and squeeze space. That stretching is a proportional factor, so the longer the distance, the greater the change. The scaling factor is so small though that to have any hope of picking it up, we need an enormous distance. The detectors are 2.5 miles long, but on top of that, we use resonant cavities that bounce the laser beam back and forth many times:

The laser goes through two partially-reflective mirrors that concentrate the light in a small area. The mirrors have to be precisely aligned to make the light add, instead of cancel out:

The laser we use in LIGO has a wavelength of 1064 nm, which means the difference between making the peaks add or cancel is only 0.000000266 meters!

This past week I was attending some (virtual) workshops on a tool we use to simulate cavities called Finesse. Adapting some of the code my colleague Luis Ortega wrote, I made a plot of the laser power that builds up in the cavity if we send a 1 Watt laser in, and have 85% reflective mirrors on either end:

One quality of a resonant cavity is how quickly the power falls off when the length is changed. Here you can see that if we have things just right, we can increase the power by 6.5x, but any error, and the output quickly drops to near zero.

The part that I'm working on is making simulations of the suspensions that help prevent the mirrors from moving out of alignment. Maybe in a future post I can discuss that, but I wanted to start with the basics, since I haven't been thinking about this stuff for as long as my coworkers.

Remote Medicine

I missed posting last week, since we were busy driving to our new home of Gainesville, FL! I have a new postdoc with the university, which I'll talk about in a later post. This week though I wanted something a bit shorter, since we've got several rental viewings to go to. I thought I'd try to expand on something I wrote on Facebook a few months ago, at the start of the lockdowns.

My friend Seth wrote that he didn't have a thermometer on hand, and most stores were sold out. I offered this suggestion:

Here's an idea I just had that could work in theory: If you have a TV remote and a voltmeter, you could press the remote against your head, and measure the voltage across the LED, which will be proportional to the infrared light emitted by your skin, which in turn is proportional to your temperature. Physicist Survival Skills!

I've mentioned TV remotes before, but this idea doesn't use the IR LED in the typical way. While LEDs are designed to emit light (as the name implies), they can also generate voltage when they absorb light near their own wavelength. You can see a demonstration of both uses in this video:

The question is how to calibrate the "thermometer" so we can convert between voltage and temperature. We need some known temperatures we can sample. Two possibilities I thought of were boiling water, and (assuming you haven't fully switched to fluorescents) incandescent light bulbs. Depending on your elevation, water will boil around 373.15 Kelvin, and bulbs emit light according to the blackbody distribution at 2400 K. That means your temperature (in Kelvin) will be

As usual though, ramblings from a crazy physicist are no substitute for healthcare, so I do not endorse the use of improvised TV thermometers. Thanks for a great idea, Seth, and stay healthy everyone!