Thursday, March 27, 2014

Red Rocks and the Two Body Problem

Roxborough State Park, Colorado (Nov 16, 2013)

There are only two problems in physics that we know how to solve: the pendulum and the planetary orbits. Everything else is calculated either by assuming that behaves like one of the two problems above, or by solving in some approximate way more complicated equations using a calculating device.

The difference between the physics of a pendulum and the one of a planet orbiting the Sun is in the relationship between the force and the distance. In the case of the pendulum there is an attractive force that doubles every time the distance double from the center of the oscillation. Galileo figured that out by looking at the chandeliers swinging in the Cathedral of Pisa. In the case of a planet orbiting the Sun the force on the planet decreases as the square of the distance from the Sun. Double the distance and the force will become one fourth. This was understood by Newton, which also had the epiphany that an apple falling towards the center of Earth would feel the same kind of force as a planet continually falling around the Sun. That's why the theory of gravitation is universal: applies to apples as well as planets.

The rest of physics can generally be approximated as one of these two basic problems. The motion of the atoms in a solid crystal? Nothing else than masses connected by springs; and springs also produce a force proportional to the displacement, making the crystal vibrate like a coupled pendulum. This generalized problem is called "harmonic oscillator". Charges into an electric field? The electric force also goes like the inverse square of distance: exactly like gravity (the only complication is that it can be either attractive or repulsive depending on the sign of the charges). The generalized problem of two masses subjected to a reciprocal force going like the inverse square of distance is called "two body problem". If the masses were three it would be a "three body problem". Four masses? You get it... it keeps going until you get the generic "n-body problem".  The kicker is that we only have an exact solution (the one found by Newton) for the two body problem: higher order problems can only be solved numerically, using a computer. Even worse, for a three (or more) body problem the solution is chaotic: this is a mathematical term that means that even an infinitesimal small change in the initial conditions will cause an unpredictable effect to the solution. You don't believe me? Look at this flash applet that simulates the orbit of one planet (the third body) around two stars.

Roxborough State Park
At this point you may ask what the two body problem has to do with red rocks in Colorado. As it turned out, last time we visited Colorado we met with some friends that work in the University there. Both scientists, a physicist and an astronomer, like Mayli and me. Academic couples are quite common, considering that people tend to pair with the people they socialize with. When you are in graduate school with other future scientists... well these are the people you tend to hook-up. This is all good until you graduate and look for a job. That's when your situation becomes a "two body problem".

Two body problems are not unique to academia, but the difficulty to find two academic positions within commuting distance makes them particularly difficult to solve. A recent article on Scientific American reports the results of a survey of two body problems in and out academia. The results? The large majority of the respondent (90%) said that they either had or think they will have to face a two body problem in the future. Of those that have already found the problem, less than half managed to negotiate a double position to solve it. The others? Either one of them had to leave a dream job to accommodate the career of the spouse, or they accepted to live at a non-commuting distance, with divorce or split-up as most common long term outcome.

Depressing? Well, if you have followed what I have written above, the two body problem (the physics version) is in principle solvable without recourse to chaos. There is hope. We got lucky, and after one failed attempt and two years of separation (Mayli in Chicago while I was still in Boston) we managed to get two positions in the same university. It was not easy, and every case is different, but I know many other couples that have found similar arrangements. What typically works best are mid-sized universities: top University can pick whatever faculty they need, and may not care enough to try creating a spousal accommodation. Too small places may not have the resources to find a solution. It also depends on the culture of the university. In our case we could count on funds that were explicitly set aside for this kind of situations: many university have come to the realization that the two body problems is actually a two body opportunity, a way to attract better candidates by offering a very special perk that fancier institutions would not even dream to contemplate.

Roxborough State Park, Colorado (Nov 16, 2013)

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