First of all, happy Thanksgiving everyone! I hope you spend the day happily with the people you care about, and remember to spend a moment or two reflecting on the things for which you’re thankful this year. Now on with the show:
Back when I first started writing this blog, I focused mostly on problem solving. The goal was to bridge the gap between popularization and textbook. I was always doubtful there was much of a market for this, but of course there are at least some interested people and especially since writing is so fun I was and am I’m more than happy to fill that gap. Over the last few months though, general grad student busyness has greatly reduced the time available for those kinds of posts. And so there’s been more soft-physics kind of posts around here. Still interesting, I hope, but there’s really no shortage of that sort of thing elsewhere. As such I’m going to try to improve the ratio of more in-depth fare a bit. I can’t promise I’ll be super consistent about it, but here’s hoping y’all will bear with me!
Let’s kick it off with perhaps the most important model in physics: the simple harmonic oscillator. It’s ubiquitous in everything from solid state physics to quantum field theory, but when it comes right down to it, the harmonic oscillator is a spring. Its defining property is that the force acting on the spring is proportional to the displacement of the mass from equilibrium. Move the mass farther from its resting point, and the restoring force is proportionally stronger. Wikipedia has a nice image:
We can write “the force is proportional to the stretch” mathematically in the following way:
The variable x is the position of the mass on the spring, and it’s a function of time. Dots denote differentiation with respect to time, so x-dot-dot is the rate of change in the rate of change of position. Sounds bad, but that’s just another name for the acceleration. The spring constant (the force produced by the spring per unit of stretch beyong equilibrium) is k, the mass of the object is m. Now if you know about solving differential equations, we can actually find the particular function x(t) that satisfies that equation. Physicists usually solve this kind of equation by the method of recognition – we’ve seen it so much we just know what the solution is. For those who haven’t seen it so much, it’s the pretty much the first thing you’ll learn in differential equations class, and if you don’t want to take the class it’s ok because the problem is not difficult and either way I’m just going to tell you the solution.
The solution is thus:
Actually the solution contains k and m, but to make the equation look simpler, I’ve just substituted in omega, where it’s an abbreviation for a slightly clumsier expression:
And that’s the general solution, for arbitrary constants A and B. Now really that’s not good enough. We might have started this oscillator off close or far from equilibrium, or we might have just let it go or given it a good shove. These initial conditions – the initial position and initial velocity – determine what A and B are for our specific physical situation. Let’s go ahead and nail the situation down. Call the initial position x0. That’s the value of the position at t = 0 when the clock started and the system started oscillating. At t = 0, the value being plugged into sin and cos is 0. But sin(0) = 0 and cos(0) = 1, so we see that x0 = A. Nice, that pegs one of our constants. Now do the other. To find the velocity from the position equation, we differentiate with respect to time. Doing this with the knowledge of A, we see that:
Now we’re working with initial conditions, so set t = 0 again. The sin term goes to zero, the cos term goes to 1, and therefore the initial velocity v0 = B*omega. Solve for B, substitute into the general solution:
Ok, that’s great but what does it mean? For starters, it means that the simple harminic oscillator oscillates. It repeats its motion over and over with an angular frequency of omega. We might have guessed that it would oscillate, but thanks to the math we know it does so in an exactly sinusoidal way. Further, we now know the timing of this oscillation in terms of other physical constants, and we can relate amplitudes and velocities to the initial conditions. It’s quite a bit of an improvement over the qualitative “back and forth” description we might have managed without math.
Why is this post given a #1 in the title, by the way? It’s because there’s about a zillion different and important ways to so physics with the harmonic oscillator. Even from a purely classical perspective, there’s this, the Lagrangian formulation, the Hamiltonian formulation, the Poisson bracket formulation, action-angle variables, you name it. Plenty of these are graduate level, but I think I can make the interesting in a guided-tour way for those who aren’t fluent in math-speak. I plan to tackle many of these methods over the next weeks.