Professor Politzer and the Rho Mesons: Simple Harmonic Oscillator

I want to talk today about things that shake and I hope my words aren't too opaque. One degree of freedom moving to and fro just how it moves we'd like to know we can represent all kinds of things by a single mass between ideal springs. Each spring's connected to a wall so the outer ends don't move at all
Let the mass be $m$ spring constant $k$ but don't let friction get in the way use Newton's laws and what have we got $F$ equals $m$ psi double dot that is also minus escape psi times 2 so now we have a diff eq and we can write down the general solution for the simple harmonic time evolution
Let omega be root 2 $k$ over $m$ here's the answer won't repeat again size a cosine omega t plus a phase call it $b$ so it's all very simple and you can see for any initial psi and velocity we can find the constants $a$ and $b$ and the equations exact for all time $t$
Now look again at the diff eq. It's homogeneous and linear too so if you add two solutions together there sums a solution that's even better. We call it the principle of superposition. You can use it to fit the boundary condition in fact there is no contradiction if we use it in a system that does have friction
In a real system nothing's perfect of course we have to include the frictional force suppose it goes as the velocity right minus $m$ gamma d psi dt now if the damping is not too strong our old solution is close but wrong see it starts out with some amplitude $a$ but after a while it just dies away
The amplitude decays exponentially as you can see experimentally as $e$ to the minus half gamma $t$. Now it's almost right but you see the frequency is lower as we can compute omega is now given by the square root of the quantity $k$ over $m$ times two minus quarter gamma squared now we're through
So now we have the complete solution for an oscillator's time evolution and when there's damping as everyone knows the amplitude decays and the frequency slows if we have two solutions no matter how chose you know we can always superpose and since you all find physics such fun to problems 12 18 and 21.

Class dismissed
In the video I use the notation used on (audio source)

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