About two months ago, I wrote a study notes named Symmetry and Harmonic Motion on how to exploit symmetry of a system doing harmonic motion. I use a baby problem: two particles in 1D, to demonstrate it is possible to determine the eigenmodes from pure group theoretical analysis. The baby problem is too trivial to show the power of group theory, so today we continue with the kindergarten problem: three particles in 2D. To fully understand this study notes, it is recommended to go through my previous post about the baby problem, where various useful mathematical tools are introduced. Two of the tools are especially important for today’s discussion. The first is Schur’s lemma, discussed here, and the second is the projection operator into desired mode, discussed here.
I want to impose
The Kindergarten Problem: Three Particles in 2D
The problem is shown in the figure above. We have three particles connected by springs, moving in a plane. Imagine we start from equilibrium configuration, and focus on small deviation from it, so the dynamic variables are displacements from the equilibrium, as is shown in the figure. We can put them together to form a vector with
