My original goal that I proposed to my mentor Chris was solving boundary value problems (a.k.a. BVPs) which were determined from second order ordinary differential equations (a.k.a. ODEs). I started the BVP path, built a shooting method to solve BVPs from initial value problems (a.k.a. IVPs). Then, I built the beginning of a mono-implicit-Runge-Kutta (a.k.a. MIRK) method, but I don't have all of the bells and whistles. Both of these solvers are in the BoundaryValueDiffEq.jl repository. Instead of trying to jump directly to the end point, and talk about how to do every detail in MIRK, I went to explore how those details naturally arise in second order ODEs.
First, there the idea of symplecticity, because the Labatto MIRK (Lobatto IIIA-IIIB) tableaux are actually symplectic integrators. Symplecticity basically is another way to say energy conservation, so symplectic integrators are specialized for solving second order ODEs that are raise