What is a closed form solution? And why don't we have it in general?
A closed form solution is simple equation that describes the position of the system at every point of time. For example for a rock falling under the influence of gravity, there's a simple equation that gives the position of the rock as a function of time that we can get by solving the Lagrangian = 0. But in general this isn't something we can get simply because the differential equations become too difficult/impossible to solve. Hence the need for numerical integration