Using Monte Carlo and self-consistency to solve Newton's 2nd Law

In working with Newton's 2nd law, given a force, one seeks to solve for the position, x(t), and the velocity, v(t), as a function of time, t. Obtaining an analytic solution is, of course, of great value, when the problem is solvable. However, a numerical solution is a common route to difficult problems and methods to effect them exist. Two different numerical approaches of interest here are a Monte Carlo (MC) approach [1] and a self-consistent (SC) method. The idea, in the MC case, is to make a random guess for the acceleration, a(t), and to use the trapezoid method to get the velocity versus time, v(t). The time dependent position, x(t), follows by standard numerical integration of v(t). Depending on the situation, sometimes this method is not very efficient. In the SC method, one starts by making an initial numerical guess for a(x(t)) to obtain a v(t), which is used to obtain an x(t), which results in a new a(x(t)), etc. The process is repeated until there is no change in x(t). Here both methods are applied to simple systems and compared to the known analytic solutions for comparison and assessment purposes.
[1] "Solving initial value ordinary differential equations by Monte Carlo Method," M. N. Akhtar, M. H. Drad, and A. Ahmed, Proceeding of IAM, V.4, N.2 (2015), pp 149-174.