Partnership for Integration of Computation into Undergraduate Physics

Alan Cromer's 1981 paper entitled ``Stable solutions using the Euler approximation,'' is an excellent introduction to computational physics modeling for students and educators. In the paper Cromer details several discrete approximations to the equations of motion, demonstrates their use computationally, follows with a seemingly daunting proof of stability for the Last Point Approximation (LPA) among others, and ends with a provocative statement: >This analysis shows that the difference between the different linear approximations is not how well they approximate the derivative at each point, but how well they approximate the first integral of motion. Our goal in this document is to fully understand the proof of stability for the LPA as well as what Cromer means by the quote above and how that connects to a wider modern approach to numerical modeling of physical systems called *symplectic integrators*.