Elementary algorithms from advanced mechanics: http://arxiv.org/abs/1607.03882
Most elementary numerical schemes found useful for solving classical trajectory problems are canonical transformations. This fact should be make more widely known among teachers of computational physics and Hamiltonian mechanics. It is very surprising that in order to solve a simple second-order differential equation, one has to invoke the deepest part, the Poissonian formulation, of classical mechanics. From the perspective of advanced mechanics, there are no bewildering number of seemingly arbitrary elementary schemes based on Taylor's expansion. There are only two canonical second-order algorithms, on the basis of which one can build numerical schemes of any order.