Structure of numerical algorithms and advanced mechanics

TL;DR

This paper reveals that most elementary numerical schemes for classical trajectory problems are canonical transformations, and demonstrates how all such algorithms from first to fourth order can be derived from principles of advanced mechanics.

Contribution

It clarifies that elementary numerical schemes are canonical transformations and shows their derivation from advanced mechanics principles, unifying the understanding of these algorithms.

Findings

Most elementary schemes are canonical transformations.

All algorithms from first to fourth order derive from canonical transformations.

Provides a unified framework for understanding numerical algorithms in mechanics.

Abstract

Most elementary numerical schemes found useful for solving classical trajectory problems are {\it canonical transformations}. This fact should be make more widely known among teachers of computational physics and Hamiltonian 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 {\it two} canonical first and second order algorithms, on the basis of which one can comprehend the structures of higher order symplectic and non-symplectic schemes. This work shows that, from the most elementary first-order methods to the most advanced fourth-order algorithms, all can be derived from canonical transformations and Poisson brackets of advanced mechanics.