On the Poincar\'e's generating function and the symplectic mid-point rule
Mathematical Physics
2019-10-30 v4 Numerical Analysis
math.MP
Numerical Analysis
Symplectic Geometry
Abstract
The use of Liouvillian forms to obtain symplectic maps for constructing numerical integrators is a natural alternative to the method of generating functions, and provides a deeper understanding of the geometry of this procedure. Using Liouvillian forms we study the generating function introduced by Poincar\'e (1899) and its associated symplectic map. We show that in this framework, Poincar\'e's generating function does not correspond to the symplectic mid-point rule, but to the identity map. We give an interpretation of this result based on the original framework constructed by Poincar\'e.
Cite
@article{arxiv.1508.07743,
title = {On the Poincar\'e's generating function and the symplectic mid-point rule},
author = {Hugo Jiménez-Pérez and Jean-Pierre Vilotte and Barbara Romanowicz},
journal= {arXiv preprint arXiv:1508.07743},
year = {2019}
}
Comments
11 pages, 2 figures. A better exposition of the method of Liouvillian forms including most recent results