Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On the Geometry of the Hamilton–Jacobi Equation and Generating Functions

On the Geometry of the Hamilton–Jacobi Equation and Generating Functions In this paper we develop a geometric version of the Hamilton–Jacobi equation in the Poisson setting. Specifically, we “geometrize” what is usually called a complete solution of the Hamilton–Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80–88, 1991), Ge (Indiana Univ. Math. J. 39(3):859–876, 1990), Ge and Marsden (Phys Lett A 133(3):134–139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Archive for Rational Mechanics and Analysis Springer Journals

On the Geometry of the Hamilton–Jacobi Equation and Generating Functions

Download PDF
60 pages

Loading next page...
1
 
/lp/springer_journal/on-the-geometry-of-the-hamilton-jacobi-equation-and-generating-1GGQwFZXF8

References (69)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Physics; Classical Mechanics; Physics, general; Theoretical, Mathematical and Computational Physics; Complex Systems; Fluid- and Aerodynamics
ISSN
0003-9527
eISSN
1432-0673
DOI
10.1007/s00205-017-1133-0
Publisher site
See Article on Publisher Site

Abstract

In this paper we develop a geometric version of the Hamilton–Jacobi equation in the Poisson setting. Specifically, we “geometrize” what is usually called a complete solution of the Hamilton–Jacobi equation. We use some well-known results about symplectic groupoids, in particular cotangent groupoids, as a keystone for the construction of our framework. Our methodology follows the ambitious program proposed by Weinstein (In Mechanics day (Waterloo, ON, 1992), volume 7 of fields institute communications, American Mathematical Society, Providence, 1996) in order to develop geometric formulations of the dynamical behavior of Lagrangian and Hamiltonian systems on Lie algebroids and Lie groupoids. This procedure allows us to take symmetries into account, and, as a by-product, we recover results from Channell and Scovel (Phys D 50(1):80–88, 1991), Ge (Indiana Univ. Math. J. 39(3):859–876, 1990), Ge and Marsden (Phys Lett A 133(3):134–139, 1988), but even in these situations our approach is new. A theory of generating functions for the Poisson structures considered here is also developed following the same pattern, solving a longstanding problem of the area: how to obtain a generating function for the identity transformation and the nearby Poisson automorphisms of Poisson manifolds. A direct application of our results gives the construction of a family of Poisson integrators, that is, integrators that conserve the underlying Poisson geometry. These integrators are implemented in the paper in benchmark problems. Some conclusions, current and future directions of research are shown at the end of the paper.

Journal

Archive for Rational Mechanics and Analysis – Springer Journals

Published: Jun 9, 2017

There are no references for this article.