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

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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.

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Authors and Affiliations

  1. Departamento de Matemática, Universidad Nacional del Sur, CONICET, Av. Alem 1253, 8000, Bahía Blanca, Argentina

    Sebastián Ferraro

  2. Instituto de Ciencias Matemáticas, ICMAT, c/ Nicolás Cabrera, no 13-15, Campus Cantoblanco, UAM., 28049, Madrid, Spain

    Manuel de León, David Martín de Diego & Miguel Vaquero

  3. Unidad Asociada ULL-CSIC “Geometría Diferencial y Mecánica Geométrica” y, Departamento de Matemáticas, Estadística e IO, Facultad de Ciencias, ULL, c/ Astrofísico Francisco Sánchez, s/n, 38206, La Laguna - Tenerife, Canary Islands, Spain

    Juan Carlos Marrero

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  1. Sebastián Ferraro
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  2. Manuel de León
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Correspondence to Juan Carlos Marrero.

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Communicated by M. Ortiz

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Ferraro, S., de León, M., Marrero, J.C. et al. On the Geometry of the Hamilton–Jacobi Equation and Generating Functions. Arch Rational Mech Anal 226, 243–302 (2017). https://doi.org/10.1007/s00205-017-1133-0

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