A Morse-Family Integrator for Hamilton--Jacobi Dynamics Across Caustics
Abstract
We develop a geometric framework for implicit discrete Hamiltonian systems based on discrete Morse families, Lagrangian relations, and discrete analogues of Tulczyjew's triple. The main idea is to regard the Lagrangian submanifold defining the discrete dynamics, rather than an explicit symplectic evolution map, as the fundamental geometric object. This viewpoint naturally accommodates implicit, constrained, and degenerate discrete systems. Within this framework, we formulate a Type--II discrete Hamilton--Jacobi theory in terms of the propagation of Lagrangian submanifolds between consecutive discrete steps. When these submanifolds are locally represented by exact one-forms dWk and dWk+1, the resulting equations provide a discrete Hamilton--Jacobi relation between consecutive generating functions. More generally, when the projection onto configuration space becomes singular and a single-valued generating function ceases to exist, we show that the evolution can be described by the composition of Type--II discrete dynamics with Morse families. This yields a generating family for the propagated Lagrangian submanifold without requiring the dynamics to be represented as a graph. As an application, we consider the propagation of optical wavefronts through fold caustics. A Type--II discrete Hamiltonian yields a symplectic ray integrator, while a Morse family represents the multivalued wavefront near the caustic. Their composition provides a discrete propagation rule for the complete Lagrangian manifold across the singularity. In this way, the same geometric construction simultaneously provides a discrete Hamiltonian integrator and a regular representation of multivalued Hamilton--Jacobi solutions.
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BibTeX
@misc{alburquerque2026morsefamily,
title = {{A Morse-Family Integrator for Hamilton--Jacobi Dynamics Across Caustics}},
author = {F. Jiménez Alburquerque and M. Leok and C. Sardón and X. Zhao},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.29645},
eprint = {2608.29645},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.29645},
}
RIS
TY - UNPB
TI - A Morse-Family Integrator for Hamilton--Jacobi Dynamics Across Caustics
AU - F. Jiménez Alburquerque
AU - M. Leok
AU - C. Sardón
AU - X. Zhao
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.29645
UR - https://arxiv.org/abs/2608.29645
AB - We develop a geometric framework for implicit discrete Hamiltonian systems based on discrete Morse families, Lagrangian relations, and discrete analogues of Tulczyjew's triple. The main idea is to regard the Lagrangian submanifold defining the discrete dynamics, rather than an explicit symplectic evolution map, as the fundamental geometric object. This viewpoint naturally accommodates implicit, constrained, and degenerate discrete systems. Within this framework, we formulate a Type--II discrete Hamilton--Jacobi theory in terms of the propagation of Lagrangian submanifolds between consecutive discrete steps. When these submanifolds are locally represented by exact one-forms dW_k and dW_k+1, the resulting equations provide a discrete Hamilton--Jacobi relation between consecutive generating functions. More generally, when the projection onto configuration space becomes singular and a single-valued generating function ceases to exist, we show that the evolution can be described by the composition of Type--II discrete dynamics with Morse families. This yields a generating family for the propagated Lagrangian submanifold without requiring the dynamics to be represented as a graph. As an application, we consider the propagation of optical wavefronts through fold caustics. A Type--II discrete Hamiltonian yields a symplectic ray integrator, while a Morse family represents the multivalued wavefront near the caustic. Their composition provides a discrete propagation rule for the complete Lagrangian manifold across the singularity. In this way, the same geometric construction simultaneously provides a discrete Hamiltonian integrator and a regular representation of multivalued Hamilton--Jacobi solutions.
ER -