AI 中文总结
该研究开发了基于离散莫尔斯族等的几何框架，构建了II型离散哈密顿-雅可比理论，可用于焦散面间的多值拉格朗日子流形传播，同时提供离散哈密顿积分器与多值解的正则表示。
AI 中文摘要
我们基于离散莫尔斯族、拉格朗日关系以及图尔齐耶夫三重态（Tulczyjew's triple）的离散类似物，开发了隐式离散哈密顿系统的几何框架。核心思路是将定义离散动力学的拉格朗日子流形而非显式辛演化映射视为基本几何对象，该视角自然适配隐式、受约束及退化离散系统。在该框架内，我们基于连续离散步间拉格朗日子流形的传播，构建了II型离散哈密顿-雅可比理论。当这些子流形由恰当1-形式$dW_k$和$dW_{k+1}$局部表示时，所得方程给出了连续生成函数间的离散哈密顿-雅可比关系。更一般地，当到构型空间的投影变为奇异、单值生成函数不复存在时，我们证明演化可通过II型离散动力学与莫尔斯族的组合来描述，这会生成传播拉格朗日子流形的生成族，无需将动力学表示为图。作为应用，我们考虑光学波前穿过折叠焦散面的传播：II型离散哈密顿量产生辛射线积分器，而莫尔斯族表示焦散附近的多值波前，二者的组合为穿过奇点的完整拉格朗日流形提供了离散传播规则。由此，同一几何构造同时提供了离散哈密顿积分器与多值哈密顿-雅可比解的正则表示。
英文摘要
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.
