A Morse-Family Integrator for Hamilton--Jacobi Dynamics Across Caustics
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.