On Lagrangian tangent sweeps and Lagrangian outer billiards

TL;DR

This paper explores the geometric properties of Lagrangian submanifolds in symplectic space, introducing tangent sweeps, tangent clusters, and a symplectic outer billiard correspondence, extending classical billiard dynamics to higher dimensions.

Contribution

It defines and analyzes the outer billiard correspondence for Lagrangian submanifolds, proving its symplectic nature and establishing the existence of periodic orbits, generalizing known 2D results.

Findings

The tangent map between tangent sweep and tangent cluster is a local symplectomorphism.

The outer billiard correspondence for Lagrangian submanifolds is symplectic.

Periodic orbits exist in the Lagrangian outer billiard system.

Abstract

Given a Lagrangian submanifold in linear symplectic space, its tangent sweep is the union of its (affine) tangent spaces, and its tangent cluster is the result of parallel translating these spaces so that the foot point of each tangent space becomes the origin. This defines a multivalued map from the tangent sweep to the tangent cluster, and we show that this map is a local symplectomorphism (a well known fact, in dimension two). We define and study the outer billiard correspondence associated with a Lagrangian submanifold. Two points are in this correspondence if they belong to the same tangent space and are symmetric with respect to its foot pointe. We show that this outer billiard correspondence is symplectic and establish the existence of its periodic orbits. This generalizes the well studied outer billiard map in dimension two.