Outer symplectic billiards

TL;DR

This work extends outer billiards to outer symplectic billiards by defining a symplectic, partially defined correspondence for immersed submanifolds M⊂R^{2d} and establishing fundamental structural properties via conormal geometry. It proves the existence of odd-periodic orbits through a variational framework, provides a 4D counterexample to even-periodic nonexistence, and shows that under dimension or convexity-type conditions, multiple n-reflection trajectories exist between affine Lagrangian subspaces, with a lower bound on 1-shot counts tied to critical points on M. The paper further analyzes the symplectic meaning of generating functions, relates midpoints to symplectic area, and derives a map criterion in the Lagrangian case. In dimension four, cubic generating functions yield minimal walls or richer walls, and notably, cubic Lagrangian submanifolds give completely integrable outer symplectic billiards, with commuting integrals provided by Poincaré-type invariants. Overall, the work blends variational methods, generating-function formalism, and Lagrangian geometry to advance the understanding of symplectic billiard dynamics in higher dimensions and establish integrability in key cases.

Abstract

A submanifold of the standard symplectic space determines a partially defined, multi-valued symplectic map, the outer symplectic billiard correspondence. Two points are in this correspondence if the midpoint of the segment connecting them is on the submanifold, and this segment is symplectically orthogonal to the tangent space of the submanifold at its midpoint. This is a far-reaching generalization of the outer billiard map in the plane; the particular cases, when the submanifold is a closed convex hypersurface or a Lagrangian submanifold, were considered earlier. Using a variational approach, we establish the existence of odd-periodic orbits of the outer symplectic billiard correspondence. On the other hand, we give examples of curves in 4-space which do not admit 4-periodic orbits at all. If the submanifold satisfies certain conditions (which are always satisfied if its dimension is at least half of the ambient dimension) we prove the existence of two $n$-reflection orbits connecting two transverse affine Lagrangian subspaces for every $n\geq1$. In addition, for every immersed closed submanifold, the number of single outer symplectic billiard ``shots" from one affine Lagrangian subspace to another is no less than the number of critical points of a smooth function on this submanifold. We study, in detail, the behavior of this correspondence when the submanifold is a curve or a Lagrangian submanifold. For Lagrangian submanifolds in 4-dimensional space we present a criterion for the outer symplectic billiard correspondence to be an actual map. We show, in every dimension, that if a Lagrangian submanifold has a cubic generating function, then the outer symplectic billiard correspondence is completely integrable in the Liouville sense.

Figures (5)

• Figure 1: Outer symplectic billiard map in ${\mathbb R}^2$.
• Figure 2: A cubic and a quartic parabolas.
• Figure 3: Left: two central ellipses; right: a central ellipse and a hyperbola.
• Figure 4: Left: nested and interlacing pairs of asymptotes; right: interlacing hyperbolas.
• Figure 5: Nested hyperbolas: four intersections on the right and none on the left.

Theorems & Definitions (54)

• Definition 2.1
• Proposition 2.2
• Example 2.3
• Lemma 2.4
• Lemma 2.5
• Remark 2.6
• Definition 2.7
• Theorem 1
• Corollary 2.8
• Example 2.9