The Integrable Cone

Billiard systems — point particles bouncing inside a closed boundary — are one of the simplest dynamical systems. A ball reflects specularly off the wall, travels in a straight line, reflects again. The behavior depends entirely on the shape of the boundary. For most shapes, the dynamics are chaotic. For ellipses (and their degenerate cases — circles, line segments), the dynamics are integrable: the system has enough conserved quantities to confine trajectories to invariant sets, and the motion is regular.

The Birkhoff conjecture proposes that ellipses are the only smooth convex boundaries in the Euclidean plane that produce integrable billiards. Despite a century of work, the conjecture remains open, but partial results strongly suggest that integrability in planar billiards is rare and tied to quadric geometry.

Mironov and Yin (arXiv:2603.28347, March 2026) show that billiards inside cones over strictly convex manifolds are completely integrable as discrete-time Hamiltonian systems. The billiard table is a cone — the boundary is not a smooth convex curve in a plane but the surface of a cone in higher-dimensional space, whose cross-section is a strictly convex manifold. The system admits n-1 independent first integrals in involution, where n is the dimension.

This breaks the expectation from the Birkhoff conjecture. The integrable billiard table is not a quadric. Its boundary is a cone over a convex manifold, and convex manifolds are a vast class — far larger than the quadrics that Birkhoff's conjecture identifies as the only integrable cases in the plane. The integrability lives in the conical structure, not in the cross-sectional geometry.

The structural observation: the class of integrable billiard tables is larger than the planar case suggests. The Birkhoff conjecture, if true, constrains integrability in the plane to quadrics. But lifting the problem to higher-dimensional cones reveals a different structure: the conical geometry provides conserved quantities that the planar geometry cannot. The restriction to the plane hides a family of integrable systems that become visible only in the cone.