Conical Billiards: Dynamics & Integrability

• Conical billiards are models where conical geometry dictates reflection laws, conserved integrals, and caustic formations across varying dynamical regimes.
• In smooth convex cones, universal quadratic integrals and spherical caustics ensure finite reflections, while less regular settings can exhibit infinitely many reflections.
• Extensions to conical surfaces and constant-curvature spaces reveal chaotic dynamics and polynomial integrability, unifying models via confocal-conical boundary classifications.

Searching arXiv for the specified papers and closely related work on conical billiards. Conical billiards is a collective label for several distinct billiard theories in which conical geometry is the organizing feature. In recent work, the phrase has referred to ordinary Euclidean Birkhoff billiards inside cones in $\mathbb{R}^n$, billiards on the surface of a cone with a reflective boundary, billiards in convex bodies with corner or cone-type singularities described by tangent cones, and polynomially integrable billiards on constant-curvature surfaces whose boundaries are unions of confocal conical arcs (Mironov et al., 22 Jan 2025, Braverman et al., 4 Aug 2025, Akopyan et al., 2015, Glutsyuk, 2017). Across these settings, the cone may appear as the billiard table itself, as a singularity of the boundary, or as an ambient quadric construction governing integrability.

1. Geometric scope of the term

In the Euclidean Birkhoff setting, a cone is written as

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$

where $\gamma$ is a smooth closed section in a hyperplane such as $\{x^n=1\}$. The particle moves along straight lines in the region inside the cone and reflects specularly from the smooth lateral boundary. This is the model developed in the integrability results for smooth convex cones (Mironov et al., 22 Jan 2025).

A second usage concerns motion on a conical surface rather than inside a conical domain. There the particle moves along cone geodesics and reflects from a boundary curve obtained by intersecting the cone with a plane, typically tilted. The cone apex is a curvature singularity, and the boundary becomes nontrivial after unfolding the cone to a planar sector (Braverman et al., 4 Aug 2025).

A third usage arises in non-smooth convex billiards. There the relevant local object is the tangent cone $T_K(q)$ at a non-smooth boundary point $q\in\partial K$. The existence and nature of periodic billiard trajectories depend on whether these tangent cones are acute in a precise sense (Akopyan et al., 2015).

A fourth usage appears in the algebraic theory of polynomially integrable billiards on the plane, sphere, and hyperbolic plane. In that context, billiard boundaries are “conical” because their smooth pieces are conics on the surface obtained as intersections with ambient quadrics or cones in $\mathbb{R}^3$, and all such pieces belong to a single confocal pencil (Glutsyuk, 2017).

2. Euclidean Birkhoff billiards inside smooth convex cones

For a cone $K\subset\mathbb{R}^n$ with vertex at the origin $O$, the billiard reflection law at a smooth boundary point $p_k$ with inward unit normal $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$0 is

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$1

where

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$2

The decisive geometric identity is that the radial vector belongs to the tangent hyperplane of the cone, so $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$3 at every smooth boundary point. As a consequence, the billiard preserves the squared distance from the trajectory line to the vertex:

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$4

and

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$5

This gives a universal quadratic first integral for billiards inside any cone. It follows in particular that spheres centered at the vertex are caustics: if one segment is tangent to such a sphere, every reflected segment remains tangent to the same sphere (Mironov et al., 22 Jan 2025).

For a $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$6 convex cone, meaning that a hyperplane section $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$7 is a strictly convex closed $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$8 submanifold with nondegenerate second fundamental form, every billiard trajectory has a finite number of reflections (Mironov et al., 22 Jan 2025). The phase space is an open subset

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$9

consisting of oriented lines intersecting the cone transversally. On this phase space there exist continuous first integrals

$\gamma$0

that are invariant under the billiard map, smooth almost everywhere, and whose joint values determine a unique billiard trajectory. The survey literature adds that there is a smooth submanifold $\gamma$1 such that the $\gamma$2 are smooth on $\gamma$3, and that globally smooth integrals $\gamma$4 can be chosen to vanish on $\gamma$5 while still determining trajectories on $\gamma$6 (Bialy et al., 4 Oct 2025).

The integrability notion used here is not Liouville–Arnold integrability. The construction yields orbit separation by first integrals rather than a theorem about commuting integrals and invariant tori. The paper presenting this theory states that this is “the first example of an integrable billiard where the billiard table is neither a quadric nor composed of pieces of quadrics” (Mironov et al., 22 Jan 2025).

3. Regularity thresholds and reflection-count estimates

The $\gamma$7 finite-reflection theorem is sharp with respect to regularity in the sense currently known. There exist $\gamma$8-smooth convex cones with billiard trajectories having infinitely many reflections in finite time (Mironov et al., 4 Feb 2025). The construction begins from the circular cone

$\gamma$9

and prescribes impact directions

$\{x^n=1\}$0

The associated angular gaps satisfy

$\{x^n=1\}$1

so $\{x^n=1\}$2. Radial factors are then chosen by

$\{x^n=1\}$3

which yields a polygonal line with

$\{x^n=1\}$4

for every segment and finite total length when $\{x^n=1\}$5. A Halpern-type interpolation argument is then used to reconstruct a strictly convex closed $\{x^n=1\}$6 cross-section with the required normals, producing an actual billiard trajectory with infinitely many reflections in finite time (Mironov et al., 4 Feb 2025).

For smooth cones, however, finiteness does not imply a uniform bound over all initial data. The survey formulation is explicit: “for smooth cones the number of reflections cannot be uniformly bounded” (Bialy et al., 4 Oct 2025). This separates the smooth case from the polyhedral case, where Sinai’s theorem provides a cone-dependent uniform bound.

A quantitative bound is known for elliptic cones in $\{x^n=1\}$7,

$\{x^n=1\}$8

Besides the universal integral

$\{x^n=1\}$9

these cones admit

$T_K(q)$0

For a trajectory with fixed values $T_K(q)$1 and $T_K(q)$2, the number of reflections is bounded by

$T_K(q)$3

The same analysis identifies a second family of caustics: in addition to spheres, the trajectories are tangent to cones

$T_K(q)$4

(Mironov et al., 4 Feb 2025).

4. Billiards on conical surfaces and gravitational cone dynamics

A different conical model places the motion on the surface of a cone. In the tilted-plane problem, the cone is characterized by half-angle $T_K(q)$5 or deficit-angle parameter $T_K(q)$6,

$T_K(q)$7

so the apex carries a concentrated curvature $T_K(q)$8. The reflecting boundary is the intersection of the cone with a plane tilted by angle $T_K(q)$9. After cutting and unrolling the cone, geodesics become straight lines in a planar sector of angle $q\in\partial K$0, and the billiard is represented by a Poincaré map in coordinates $q\in\partial K$1, where $q\in\partial K$2 is the boundary position in unfolded polar angle and $q\in\partial K$3 is the angle made by the outgoing geodesic with the vector from the collision point to the apex (Braverman et al., 4 Aug 2025).

For the untilted cone, $q\in\partial K$4, the system is integrable:

$q\in\partial K$5

$q\in\partial K$6

Every trajectory is then a rim trajectory and leaves an excluded inner region bounded by a ring caustic. For $q\in\partial K$7, the extra conservation law is lost. The paper identifies three orbit classes: Type A or rim trajectories, which sample the full boundary while avoiding the apex region; Type B or hourglass trajectories, which sample only part of the boundary; and Type C or mixed trajectories, which explore the cone much more uniformly and are the candidate chaotic or ergodic trajectories. A key geometric threshold is the onset of partial concavity of the unfolded boundary:

$q\in\partial K$8

The authors report a KAM-like transition in which rim trajectories persist for small $q\in\partial K$9, hourglass families appear, and mixed trajectories expand into a chaotic sea. For strongly mixed cases they measure a positive Lyapunov exponent, with

$\mathbb{R}^3$0

for $\mathbb{R}^3$1, and they present evidence of strong mixing for $\mathbb{R}^3$2 and $\mathbb{R}^3$3 (Braverman et al., 4 Aug 2025).

A further conical model introduces gravity inside a linear cone. There the particle moves ballistically under a constant gravitational field and reflects elastically from the cone. With cone half-angle $\mathbb{R}^3$4 and conserved axial angular momentum $\mathbb{R}^3$5, the Hamiltonian is

$\mathbb{R}^3$6

After scaling to

$\mathbb{R}^3$7

the dynamics reduces to a two-parameter, two-dimensional Poincaré map in $\mathbb{R}^3$8, with parameters $\mathbb{R}^3$9 and a normalized angular-momentum parameter $K\subset\mathbb{R}^n$0. In these coordinates the map is area-preserving. The model has several integrable limits: $K\subset\mathbb{R}^n$1, where it reduces exactly to the wedge billiard; $K\subset\mathbb{R}^n$2, where the cone becomes a horizontal plane; and $K\subset\mathbb{R}^n$3 with $K\subset\mathbb{R}^n$4, where the effective Hamiltonian becomes central. A fixed point of the reduced map corresponds to collisions at constant radius on a circle of the cone, but the full spatial orbit is closed only when the azimuthal increment per bounce is rationally commensurate with $K\subset\mathbb{R}^n$5. For small $K\subset\mathbb{R}^n$6 the phase space resembles the wedge billiard, whereas increasing $K\subset\mathbb{R}^n$7 makes the dynamics on the whole less chaotic (Langer et al., 2015).

5. Acute tangent cones and non-smooth convex billiards

In the theory of non-smooth convex billiards, conical geometry enters through tangent cones at singular boundary points. A generalized billiard trajectory may reflect at a non-smooth point $K\subset\mathbb{R}^n$8 provided the bisector of the angle $K\subset\mathbb{R}^n$9 is orthogonal to some support hyperplane at $O$0, or equivalently, the difference of unit velocities is proportional to some outer normal. The variational characterization used in this setting is

$O$1

where the minimum is attained for

$O$2

Here $O$3 is the length of the shortest closed generalized billiard trajectory, and $O$4 consists of $O$5-tuples whose vertex set cannot be translated into $O$6 (Akopyan et al., 2015).

At a non-smooth point $O$7, the relevant local data are the normal cone $O$8 and the tangent cone

$O$9

The acuteness condition is

$p_k$0

where $p_k$1 is a linear subspace orthogonal to $p_k$2, and

$p_k$3

If every non-smooth point satisfies this condition, the convex body is called acute. The main theorem then states: in an acute convex body $p_k$4, there exists a closed classical billiard trajectory with no more than $p_k$5 bounces (Akopyan et al., 2015).

The mechanism is exclusionary rather than constructive at the singular point. If a shortest generalized trajectory passed through a non-smooth point, the acute-cone geometry would allow that single generalized reflection to be replaced by two nearby reflections on supporting hyperplanes, giving a strictly shorter admissible polygon. Thus the minimizing closed trajectory avoids singular points and is classical. The paper also proves a more general statement: if the shortest closed generalized trajectory in $p_k$6 has precisely $p_k$7 bounces, then it is classical even without an acute-cone assumption (Akopyan et al., 2015).

Polyhedral examples are immediate. In a simplex with all acute dihedral angles, there exists a closed classical billiard trajectory with $p_k$8 bounces. Conversely, the same paper explicitly states that it has “nothing to say about obtuse triangles,” so wide cone openings remain outside this method (Akopyan et al., 2015).

6. Polynomial integrability and confocal-conical boundaries on constant-curvature surfaces

On the Euclidean plane, sphere, and hyperbolic plane, polynomially integrable Birkhoff billiards admit a complete geometric classification in which “conical” refers to the ambient quadric geometry of the boundary. A billiard in a domain $p_k$9 is polynomially integrable if its flow has a first integral on $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$00 that is a polynomial in the velocity and non-constant on the unit-energy hypersurface. Bolotin’s theorem permits such an integral to be chosen as a homogeneous polynomial in the moment vector

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$01

The classification result states that a billiard with countably piecewise $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$02-smooth boundary is polynomially integrable if and only if it is countably confocal, meaning that the regular part of the boundary is a union of arcs of conics from one confocal pencil, together possibly with admissible geodesic segments (Glutsyuk, 2017).

For a $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$03-smooth connected boundary not contained in a geodesic, the boundary itself is a conic or a connected component of a conic. In the bounded Euclidean planar case, this reduces to the statement that every bounded polynomially integrable planar billiard with $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$04-smooth connected boundary is an ellipse. The confocal pencil is written

$$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$05

with special degenerate parameters producing limiting geodesics. On the sphere and hyperbolic plane, the boundary arcs are literally intersections of the surface with ambient quadratic cones or quadrics in $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$06 (Glutsyuk, 2017).

This theory solves the algebraic Birkhoff conjecture on simply connected complete surfaces of constant curvature. The proof uses tautological projection to $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$07, orthogonal-polar duality, Bolotin’s theorem, and the result that every nonlinear irreducible component of the dual curve generating a rationally integrable angular billiard must be a conic. In this setting, “conical billiards” does not mean motion inside a Euclidean cone; it means that the admissible smooth boundary pieces are conics on $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$08 cut out by a single confocal family of ambient quadratic cones or quadrics (Glutsyuk, 2017).

7. Limits, unresolved questions, and conceptual distinctions

Several limitations of the current theory are explicit. In the Euclidean inside-cone problem, the finite-reflection theorem is known under $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$09 smoothness, strict convexity, and positive definite second fundamental form, but the survey literature asks for which submanifolds $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$10 every trajectory inside the cone over $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$11 has a finite number of reflections. The same discussion asks whether positive definiteness can be weakened to positive semidefiniteness or even indefiniteness, and whether the cross-section can have a topology different from a sphere, for example a torus (Bialy et al., 4 Oct 2025).

In the non-smooth convex-body theory, the acute tangent-cone hypothesis is a sufficient condition, not a necessary one, and the difficult obtuse-corner regime remains open. The existence theorem yields a classical periodic orbit avoiding the singular set; it does not classify periodic trajectories that pass through singular points (Akopyan et al., 2015).

In the conical-surface model with tilted boundary, the strongest chaotic behavior occurs between two integrable limits: the untilted cone $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$12 and the nearly flat ellipse limit $$K=\{tp\mid p\in \gamma,\ t>0\}\subset \mathbb{R}^n,$$13. The paper’s parameter diagram divides the problem into a convex unfolded-boundary region, a partially concave region where stable hourglass structures can still obstruct ergodicity, and a subregion where mixed trajectories dominate and strong mixing is plausible (Braverman et al., 4 Aug 2025).

These distinctions are essential because the same phrase covers genuinely different dynamical mechanisms. In smooth Euclidean cones, the dominant structures are the universal quadratic integral, spherical caustics, escape after finitely many reflections, and asymptotic orbit classification (Mironov et al., 22 Jan 2025). On conical surfaces, the apex curvature singularity and symmetry-breaking tilt generate KAM-type breakup and chaotic transport (Braverman et al., 4 Aug 2025). In non-smooth convex bodies, tangent-cone acuteness suppresses minimizing singular reflections (Akopyan et al., 2015). On constant-curvature surfaces, confocal conical arcs characterize polynomial integrability (Glutsyuk, 2017). Taken together, these results make conical billiards a broad research area rather than a single model, unified by the fact that conical geometry controls reflection, caustics, and integrability in a mathematically decisive way.