Dynamical Systems · Chapter VI, §6

Billiards as a discrete variational problem

Original

6. The billiard ball problem.*

In order to see how the theorem of Poincaré and its generalization can be applied, we will consider first a special but highly typical system of this sort, namely that afforded by the motion of a billiard ball upon a convex billiard table. This system is very illuminating for the following reason: Any Lagrangian system with two degrees of freedom is isomorphic with the motion of a particle on a smooth surface rotating uniformly about a fixed axis and carrying a conservative field of force with it.† In particular if the surface is not rotating and if the field of force is lacking, the paths of the particle will be geodesics. If the surface is now flattened to the form of a plane convex curve $C$, the ‘billiard ball problem’ results. But in this problem the formal side, usually so formidable in dynamics, almost completely disappears, and only the interesting qualitative questions need to be considered. If $C$ is an ellipse an integrable problem results, namely the limiting case of an ellipsoid treated by Jacobi.

The sections 6–9 are taken from my paper On the Periodic Motions of Dynamical Systems, about to appear in the Acta Mathematica.

† See my paper, Dynamical Systems with Two Degrees of Freedom, Trans. Amer. Math. Soc., vol. 18 (1917), section 7. It is assumed that the Lagrangian principal function is quadratic in the velocities.

In the billiard ball problem one can arrive at the existence of certain periodic motions by direct maximum-minimum methods. As of interest in itself I wish to show how this can be done. Results which are being obtained by Morse (see chapter V, section 8) indicate that the scope of these methods, already developed to some extent by Poincaré, Hadamard, Whittaker and myself, can be further extended. Thus the power of such maximum-minimum considerations in the billiard ball problem is likely to prove typical of the general case.

The longest chord of the boundary $C$ of the billiard table, when traversed in both directions, evidently yields one of the simplest periodic motions. The billiard ball moving along this chord strikes the curved boundary at right angles and recoils along it in the opposite direction. If we seek to vary this chord continuously, while diminishing its length as little as possible, so as finally to interchange its two ends, there will be an intermediate position of least length, which will be the chord crossing $C$ where $C$ is of least breadth. Detailed computation of the slightly perturbed motions indicates that the first of these two periodic motions is unstable, while the second may be stable, or unstable.

Next we ask for the triangle of maximum length inscribed in $C$. Evidently at least one such triangle will exist, and can have no degenerate side of length $0$. At each of its vertices the tangent will of course make equal angles with the two sides passing through the vertex. Hence a ‘harmonic triangle’ is obtained which will correspond to two distinct motions, one for each of the two possible senses of description.

Moreover, if we seek to vary this triangle continuously, not changing the order of its vertices and diminishing the perimeter as little as possible, so as finally to advance the vertices cyclically, we discover a second harmonic triangle, also corresponding to two periodic motions.

In this way the existence of two harmonic $n$-sided polygons which make $k\leq n/2$ circuits of the curve $C$ ($k$ prime to $n$) can be proved. The two periodic motions corresponding to the polygon of maximum type will be unstable, while the others of minimax type may be stable or unstable.

In the case of a circular boundary the totality of regular inscribed polygons (simple or cross) form the harmonic polygons.

Modernized

The billiard model

A harmonic polygon is a stationary inscribed polygon satisfying the equal-angle reflection law at each vertex. Two orientations of one polygon give two directed motions, not two different geometric polygons. The max–min existence and stability statements below retain the source’s scope and level of justification.

The sections 6–9 are taken from Birkhoff’s paper On the Periodic Motions of Dynamical Systems, announced at the time as forthcoming in Acta Mathematica.

† See Birkhoff’s paper, Dynamical Systems with Two Degrees of Freedom, Trans. Amer. Math. Soc., vol. 18 (1917), section 7. It is assumed that the Lagrangian principal function is quadratic in the velocities.

In the billiard ball problem one can arrive at the existence of certain periodic motions by direct max–min methods. As of interest in itself Birkhoff explains how this can be done. Results which Birkhoff reports were being obtained by Morse (see chapter V, section 8) indicate that the scope of these methods, already developed to some extent by Poincaré, Hadamard, Whittaker and Birkhoff, can be further extended. Thus the power of such max–min considerations in the billiard ball problem is likely to prove typical of the general case.

Extremal chords

Stationary inscribed polygons

In the case of a circular boundary the totality of regular inscribed polygons (simple or cross) form the harmonic polygons.

Reinterpreted

Collision polygons and perimeter

Let $C$ be a smooth strictly convex closed plane curve, with a counterclockwise arclength parametrization $\gamma(s)$ of period $L$. A billiard path moves along straight segments and reflects by preserving the tangential velocity component and reversing the normal component. At unit speed, each segment’s length equals its flight time.

For a cyclic list of boundary positions with consecutive vertices distinct, define the discrete action

$$\mathcal A(s_0,\ldots,s_{n-1})=\sum_{j=0}^{n-1}H(s_j,s_{j+1}),\qquad H(s,r)=|\gamma(r)-\gamma(s)|,\quad s_n=s_0.$$

The term “discrete action” here means the sum of chord lengths, with no prescribed common flight time. It is the fixed-energy length functional, not a time-step approximation to a kinetic-energy action.

Stationarity is the reflection law

Write $u_j=(\gamma(s_{j+1})-\gamma(s_j))/H(s_j,s_{j+1})$ for the unit vector along segment $j$, and $\tau_j=\gamma'(s_j)$. Differentiating with respect to one vertex gives

$$\frac{\partial\mathcal A}{\partial s_j}=(u_{j-1}-u_j)\cdot\tau_j.$$

Thus stationarity says that incoming and outgoing tangential components agree. Because both segments join points of a strictly convex boundary, their normal components have opposite signs; unit length then forces equal magnitudes. This is precisely specular reflection. Conversely, every non-grazing closed billiard polygon is a critical point of $\mathcal A$.

This calculation links §6 to §8: there the same endpoint derivatives, $p=-H_s$ and $p_1=H_r$, define an exact symplectic map; here their matching at adjacent chords is the discrete Euler–Lagrange equation

$$H_r(s_{j-1},s_j)+H_s(s_j,s_{j+1})=0.$$

Extremal chords

A maximum of $H(s,r)$ exists by compactness and has distinct endpoints. Its two endpoint derivatives vanish, so the chord is normal to $C$ at both ends. Traversing it back and forth is a two-collision periodic orbit.

For a second construction assume strictly positive curvature. Let $h(\phi)$ be the support function relative to a fixed origin: the supporting line with outward unit normal $n(\phi)=(\cos\phi,\sin\phi)$ has equation $x\cdot n(\phi)=h(\phi)$. Its contact point is

$$x(\phi)=h(\phi)n(\phi)+h'(\phi)t(\phi),\qquad t(\phi)=(-\sin\phi,\cos\phi).$$

The width is $w(\phi)=h(\phi)+h(\phi+\pi)$. The opposite contact points satisfy

$$x(\phi)-x(\phi+\pi)=w(\phi)n(\phi)+w'(\phi)t(\phi).$$

At an extremum of width, $w'=0$, so the joining chord is normal at both endpoints. A minimum-width direction supplies a periodic chord. If width is nonconstant, its minimum and maximum directions give distinct chords; if width is constant, every direction does. This proves existence of at least two geometric two-bounce trajectories under the stated curvature assumption, without the source’s informal path-of-chords minimax construction.

Maximal polygons and the minimax limit

For $n\geq3$, consider ordered vertices around $C$, allowing collisions of adjacent vertices in the compact closure. Perimeter attains a maximum. Such a maximum cannot have a repeated adjacent vertex: with a redundant vertex one can insert a new point on a boundary arc between two distinct neighbors, and strict convexity makes the sum of the two new chords strictly longer than the old chord. A configuration with all vertices equal has zero perimeter and cannot maximize. Therefore a maximum lies in the interior of the ordered configuration space and satisfies the reflection law at every vertex.

This establishes a winding-one stationary $n$-gon. Reversing its orientation gives the opposite directed orbit, not a second polygon. Finding a second geometric polygon, or imposing another winding class, requires a further minimax or twist argument. Birkhoff sketches the minimax route; §9 treats the periodic-orbit consequence of the annulus theorem. We do not identify a reversed orbit with the second orbit supplied by that theorem without a separate counting argument.

A variational maximum does not by itself prove strict dynamical instability. On a circular table, all rotated copies of a regular polygon are stationary, and the linearized return has neutral directions. The broad stability sentence in the historical text needs nondegeneracy qualifications; it is retained in Original and Modernized and is not asserted here in that unrestricted form.