Average Elliptic Billiard Invariants with Spatial Integrals
AAVERAGE ELLIPTIC BILLIARDINVARIANTS WITH SPATIAL INTEGRALS
JAIR KOILLER, DAN REZNIK, AND RONALDO GARCIA
Abstract.
We compare invariants of N-periodic trajectories in the ellipticbilliard, classic and new, to their aperiodic counterparts via a spatial integralsevaluated over the boundary of the elliptic billiard. The integrand is weighedby a universal measure equal to the density of rays hitting a given boundarypoint. We find that aperiodic averages are smooth and monotonic on causticeccentricity, and perfectly match N-periodic average invariants at the discretecaustic parameters which admit a given N-periodic family. Introduction
The two classic invariants of Poncelet N-periodics in the elliptic billiard areperimeter L and quantity known as Joachimsthal’s constant J ; see Figure 1. Theformer implies a billiard trajectory is an extremum of the perimeter function whilethe latter is equivalent to stating that all trajectory segments are tangent to aconfocal caustic [14].Experiments have unearthed a few additional “dependent” invariants including(i) the sum of cosines, (ii) the product of outer polygon cosines, (iii) certain ratiosof areas, etc. [12]. These have been subsequently proved [1, 3, 4]. More recently,the list of conjectured invariants has grown to many dozen [13].With a small perturbation of the caustic, an N-periodic trajectory becomesaperiodic (space-filling); see Figure 2. A key question we explore is: given a discreteinvariant computed for an N-periodic, what is its analogue in the space-filling case?In the latter case, the sum or product of a given quantity can diverge. Fortunately,in both cases we can compare their finite averages . Main Result.
Our contribution is to accurately and efficiently estimate aperiodicaverages using a spatial integral evaluated over the caustic’s boundary. We weighthe integrand by a the elliptic billiard universal measure [2, Section 51], [11, 10, 8]which yields the aperiodic density of rays hitting a particular point on the billiardboundary.Referring to Table 1, we will examine one classic (perimeter) and two “new”invariants (sum of cosines and product of exterior cosines). We will compare theiraverages (average chord length, average cosine, and geometric mean of exteriorcosines) within the continuum of aperiodic trajectories.
Date : January 2020. A billiard N-periodic is fully specified by
L, J , so any “new” invariants are ultimately dependenton them. a r X i v : . [ m a t h . D S ] F e b JAIR KOILLER, DAN REZNIK, AND RONALDO GARCIA P P P P P P' P' P' P' P' N = = = = = P Figure 1.
Left:
A 5-periodic trajectory (blue) in the elliptic billiard (black), whose vertices arebisected by the ellipse normals (black arrows). The Poncelet family remains tangent to a confocalelliptic caustic (brown). A second, same perimeter 5-periodic is also shown (dashed blue). Theouter polygon (green) has sides tangent to the elliptic billiard at the vertices of the N-periodic.
Right: discrete confocal caustics (brown) associated with N-periodics, N=3,...,7. Show are sample3-, 4-, and 5-periodics (dashed red, dashed green, solid blue, respectively) sharing one commonvertex P . Figure 2.
Two regimes of aperiodic, space-filling trajectories in an elliptic billiard, reproducedfrom [12].
Left: initial ray P P does not pass between the foci, confocal caustic is ellipse. Left: initial ray passes between the foci, confocal caustic is hyperbola. invariant formula from averageL elliptic functions classic
L/N (cid:80) cos θ i LJ − N [12] ( LJ ) /N − (cid:81) cos θ (cid:48) i ? [12] geometric mean Table 1.
Three N-periodic invariants whose averages are compared to those displayed by theiraperiodic counterparts. A “?” means no closed expression has yet been derived.
Article Structure.
In Section 2 we review preliminary concepts and definitions.In Sections 3 to 5 we calculate, via spatial integrals, the (i) average sidelength (i.e.,average perimeter), (ii) average cosine, and (iii) geometric mean of outer cosines.We then compare them with the values predicted by either closed form or numericcomputation of the original quantities in the N-periodic case, showing that theylie at consistent locations within the continuum of confocal caustics. Unansweredquestions and/or future work appear in Section 6.
VERAGE ELLIPTIC BILLIARD INVARIANTS WITH SPATIAL INTEGRALS 3 P P P c ,1 P c ,2 θ θ P c P ' θ ' Figure 3.
The tangent to a chosen point P c on the caustic intersects the elliptic billiard at P and P . Let θ (resp. θ ) be the angle between segment P P and the next P c, (resp. previous P c, ) tangent to the caustic (dashed blue). The outer angle θ (cid:48) associated with P c is measured atthe intersection P (cid:48) of the tangents to the billiard (dashed green) at P and P . Preliminaries
Let ( E , E c ) denote the outer and inner ellipses in the confocal pair given by: E : x a + y b = 1 , E c : x a c + y b c = 1 Let c = a − b = a c − b c . Let A = diag [1 /a , /b ] . Joachimsthal’s constant ata point δ on E is given by (cid:104)A .P, v (cid:105) , where v is the unit incoming vector [14]. J isalso given by: J = √ λab where λ = a − a c = b − b c .Referring to Fig. 3, let P c be a point on the caustic, and P , P be the intersectionsof the tangent through P c with the outer ellipse. These are given by: P = ( x , y ) = 1 ψ (cid:2) a c a ( ab c x c − ζby c ) , b c b ( ba c y c + ζax c ) (cid:3) (1) P = ( x , y = 1 ψ (cid:2) a c a ( ab c x c + ζby c ) , b c b ( ba c y c − ζax c ) (cid:3) (2) ζ = (cid:112) a − a c (cid:112) b c x c + a c y c ψ = a b c x c + b a c y c Given a confocal caustic (say parametrized by its minor axis b c ) let P i be thevertices of an associated aperiodic trajectory. i = 1 , ..., ∞ . The asymptotic average g of some vertex-evaluated quantity g ( P i ) is given by: JAIR KOILLER, DAN REZNIK, AND RONALDO GARCIA (3) g = lim N →∞ N k (cid:88) i =1 g ( P i ) One can evaluate g ( s ) , for example at either intersection P or P in Figure 3.The billiard map is an involution of the pair ( P, P c ) of a point on E and E c respectively to new points There is change of variables s → x which linearizes thebilliard map, x → x + τ [5, Chapter 13], [9, 6, 15].Let ρ be the density of an invariant measure normalized such that (cid:72) ρ ( x ) ds = 1 .This can be regarded as the density of rays associated with x . The following universalmeasure has been derived for the elliptic billiard, independent of τ :(4) dx = κ / c ds The above allows us to replace (3) with the following spatial integral:(5) g = 1 (cid:72) κ / c ds (cid:73) g ( s ) κ / c ds. Auxiliary expressions.
Arc length and curvature along the caustic ellipse aregiven by: ds = (cid:0) a c sin u + b c cos u (cid:1) / duκ c =( a c b c ) / (cid:0) a c sin u + b c cos u (cid:1) − / so that: ρ = dx = κ / c ds =( a c b c ) / (cid:0) a c sin u + b c cos u (cid:1) − / = ( a c b c ) / (cid:112) a c y /b c + b c x /a c = ( a − λ ) ( b − λ ) (cid:112) a − λ − ( a − b ) cos u Below we will be also expressing certain average quantities in terms of the followingJacobi elliptic functions of the first and third kind, respectively [7, Introduction]: K ( m ) = (cid:90) π dα (cid:112) − m sin α Π( n, m ) = (cid:90) π dα (1 − n sin α ) (cid:112) − m sin α , Average Sidelength
We constuct a spatial integral to compute L , the average sidelength in an aperiodictrajectory and compare it with L/N for an N-periodic.The distance between two consecutive points P and P of a billiard orbitparametrized by the point P c = [ x c , y c ] in the confocal caustic is given by: VERAGE ELLIPTIC BILLIARD INVARIANTS WITH SPATIAL INTEGRALS 5 l = 2 ab (cid:112) a b c x c + a c b y c − a c b c (cid:112) a c y c + b c x c a b c x c + a c b y c Therefore, P c = [ √ a − λ cos u, √ b − λ sin u ] leads to l ( u ) = 2 ab √ λ c cos u − ( a − λ ) λc cos u − ( a − λ ) b l ( u ) = l κ / c ds = c √ a − λ − c cos ub ( a − λ ) − λc cos uc = 2 ab √ λ (cid:112) a − λ (cid:112) b − λ (6) L = 1 (cid:72) κ / c ds (cid:90) π l ( u ) du In terms of the elliptic integrals K and Π we have that: (cid:90) κ c ds = (cid:90) π ( a − λ ) ( b − λ ) √ s √ − s cos u du = 4( a − λ ) ( b − λ ) √ s K ( √ s ) (cid:90) π l ( u ) du = c b (2 s − s ) Π (cid:0) s , √ s (cid:1) + 2 s K (cid:0) √ s (cid:1) s s = c a − λ , s = λs b Therefore, L = 2 a √ λ K (cid:0) √ s (cid:1) (cid:18)(cid:0) − b + λ (cid:1) Π (cid:18) λ s b , √ s (cid:19) + K ( √ s ) b (cid:19) Numerical results are shown in Figure 4 for three different billiard aspect ratios.Notice points on each curve report the average perimeters
L/N obtained withN-periodics at the required caustic parameters λ .4. Average Cosine
We evaluate the average cosine C for aperiodics with a spatial integral.Using the Joachimstall invariant we obtain:(7) cos θ = J a b a y + b x ) − λa b a y + b x ) − λ d d − , d = | P − F | , d = | P − F | Let cos θ ( u ) = (cos θ ( u ) + cos θ ( u )) / . Let a c = √ a − λ , b c = √ b − λ and ( x c , y c ) = ( a c cos u, b c sin u ) .Using (1) and (2) it follows that: JAIR KOILLER, DAN REZNIK, AND RONALDO GARCIA N = = = = = = = = = = = = = = = = = = - λ L a v g ● a = ■ a = ◆ a = Figure 4.
The value of the average sidelengths vs − λ , b = 1 , and three values of a . The dotsshow agreement of the value with L/N for various non-intersecting N-periodics. When − λ iszero, the average perimeter tends to a . cos θ ( u ) = r + r cos ur + r cos u = r r − s cos u − s cos ur = ( a − λ )( a b − c λ )(2 a b − a λ + b λ ) r = − c ( a b − ( a + b ) λ )(2 a b − a λ − b λ + λ ) r = − a − λ )( a b − c λ ) r = (2 c ( a b − ( a + b ) λ ) s = − r r , s = − r r Substituting cos θ above for g in (5) and obtain the spatial integral for the averagecosine. Therefore it follows that(8) (cid:90) dx = (cid:90) π κ c ds = (cid:90) π ( a − λ ) ( b − λ ) √ s √ − s cos u du (cid:90) κ c cos θds = (cid:90) π κ c cos θdu = s (cid:90) π − s cos u (1 − s cos u ) √ − s cos u dus = a − b a − λ , s = r r √ s ( a − λ ) ( b − λ ) In terms of the elliptic integrals K and Π it follows that: C = r r ( s − s ) Π (cid:0) s , √ s (cid:1) + s K (cid:0) √ s (cid:1) s K (cid:0) √ s (cid:1) In [12, 1, 3] the following expression was presented for the invariant sum of cosinesin N-periodics:
VERAGE ELLIPTIC BILLIARD INVARIANTS WITH SPATIAL INTEGRALS 7 N = = = = = = = = = = = = = = = = = = - - - λ a v g ∑ c o s ● a = ■ a = ◆ a = Figure 5.
The average cosine vs − λ , b = 1 for three values of a , with b = 1 . The dots showagreement of the value with JL/N − for various non-intersecting N-periodics. When − λ is one(resp. zero), the average cosine tends to (resp. − ). (cid:88) cos θ i = JL − N Therefore the average cosine for N-periodics is simply
JL/N − . Figure 5shows results obtained with spatial integration, and that they agree with the valuespredicted for N-periodics at the appropriate locations. Sum of curvatures to two-thirds.
In [13] we show conservation of (cid:80) κ / i is a corollaryto the sum of cosines, where κ i denotes the curvature of the outer ellipse at the ithvertex. One can express κ / as a linear function of cos θ : κ / = ( ab ) − (cid:18) x a + y b (cid:19) − = ( ab ) d d = 4( ab ) − |∇ f | = ( ab ) − (cid:18) cosθ J (cid:19) − Therefore, the sum of κ / is also invariant and its average value will be given by: κ / = 1 (cid:72) κ / c ds (cid:73) κ / ( s ) κ / c ds. Geometric Mean of Outer Cosines
Referring to Figure 1, let θ (cid:48) i denote the ith internal angle of the outer polygonwhose sides are tangent to the elliptic billiard at the vertices of an N-periodics. Theproduct of θ (cid:48) i is invariant over N-periodics, for all N [1, 3]. The geometric mean C (cid:48) of θ (cid:48) i is given asymptotically by: C (cid:48) = lim k →∞ (cid:32) k (cid:89) i =1 cos θ (cid:48) i (cid:33) /k To work with spatial integrals we must first convert the above to a sum:
JAIR KOILLER, DAN REZNIK, AND RONALDO GARCIA - - - - b c = - λ a / b = N = = = = = Arithm Mean ∑ Cos ( θ ) Geom Mean ∏ | Cos ( θ ' )| Geom Mean ∏ | Cos ( θ ' )| reflected Figure 6.
Average cosines (red) and geometric mean of outer cosines (green) vs. b c , the minorsemiaxes of the caustic. Here a = 5 , b = 1 . Blue dashed vertical lines mark the b c for non-intersecting orbits. Dashed green: past the N = 4 caustic, the latter, the latter is reflected aboutthe x axis showing proximity to the average cosine. log C (cid:48) = lim k →∞ (1 /k ) k (cid:88) i =1 log | cos θ (cid:48) i | As before, replace the above time average by the following spatial integral: log C (cid:48) = 1 (cid:72) κ c ( s ) / ds (cid:73) log | cos θ (cid:48) ( s ) | κ / c ds A quick look on a picture shows that in order to compute cos θ (cid:48) it suffices tomake the scalar product of the normalized gradients at the points P , P .(9) cos θ (cid:48) = x x /a + y y /b ( x /a + y /b ) / ( x /a + y /b ) / cos θ (cid:48) = − c a (cid:113) − a c cos u + ( a − λ ) (cid:113) c c a cos u − (2 b λ + c a ) ( a − λ ) c a = a b − λ ( a + b ) , sign (cos θ (cid:48) ) = − sign ( c a ) Numerical results for both average cosines and geometric mean of outer cosines areshown in Figure 6 for a = 5 (smaller a make the two spatial averages become to closeto each other). For values of b c where the trajectory is periodic, results obtained withspatial averages perfectly match numerically-estimated discrete averages computednumerically with N-periodics. 6. Questions
The following questions are still unanswered: • Why is the geometric mean of outer aperiodic cosines so close to the averageaperiodic cosines?
VERAGE ELLIPTIC BILLIARD INVARIANTS WITH SPATIAL INTEGRALS 9 • Is there a universal measure expressed in terms of the outer ellipse? • Can we use this framework to estimate aperiodic averages for cases wherethe caustic is a hyperbola? • A third invariant introduced in [12] was the ratio of outer-to-orbit areas.These do not seem amenable to a discrete sum of individual quantities.Would there be counterpart be for aperiodic areal averages?
Acknowledgments
We would like to thank Sergei Tabachnikov and Arseniy Akopyan, and HellmuthStachel for invaluable discussions.The second author is fellow of CNPq and coordinator of Project PRONEX/CNPq/ FAPEG 2017 10 26 7000 508.
References [1] Akopyan, A., Schwartz, R., Tabachnikov, S. (2020). Billiards in ellipses revisited.
Eur. J.Math. doi.org/10.1007/s40879-020-00426-9 . 1, 6, 7[2] Arnold, V. (1978).
Mathematical Methods of Classical Mechanics . 60. Springer GraduateTexts in Mathematics. 1[3] Bialy, M., Tabachnikov, S. (2020). Dan Reznik’s identities and more.
Eur. J. Math. doi.org/10.1007/s40879-020-00428-7 . 1, 6, 7[4] Chavez-Caliz, A. (2020). More about areas and centers of Poncelet polygons.
Arnold Math J. doi.org/10.1007/s40598-020-00154-8 . 1[5] Flatto, L. (2009).
Poncelet’s theorem . Providence, RI: American Mathematical Society. 4[6] Glutsyuk, A. (2019). On curves with poritsky property. arXiv:1901.01881. 4[7] Gradshteyn, I. S., Ryzhik, I. M. (1965).
Table of integrals, series, and products . AcademicPress, New York-London, 4th ed. 4[8] Jovanović, B. (2011). What are completely integrable hamilton systems.
The Teaching ofMathematics , 13(1): 1–14. http://elib.mi.sanu.ac.rs/files/journals/tm/26/tm1411.pdf .1[9] Kolodziej, R. (1985). The rotation number of some transformation related to billiards in anellipse.
Studia Math. , 81(3): 293–302. 4[10] Lazutkin, V. F. (1973). The existence of caustics for a billiard problem in a convex domain.
Izv. Akad. Nauk SSSR Ser. Mat. , 3(1): 186–216. 1[11] Poritsky, H. (1950). The billiard ball problem on a table with a convex boundary – anillustrative dynamical problem.
Ann. of Math. , 2(51): 446–470. 1[12] Reznik, D., Garcia, R., Koiller, J. (2020). Can the elliptic billiard still surprise us?
MathIntelligencer , 42: 6–17. rdcu.be/b2cg1 . 1, 2, 6, 9[13] Reznik, D., Garcia, R., Koiller, J. (2021). Fifty new invariants of n-periodics in the ellipticbilliard.
Arnold Math. J.
Growing list in arXiv:2004.12497. 1, 7[14] Tabachnikov, S. (2005).
Geometry and Billiards , vol. 30 of
Student Mathematical Li-brary . Providence, RI: American Mathematical Society. . Mathematics Advanced Study Semesters, University Park, PA. 1, 3[15] Zhang, J. (2017). Suspension of the billiard maps in the lazutkin’s coordinate.
Discrete andContinuous Dynamical Systems .