Oval Billiards on Surfaces of Constant Curvature

Transcription

1 Oval Billiards on Surfaces of Constant Curvature Luciano Coutinho dos Santos Sônia Pinto-de-Carvalho Abstract arxiv: v1 [math.ds] 2 Nov 2014 In this paper we prove that the billiard problem on surfaces of constant curvature defines a 2-dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary curve is a reular, simple, closed, strictly eodesically convex curve and at least C 2 curve. 1 Introduction The plane billiard problem, oriinally defined by Birkhoff [3] in the beinnin of the XX century, consists in the free motion of a point particle in a bounded plane reion, reflectin elastically when it reachs the boundary. In this work we extend this problem to bounded reions contained on eodesically convex subsets of surfaces of constant curvature. We will show that this new billiard defines a 2- dimensional conservative and reversible dynamical system, defined by a Twist diffeomorphism, if the boundary curve is an oval, i.e., a reular, simple, closed, strictly eodesically convex curve and at least C 2 curve. This is a classical result for the Euclidean case and is proved for instance in [13]. Nevertheless, and for the sake of completeness, we will present the proof for the three cases. Billiards on the Euclidean plane were, and still are, extensively studied. Billiards on surfaces of constant curvature are much less studied and the papers focus on special properties. For instance, Veselov [16], Bolotin [5], Draovíc, Jovanovíc and Radnovíc [7], Popov and Topalov [14], [15] and Bialy [2] deal with the question of interability. B.Gutkin, Smilansky and E.Gutkin [9] looked at hyperbolic billiards on the sphere and the hyperbolic plane. E.Gutkin and Tabachnikov [10] studied eodesic polyonal billiards. Amon them, only Bialy [2] looked more closely to oval billiards, but he starts from the variational formulation of the billiard problem, takin as ranted that the eodesic distance defines a eneratin function, which is not our case here. 2 Surfaces of constant curvature For the definition of the billiard, only the behaviour of the eodesics and the measure of anles will count. So, we can take as model one of the three surfaces: the Euclidean plane E 2, iven in R 3 by {z = 1} or X (ρ, θ) = (ρ cos θ, ρ sin θ, 1), ρ 0, 0 θ < 2π; CEFET-MG, Brazil, astrofisico2@yahoo.com.br UFMG,Brazil, sonia@mat.ufm.br 1

2 an open hemisphere of the unit sphere S 2 + iven in R 3 by < (x, y, z), (x, y, z) >= 1, z > 0 where <, > is the usual inner product on R 3 or by X (ρ, θ) = (sin ρ cos θ, sin ρ sin θ, cos ρ), 0 < ρ < π, 0 θ < 2π; the upper sheet of the hyperbolic plane H 2 + iven in R 3 by << (x, y, z), (x, y, z) >>= 1, z > 0 where <<, >> is the inner product on H 2 or by X (ρ, θ) = (sinh ρ cos θ, sinh ρ sin θ, cosh ρ), ρ 0, 0 θ < 2π which will be called by S. The eodesics on S are the intersections of the surface with the planes passin by the oriin. S is eodesically convex. Then iven a point X S and unitary vector v T X S there is a unique minimisin eodesic by X on the direction of v which is iven by γ(t) = X + tv if S = E 2 X cos t + v sin t if S = S 2 + X cosh t + v sinh t if S = H 2 + The distance between two points X and Y on S is so measured by < X Y, X Y > if X, Y E 2 d S (X, Y ) = arccos(< X, Y >) if X, Y S 2 + arccosh( << X, Y >>) if X, Y H 2 + Definition 1. A reular curve Γ(t) S is said to be eodesically strictly convex if the intersection of any eodesic tanent to Γ with the curve Γ has only one point. It is proved on [12] for the spherical case, on [6] for the hyperbolic case and for instance on [1] for the Euclidean case that Lemma 1. If a curve Γ S is closed, reular, simple, C n, n 2 and has strictly positive eodesic curvature then Γ is eodesically strictly convex. So, by lemma 1, any oval is eodesically strictly convex. This property, toether with the fact that S is eodesically convex will enable us to define oval billiard maps on surfaces of constant curvature. 3 Billards on ovals Let Ω S be a reion bounded by an oval Γ. Analoously to the plane case, we can define the billiard on Γ as the free motion of a point particle inside Ω, reflectin elastically at the impacts with Γ. Since the motion is free, the particle moves alon a eodesic of S while it stays inside Ω and reflects makin equal anles with the tanent at the impact points with Γ. The trajectory of the particle is a eodesic polyonal curve, with vertices at the impact points. 2

3 As Ω is a bounded subset of a eodesically convex surface, with strictly eodesically convex boundary, the motion is completely determined by the impact point and the direction of movement immediately after each reflection. So, a parameter which locates the point of impact, and the anle between the direction of motion and the tanent to the boundary at the impact point, may be used to describe the system. Let l be the lenth of Γ, s the arclenth parameter for Γ and ψ (0, π) be anle that measures the direction of motion at the impact point. Let C be the cylinder R/l (0, π). We can define the billiard map T : C C (s 0, ψ 0 ) (s 1, ψ 1 ) which associates to each impact point and direction of motion the next impact and direction. This billiard map defines a 2-dimensional dynamical system and the orbit of any initial point (s 0, ψ 0 ) is the set O(s 0, ψ 0 ) = {T i ((s 0, ψ 0 ) = (s i, ψ i ), i Z}. 3.1 Properties of the eneratin function Let S be E 2, S 2 + or H 2 + and d S be the eodesic distance on S. Let Γ S be an oval, parameterized by the arclenth parameter s and Ω be the reion bounded by Γ. Lemma 2. Let T (s 0, ψ 0 ) = (s 1, ψ 1 ) be the billiard map on Γ and (s 0, s 1 ) = d S (Γ(s 0 ), Γ(s 1 )). Then verifies (s 0, s 1 ) = cos ψ 0 and (s 0, s 1 ) = cos ψ 1 Proof: Let τ i, i = 0 or 1, be the unitary tanent vector to the oriented eodesic joinin Γ(s 0 ) to Γ(s 1 ), at Γ(s i ). When S = E 2 we have that 2 (s 0, s 1 ) =< Γ(s 1 ) Γ(s 0 ), Γ(s 1 ) Γ(s 0 ) > and then (s 0, s 1 ) = < Γ (s 0 ), Γ(s 1) Γ(s 0 ) ((s 0 ), (s 1 )) >=< Γ (s 0 ), τ 0 >= cos ψ 0. (1) Analoously, (s 0, s 1 ) =< Γ (s 1 ), τ 1 >= cos ψ 1. When S = S 2 + we have that cos (s 0, s 1 ) =< Γ(s 0 ), Γ(s 1 ) > and then (s 0, s 1 ) = < Γ (s 0 ), Γ(s 1 ) > (s 0, s 1 ) = < Γ (s 0 ), cos (s 0, s 1 ) Γ(s 0 ) (s 0, s 1 ) τ 0 > (s 0, s 1 ) = < Γ (s 0 ), τ 0 >= cos ψ 0. (2) Analoously (s 0, s 1 ) =< Γ (s 1 ), τ 1 >= cos ψ 1. When S = H 2 + we have that cosh (s 0, s 1 ) = Γ(s 1 ), Γ(s 2 ) and then (s 0, s 1 ) = Γ (s 0 ), Γ(s 1 ) sinh (s 0, s 1 ) = Γ (s 0 ), cosh (s 0, s 1 ) Γ(s 0 ) sinh (s 0, s 1 ) τ 0 sinh (s 0, s 1 ) = Γ (s 0 ), τ 0 = cos ψ 0. (3) 3

4 Analoously (s 0, s 1 ) = Γ (s 1 ), τ 1 = cos ψ 1. Let p = cos ψ ( 1, 1). Lemma 2 implies that the arclenth s and the tanent momentum p are conjuated coordinates with eneratin function for the billiard map, or, T (s 0, p 0 ) = (s 1, p 1 ) = p 0, leadin to the variational definition of billiards. (s s 2 0, s 1 ) = i = p 1 Lemma 3. Let k i be the eodesic curvature of Γ at s i, i = 0, 1. The second derivatives of are (s 0, s 1 ) = sin 2 ψ i (s 0,s 1 ) i sin ψ i in E 2 sin 2 ψ i tan (s 0,s 1 ) i sin ψ i in S 2 + sin 2 ψ i tanh (s 0,s 1 ) i sin ψ i in H 2 + sin ψ 0 sin ψ 1 (s 0,s 1 ) in E 2 sin ψ 0 sin ψ 1 (s 0,s 1 ) in S 2 + sin ψ 0 sin ψ 1 sinh (s 0,s 1 ) in H 2 + Proof: Let τ i, η i and ν i be the unitary tanent, normal and binormal vectors, respectively, to the oriented eodesic joinin Γ(s 0 ) to Γ(s 1 ), at Γ(s i ), seen as a curve in R 3. When S = S 2 + or H 2 +, since the eodesic is contained on a plane passin by the oriin, η i = Γ(s i ), ν i is a constant unitary vector, normal to the plane, and {τ i, ν i } is an orthonormal basis for T Γ(si )S. In the case S = E 2 we differentiate equation (1) obtainin s 2 0 = 1 cos2 ψ 0 + < Γ (s 0 ), Γ(s 1) Γ(s 0 ) = sin2 ψ 0 + < k 0 sin ψ 0 τ 0 + k 0 cos ψ 0 η 0, τ 0 >= sin2 ψ 0 k 0 sin ψ 0 and = < Γ (s 0 ), Γ (s 1 ) > + cos ψ 0 cos ψ 1 = < cos ψ 0τ 0 sin ψ 0 η 0, cos ψ 1 τ 1 + sin ψ 1 η 1 > + cos ψ 0 cos ψ 1 When S = S 2 + we differentiate (2) ettin and s 2 0 = < Γ (s 0), Γ(s 1 ) > cos 2 ψ 0 cos > = sin ψ 0 sin ψ 1 = < Γ(s 0) + k 0 Γ(s 0 ) Γ (s 0 ), cos Γ(s 0 ) + τ 0 > cos 2 ψ 0 cos = cos k 0 sin ψ 0 cos 2 ψ 0 cos = cos sin2 ψ 0 = < Γ (s 0 ), Γ (s 1 ) > + cos ψ 0 cos ψ 1 cos k 0 sin ψ 0 = < cos ψ 0τ 0 sin ψ 0 ν 0, cos ψ 1 τ 1 + sin ψ 1 ν 1 > + cos ψ 0 cos ψ 1 cos = sin ψ 0 sin ψ 1 4

5 The last case is analoous. When S = H 2 + we differentiate (3) ettin = Γ (s 0 ), Γ (s 1 ) + cos ψ 0 cos ψ 1 cosh sinh = Γ(s 0) + k 0 Γ(s 0 ) Γ (s 0 ), cosh Γ(s 0 ) + sinh τ 0 cos 2 ψ 0 cosh sinh = cosh sin2 ψ 0 sinh k 0 sin ψ 0 and = Γ (s 0 ), Γ (s 1 ) + cos ψ 0 cos ψ 1 cosh sinh = cos ψ 0τ 0 sin ψ 0 ν 0, cos ψ 1 τ 1 + sin ψ 1 ν 1 + cos ψ 0 cos ψ 1 cosh sinh = sin ψ 0 sin ψ 1 sinh Finally, the calculation of 2 s 2 1 is analoous to 2. s Properties of the billiard map In this subsection we will prove that Theorem 1. Let S be E 2, S 2 + or H 2 +. Let Γ S be an oval, a closed, simple, reular C n curve, n 2, with strictly positive eodesic curvature. Then the billiard map T on Γ is a reversible, conservative, Twist, C n 1 -diffeomorphism. The proof will follow directly from the lemmas bellow. As above, s stands for the arclenth parameter for Γ, k i is the eodesic curvature of Γ at s i, d S is the distance on S and (s 0, s 1 ) = d S (Γ(s 0 ), Γ(s 1 )). Lemma 4. T is invertible and reversible. Proof: Any trajectory of the billiard problem can be travelled in both senses. So, if T (s i, ψ i ) = (s i+1, ψ i+1 ) then T 1 (s i, π ψ i ) = (s i 1, π ψ i 1 ). Let I be the involution on C iven by I(s, ψ) = (s, π ψ). Clearly I 1 = I. We have then that T 1 = I T I or I T 1 = T I, i.e., T is reversible. Lemma 5. T is a C n 1 diffeomorphism. Proof: Let T (s 0, ψ 0 ) = (s 1, ψ 1 ) and V 0 and V 1 be two disjoint open intervals containin s 0 and s 1 respectively. We define G : V 0 V 1 (0, π) R, G(s 0, s 1, ψ 0 ) = (s 0, s 1 ) cos ψ 0. G is a C n 1 function, since Γ and are C n. Then, by lemma 2, G(s 0, s 1, ψ 0 ) = 0 and G (s 0, s 1 ) = 2 (s 0, s 1 ) 0 by lemma 3, since ψ 0, ψ 1 (0, π). So we can locally define a C n 1 function s 1 = s 1 (s 0, ψ 0 ) such that G(s 0, s 1 (s 0, ψ 0 ), ψ 0 ) = 0. 5

6 Takin now ψ 1 (s 0, ψ 0 ) = arccos( (s 0, s 1 (s 0, ψ 0 ))) we conclude that T (s 0, ψ 0 ) = (s 1 (s 0, ψ 0 ), ψ 1 (s 0, ψ 0 )) is a C n 1 function. As T is invertible, with T 1 = I T I, we conclude that T is a C n 1 diffeomorphism. Differentiatin the expressions we obtain Lemma 6. Lemma 7. T is a Twist map. cos ψ 0 = (s 0, s 1 (s 0, ψ 0 )) cos ψ 1 (s 0, ψ 0 ) = (s 0, s 1 (s 0, ψ 0 )) s = 0 ψ 0 = sin ψ 0 (4) + 2 s 2 1 = sin ψ 1 = sin ψ s ψ 0 ψ 0 Proof: By equation (4) toether with lemma 3 and rememberin that ψ (0, π) we have that ψ 0 0 and T has the Twist property. Usin the formulas of lemmas 6 and 3 we obtain the derivative of the billiard map as: ) Lemma 8. DT (s 0, ψ 0 ) = ( s1 ψ 0 ψ 0 where in E 2 = k 0 sin ψ 0 sin ψ 1 = = k 0 sin ψ 1 + k 1 sin ψ 0 k 1 k 0 ψ 0 = sin ψ 1 k 1 sin ψ 1 sin ψ 1 in S 2 + = k 0 sin ψ 0 cos sin ψ 1 = = k 0 sin ψ 1 cos + sin ψ 0 sin ψ 1 k 0 k 1 + k 1 sin ψ 0 cos sin ψ 1 = sin ψ 1 cos k 1 6

7 in H 2 + = k 0 sinh sin ψ 0 cosh sin ψ 1 = sinh = k 0 sin ψ 1 cosh + sin ψ 0 sin ψ 1 sinh k 0 k 1 sinh + k 1 sin ψ 0 cosh sin ψ 1 = sin ψ 1 cosh k 1 sinh Calculatin now the determinant of DT we prove that Lemma 9. T preserves the measure dµ = sin ψdsdψ. Acknowledements We thank the Brazilian aencies FAPEMIG, CAPES and CNPq for financial support. References [1] Araújo, P. V.: Geometria Diferencial, SBM, [2] Bialy, M.: Hopf Riidity for convex billiards on the hemisphere and hyperbolic plane. DCDS, 2013, 33, [3] Birkhoff, G.D.: Dynamical Systems. Providence, RI: A. M. S. Colloquium Publications 1966 (Oriinal ed. 1927). [4] Blumen, V., KIM, K. Y., Nance, J., Zarnitsky, V.: Three-period orbits in billiards on the surfaces of constant curvature, Int.Math.Res.Not., 2012, 21, [5] Bolotin, S.V.: Interable Billiards on surfaces of constant curvature, Math. Notes, 1992, 51/2, [6] Brickell F., Hsiun C.C.: The total absolute curvature of closed curves in Riemannian manifold, J. Diff. Geom., 1974, 9, [7] Draovíc, V., Jovanovíc, B., Radnovíc, M.: On elliptical billiards in the Lobachevsky space and associated eodesic hierarchies, J.Geom.Phys, 2003, 47, [8] do Carmo, M. P.: Geometria diferencial de curvas e superfícies, SBM, [9] Gutkin, B., Smilansky, U., Gutkin, E.: Hyperbolic Billiards on surfaces of constant curvature. Comm. Math. Phys.,1999, 208, [10] Gutkin,B., Tabachnikov, S.: Complexity of piecewise convex transformations in two dimensions, with applications to polyonal billiards on surfaces of constant curvature, Mosc. Math. J., 2006, 6/4, [11] Kozlov, V.V., Treschev, D.V.: Billiards A Genetic Introduction to the Dynamics of Systems with Impacts. AMS,

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