Stability of periodic orbits in no-slip billiards
SStability of periodic orbits in no-slip billiards
C. Cox ∗ , R. Feres ∗ , H.-K. Zhang † November 16, 2018
Abstract
Rigid bodies collision maps in dimension two, under a natural set of physicalrequirements, can be classified into two types: the standard specular reflection mapand a second which we call, after Broomhead and Gutkin, no-slip.
This leads tothe study of no-slip billiards —planar billiard systems in which the moving particleis a disc (with rotationally symmetric mass distribution) whose translational androtational velocities can both change at each collision with the boundary of thebilliard domain.In this paper we greatly extend previous results on boundedness of orbits (Broom-head and Gutkin) and linear stability of periodic orbits for a Sinai-type billiard(Wojtkowski) for no-slip billiards. We show among other facts that: (i) for billiarddomains in the plane having piecewise smooth boundary and at least one corner ofinner angle less than π , no-slip billiard dynamics will always contain elliptic period- orbits; (ii) polygonal no-slip billiards always admit small invariant open sets andthus cannot be ergodic with respect to the canonical invariant billiard measure; (iii)the no-slip version of a Sinai billiard must contain linearly stable periodic orbits ofperiod and, more generally, we provide a curvature threshold at which a commonlyoccurring period- orbit shifts from being hyperbolic to being elliptic; (iv) finally,we make a number of observations concerning periodic orbits in a class of polygonalbilliards. No-slip billiard systems have received so far very little attention despite some interestingfeatures that distinguish them from the much more widely studied standard billiards.These non-standard types of billiards are discrete time systems in dimension (aftertaking the natural Poincaré section of a flow) representing a rotating disc with unit kineticenergy that moves freely in a billiard domain with piecewise smooth boundary. Althoughnot Hamiltonian, these systems are nevertheless time reversible and leave invariant thecanonical billiard measure. They also exhibit dynamical behavior that is in sharp contrastwith standard billiards. For example, they very often contain period- orbits having smallelliptic islands. These regions exist amid chaos that appears, in numerical experiments,to result from the usual mechanisms of dispersing and focusing. ∗ Department of Mathematics, Washington University, Campus Box 1146, St. Louis, MO 63130 † Department of Mathematics, University of Massachusetts in Amherst a r X i v : . [ m a t h . D S ] D ec e are aware of only two articles on this subject prior to our [7, 8]: one by Broomheadand Gutkin [2] showing that no-slip billiard orbits in an infinite strip are bounded, andanother by Wojtkowski, characterizing linear stability for a special type of period- orbit.In this paper we extend their results as will be detailed shortly, and develop the basictheory of no-slip billiards in a more systematic way. In this section we explain theorganization of the paper and highlight our main new results.Section 2 gives preliminary information and sets notation and terminology concerningrigid collisions. It specializes the general results from [7] (stated in that paper in arbitrarydimension for bodies of general shapes and mass distributions) to discs in the planewith rotationally symmetric mass distributions. The main fact is briefly summarizedin Proposition 2.1. Although the classification into specular and no-slip collisions isthe same as in [2], our approach is more differential geometric in style and may havesome conceptual advantages. For example, we derive this classification (in [7]) from anorthogonal decomposition of the tangent bundle T M restricted to the boundary ∂M (orthogonal relative to the kinetic energy Riemannian metric in the system’s configurationmanifold M ) into physically meaningful subbundles. This orthogonal decomposition isexplained here only for discs in the plane.By a planar no-slip billiard system we mean a mechanical system in R in which oneof the colliding bodies, which may have arbitrary shape, is fixed in place, whereas thesecond, moving body is a disc with rotationally symmetric mass distribution; post-collisionvelocities (translational and rotational) are determined from pre-collision velocities via theno-slip collision map and between consecutive collisions the bodies undergo free motion.Contrary to the standard case, the moving particle’s mass distribution influences thecollision properties (via an angle parameter which is denoted β throughout the paper).The main definitions and notations concerning no-slip billiard systems, in particular thenotions of reduced phase space , velocity phase space , and the product, eigen- and wavefront frames, are introduced in Section 3.A special feature of no-slip billiards around which much of the present study is based,is the ubiquitous occurrence of period- trajectories. The general description of thesetrajectories is given in Section 4. In Section 5 we obtain the differential of the no-slipbilliard map and show the form it takes for period- trajectories.For general collision systems as considered in [7] (satisfying energy and momentaconservation, time reversibility, involving impulsive forces that can only act at the contactpoint between colliding bodies), the issue of characterizing smooth invariant measures stillneeds much further investigation, although it is shown there that the canonical (Liouville)measure is invariant if a certain field of subspaces defined in terms of the collision mapsat each collision configuration q ∈ ∂M is parallel with respect to the kinetic energy metric.This is the case for planar no-slip billiards, so that the standard billiard measure is stillinvariant. (Note, however, that the configuration manifold is now -dimensional.) Adetailed proof of this fact, in addition to comments on time reversibility are given inSection 6.Dynamics proper begins with Sections 7 and 8, which are concerned with no-slip billiardsystems in wedge-shaped regions and polygons. Here we generalize the main result from[2]. In that paper it is shown that orbits of the no-slip billiard system in an infinite stripdomain are bounded. By extending and refining this fact to wedge regions we obtain2ocal stability for periodic orbits in no-slip polygonal billiards. This is Theorem 8.1. Wealso give in Section 8 an exhaustive description of periodic orbits in wedge billiards.Finally, in Section 9 we consider linear stability of period- orbits in the presence ofcurvature. Our results here extend those of [16] for no-slip Sinai billiards. Wojtkowskimakes in [16] the following striking observation: for a special period- orbit in a Sinaibilliard (corresponding in our study to angle φ = ) there is a parameter defined in terms ofthe curvature of the circular scatterer that sets a threshold between elliptic and hyperbolicbehavior. This is based on an analysis of the differential of the billiard map at the periodictrajectory. Here we derive similar results for period- orbits in general. Although theanalysis is purely linear, we observe the occurrence of elliptic islands and stable behaviorin systems that are the no-slip counterpart of fully chaotic standard billiards. The fasttransition between stability and chaos near the threshold set by the curvature parameterobtained from the linear analysis is also very apparent numerically. As already observedby Wojtkowski in [16], proving local stability in the presence of curvature would require adifficult KAM analysis (for reversible, but not Hamiltonian systems; cf. [14]). For a much more general treatment of the material of this section (not restricted todiscs and valid in arbitrary dimension) see [7]. See also [2, 16]. Let x = ( x , x ) be thestandard coordinates in R . A mass distribution on a body B ⊂ R is defined by a finitepositive measure µ mass on B ⊂ R . We assume without loss of generality that the firstmoments ∫ B x i dµ mass ( x ) vanish. Let m be the total mass: m = µ mass ( B ) . The secondmoments of µ mass (divided by m ) will be denoted by (cid:96) rs = m ∫ B x r x s dµ mass ( x ) . When B is a disc of radius R centered at the origin of R , it will be assumed that µ mass isrotationally symmetric, in which case the symmetric matrix L = ( (cid:96) rs ) is scalar: L = λI .Also in this case, ≤ λ ≤ R / , where corresponds to all mass being concentrated at theorigin and the upper bound corresponds to having all mass concentrated on the circle ofradius R . For the uniform mass distribution, λ = R / . It will be useful to introduce theparameter γ ∶= √ λ / R . The moment of inertia of a disc of radius R is given in terms of γ by I = m ( γR ) . For the uniform mass distribution on the ball γ = /√ . In general, ≤ γ ≤ .From now on B will be the disc of radius R in R centered at the origin. A configuration of the moving disc is an image of B under a Euclidean isometry; it is parametrized by thecoordinates of the center of mass ( y, z ) of the image disc and its angle of orientation θ .Introducing the new coordinate x ∶= γRθ and denoting by v = ( ˙ x, ˙ y, ˙ z ) the velocity vectorin configuration space, the kinetic energy of the body takes the form m ∣ v ∣ , where ∣ v ∣ isthe standard Euclidean norm.We consider now a system of two discs with radii R , R , denoted by B , B when intheir reference configuration, that is, when centered at the origin of R . The configurationmanifold M of the system, which is the set of all non-overlapping images of B i underEuclidean isometries of the plane, is then the set M ∶= {( a , a ) ∈ ( R × T ) × ( R × T ) ∶ ∣ a − a ∣ ≥ R + R } , T i = R /( πγ i R i ) and a indicates the coordinate projection of a in R . Givenmass distributions µ i , i = , , the kinetic energy of the system at a state ( a , v , a , v ) is K = ( m ∣ v ∣ + m ∣ v ∣ ) . The manifold M becomes a Riemannian manifold withboundary by endowing it with the kinetic energy metric ⟨( u , u ) , ( v , v )⟩ = m u ⋅ v + m u ⋅ v where the dot means ordinary inner product in R .Figure 1: Definition of the ( e , e , e ) frame at the contact point Q . The unit normal vector ν points away from the center of (the image in the given configuration of) body B . Let τ and ν denote the unit tangent and normal vectors at the contact point Q ∈ R ofthe bodies in a boundary configuration q ∈ ∂M as indicated in Figure 1. We introducethe orthonormal frame e , e , e of R as defined in the figure. If the system is at a state ( a , v , a , v ) , then Q will have velocity V i ( Q ) when regarded as a point in body B i inthe given configuration a i . One easily obtains V ( Q ) = [ v ⋅ e − γ − v ⋅ e ] e + v ⋅ e e , V ( Q ) = [ v ⋅ e − γ − v ⋅ e ] e + v ⋅ e e . The unit normal vector to ∂M at the configuration q = ( a , a ) pointing towards theinterior of M will be denoted n q . Note that this is defined with respect to the kineticenergy metric. Explicitly, letting m = m + m , n q = (−√ m m m e , √ m m m e ) . We also define the no-slip subspace S q of the tangent space to ∂M at q : S q = {( v , v ) ∶ V ( Q ) = V ( Q )} , and the orthogonal complement to S q in T q ∂M , which we write as C q . The orthogonaldirect sum of the latter and the line spanned by n q was denoted by C q in [7] and calledthe impulse subspace. These two spaces are given by(2.1) C q = {(− γ v ⋅ e , v ⋅ e ,
0; 1 γ v ⋅ e , v ⋅ e , ) ∶ m v + m v = } S q = {( v , v ) ∶ ( v − v ) ⋅ e = , ( v − v ) ⋅ e = ( v γ + v γ ) ⋅ e } . Figure 2 shows typical vectors in these orthogonal subspaces.Figure 2:
Tangent vectors in T q M decompose orthogonally (relative to the kinetic energy metric)into vectors of the above three types. In the diagram on the left the bodies are notrotating, and in the middle diagram the rotation velocities are determined by thecenter of mass velocities as given by (2.1). We can now state in the present very special case one of the main results of [7]. Seealso [2]. In the interior of M , the motion of the system, in the absence of potential forces,is geodesic relative to the kinetic energy metric. In dimension two this amounts simply toconstant linear and angular velocities. For the motion to be fully specified it is necessaryto find for any given v − = ( v − , v − ) at q ∈ ∂M such that ⟨ v − , n q ⟩ < (representing thebodies’ velocities immediately before a collision) a v + such that ⟨ v + , n q ⟩ > (representingthe bodies’ velocities immediately after). This correspondence should be given by a map C q ∶ v − ↦ v + . We call such correspondence a collision map C q at q ∈ ∂M . Proposition 2.1.
Linear collision maps C q ∶ T q M → T q M at q ∈ ∂M describing energypreserving, (linear and angular) momentum preserving, time reversible collisions havingthe additional property that impulsive forces between the bodies can only act at the singlepoint of contact (denoted by Q above) are given by the linear orthogonal involutions thatrestrict to the identity map on S q and send n q to its negative. Thus C q is fully determinedby its restriction to C q , where it can only be (in dimension ) the identity or its negative. We refer to [7] for a more detailed explanation of this result. The key point for ourpresent purposes is that, in addition to the standard reflection map given by specularreflection with respect to the kinetic energy metric, there is only one (under the statedassumptions) other map, which we refer to as the no-slip collision map.
The explicit form of the no-slip collision map C q can thus be obtained by first decompos-ing the pre-collision vector v − according to the orthogonal decomposition S q ⊕ C q and thenswitching the sign of the C q component to obtain v + . If Π q is the orthogonal projection to C q , then C q = I − q . Setting [ v ] = [( v , v )] ∶= ( v ⋅ e , v ⋅ e , v ⋅ e , v ⋅ e , v ⋅ e , v ⋅ e ) t , the pre- and post-collision velocities are related by [ v + ] = [ C q ][ v − ] where 5 C q ] = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ − δm γ δm γ − δm γ γ − δm γ δm γ − δm δm γ δm
00 0 1 − m m m m − δm γ γ δm γ − δm γ − δm γ − δm γ δm − δm γ − δm
00 0 m m − m m ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ . Here m = m + m and δ = { m [ + γ ] + m [ + γ ]} − . We record two special cases.First suppose that the two discs have the same mass distribution and γ = γ i . Then [ C q ] = ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ γ + γ γ + γ − + γ − γ + γ γ + γ + γ γ + γ γ + γ
00 0 0 0 0 1 − + γ γ + γ γ + γ − γ + γ − γ + γ γ + γ − γ + γ + γ
00 0 1 0 0 0 ⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠
The second case of interest assumes that one body, say B is fixed in place. It makessense to pass to the limit in which the mass and moment of inertia of B approach infinityand its velocity (which does not change during collision process as a quick inspection of C q shows) is set equal to . In this case only the velocity of B changes and we write m = m , v = v , γ = γ and v = v . Then n q = ( , , m − / ) , S q = {(− γs, s, ) ∶ s ∈ R } , C q = {( s, γs, ) ∶ s ∈ R } where vectors are expressed in the frame ( e , e , e ) . The (lower right block of the) matrix [ C q ] is now(2.3) [ C q ] = ⎛⎜⎜⎜⎜⎝ − − γ + γ − γ + γ − γ + γ − γ + γ
00 0 − ⎞⎟⎟⎟⎟⎠ = ⎛⎜⎜⎜⎜⎝ − cos β − sin β − sin β cos β
00 0 − ⎞⎟⎟⎟⎟⎠ . As noted earlier, γ = tan ( β / ) can take any value between and (equivalently, ≤ β ≤ π / ) in dimension ; thus it makes sense to define the angle β as we did above.When the mass distribution is uniform, γ = /√ and cos β = / , sin β = √ / . Noticethat these expressions still hold regardless of the shape of the fixed body B . In whatfollows we denote the above -by- matrix by C ∶= [ C q ] . 6 No-slip planar billiards
We focus attention on the last case indicated at the end of Section 2, which we call aplanar no-slip billiard : the billiard table is the complement of the fixed body B , nowany set in R with non-empty interior and piecewise smooth boundary, and the billiardball is the disc B of radius R with a rotationally symmetric mass distribution. Theconfiguration manifold is then the set M consisting of all q = ( q, x ) ∈ R × T for which thedistance between q and B is at least R . We define T ∶= R /( πγR ) . The projection q ↦ q maps M onto the billiard table , denoted B . The boundary of M is ∂ B × T and the frame ( e ( q ) , e ( q ) , e ( q )) at q ∈ ∂M is as indicated in Figure 3. We also view this q -dependentframe as the orthogonal map σ q ∶ R → T q R that sends the standard basis vectors (cid:15) i of R to e i ( q ) , for i = , , . This allows us to write C q = σ q C σ − q for each q .Due to energy conservation, the norms of velocity vectors are not changed duringcollision or during the free motion between collisions; we restrict attention to vectorsof unit length (in the kinetic energy metric, which agrees with the standard Euclideanmetric in R under the choice of coordinate x = γRθ for the disc’s angle of orientation). velocity phase space r e du ce d ph a s e s p ace billiard table Figure 3:
Definition of the product frame ( e , e , e ) , the reduced phase space, and the velocityphase space. Throughout the paper the notations ⟨ u, v ⟩ and u ⋅ v are both used for the standardinner product in R , the choice being a matter of typographical convenience. (Recall thatthe kinetic energy metric has been reduced to the standard inner product in R by ourdefinition of the variable x .) The phase space of the billiard system is N ∶= N + ∶= {( q, v ) ∈ T R ∶ q ∈ ∂M, ∣ v ∣ = , v ⋅ e ( q ) > } . We refer to vectors in N + as post-collision velocities; we similarly define the space N − of pre-collision velocities. The billiard map T , whose domain is a subset of N , is the7omposition of the free motion between two points q , q in ∂M and the billiard map C q at the endpoint. Thus T ∶ N → N is given by ( ˜ q, ˜ v ) = T ( q, v ) = ( q + tv, C ˜ q v ) where t ∶= inf { s > ∶ q + sv ∈ N } . (We assume that the shape of the billiard table B issuch that T makes sense and is smooth for all ξ in some big subset of N , say open of fullLebesgue measure.) Let ξ = ( q, v ) ↦ ˜ ξ − = ( ˜ q, v ) ↦ ˜ ξ = ˜ ξ + = ( ˜ q, C ˜ q v ) . The first map in this composition, which we denote by Φ , is parallel translation of v from q to ˜ q , and the second map, denoted C , applies the no-slip reflection map to thetranslated vector, still denoted v , at ˜ q . Hence T = C ○ Φ .Figure 4: Left: projection from R to R of an orbit of the no-slip Sinai billiard, to be discussedin more detail later on. Middle: the same orbit shown in the reduced phase space and,on the right, in the velocity phase space. Taking into account the rotation symmetry of the moving disc, we may for mostpurposes ignore the angular coordinate (but not the angular velocity!) and restrictattention to the reduced phase space . This is defined as ∂ B × { u ∈ R ∶ ∣ u ∣ < } , wherean element u of the unit disc represents the velocity vector at q ∈ ∂ B (pointing into thebilliard region) given by σ q ( u , u , √ − ∣ u ∣ ) = u e ( q ) + u e ( q ) + √ − ∣ u ∣ e ( q ) . By velocity phase space we mean this unit disc. Figure 3 summarizes these definitionsand Figure 4 shows what an orbit segment looks like in each space. On the left of thelatter figure is shown the two dimensional projection of the orbit segment defined in the -dimensional space M .The rotation symmetry that justifies passing from the -dimensional phase space tothe -dimensional reduced phase space may be formally expressed by the identity T ( q + λe , v ) = T ( q, v ) + λe . e is a parallel vector field (independent of q ). In particular, e isinvariant under the differential map: dT ξ e = e for all ξ = ( q, v ) .In addition to the orthonormal frames σ q it will be useful to introduce a frame consistingof eigenvectors of the collision map C q . We define u ( q ) = sin ( β / ) e ( q ) − cos ( β / ) e ( q ) u ( q ) = cos ( β / ) e ( q ) + sin ( β / ) e ( q ) u ( q ) = e ( q ) . (3.1)See Figure 5. Then C q u ( q ) = u ( q ) , C q u ( q ) = − u ( q ) , C q u ( q ) = − u ( q ) . Yet a third orthonormal frame will prove useful later on in our analysis of period- trajectories. Let ξ = ( q, v ) ∈ N . Then w ( ξ ) , w ( ξ ) , w ( ξ ) is the orthonormal frame at q such that w ( ξ ) ∶= e ( q ) − e ( q ) ⋅ vv ∣ e ( q ) − e ( q ) ⋅ vv ∣ , w ( ξ ) ∶= v × w ( ξ ) , w ( ξ ) ∶= v. Note that w ( ξ ) and w ( ξ ) span the -space perpendicular to v .Figure 5: Some angle and frame definitions: the q -dependent product frame ( e i ( q )) , the eigen-frame ( u i ) for the collision map C q at a collision configuration q ∈ ∂M , the character-istic angle β (a function of the mass distribution of the disc), and the angles φ ( q, v ) and ψ ( q, v ) . Definition 3.1 (Special orthonormal frames) . For any given ξ = ( q, v ) ∈ N we refer to ( e ( q ) , e ( q ) , e ( q )) , ( u ( q ) , u ( q ) , u ( q )) , ( w ( ξ ) , w ( ξ ) , w ( ξ )) as the product frame , the eigenframe , and the wavefront frame , respectively. orbits It appears to be a harder problem in general to show the existence of periodic orbits forno-slip billiards than it is for standard billiard systems in dimension , despite numerical9vidence that such points are common. A few useful observations can still be made forspecific shapes of B . We begin here with the general description of period- orbits. Thereader should bear in mind that, when we represent billiard orbits in figures such as 11,we often draw their projections on the plane, even though periodicity refers to a propertyof orbits in the -dimensional reduced phase space.Figure 6: Period orbit. The indicated parameters are: the disc’s mass m , its moment of inertia I , and radius R . The velocity of the center of mass is u and the angular velocity is ˙ θ . Let ξ = ( q, v ) be the initial state of a periodic orbit of period , ˜ ξ = ( ˜ q, ˜ v ) = T ( ξ ) , and t the time of free flight between collisions. Then, clearly, ( q, v ) = ( ˜ q + tC ˜ q v, C q ˜ v ) = ( q + t ( v + C ˜ q v ) , C q C ˜ q v ) so that C ˜ q v = − v and v = C q C ˜ q v . Because v and u ( q ) (respectively, u ( ˜ q ) ) are eigenvectorsfor different eigenvalues of the orthogonal map C q (respectively, C ˜ q ), v is perpendicular toboth u ( q ) and u ( ˜ q ) . It follows from (3.1) that u ( q ) ⋅ e = u ( ˜ q ) ⋅ e . Thus the projectionof e to v ⊥ is proportional to u ( q ) + u ( ˜ q ) . By the definition of the wavefront vector w ( ξ ) (and the angle φ , cf. Figure 5) we have w ( ξ ) = w ( ˜ ξ ) = u ( q ) + u ( ˜ q )∣ u ( q ) + u ( ˜ q )∣ = u ( q ) + u ( ˜ q ) √ − cos ( β / ) cos φ . Now observe that u ( ˜ q ) − u ( q ) is perpendicular to u ( q ) + u ( ˜ q ) . It follows from thisremark and a glance at Figure 5 (to determine the orientation of the vectors) that w ( ξ ) = − w ( ˜ ξ ) = u ( ˜ q ) − u ( q )∣ u ( ˜ q ) − u ( q )∣ = u ( ˜ q ) − u ( q ) ( β / ) cos φ . Notice, in particular, that v is a positive multiple of u ( q ) × u ( ˜ q ) . An elementarycalculation starting from this last observation gives v in terms of the product frame: v = cos ( β / ) sin φe + sin ( β / ) [ sin φe ( q ) + cos φe ( q )]√ − cos ( β / ) cos φ . A more physical description of the velocity v of a period- orbit is shown in Figure 6 interms of the moment of inertia I .Equally elementary computations yield the collision map C q in the wavefront frameat q , for a period- state ξ = ( q, v ) . We register this here for later use. To shorten the10quations we write c β / = cos ( β / ) and c φ = cos φ . C q w ( ξ ) = ( − c β / c φ ) w ( ξ ) − c β / c φ √ − c β / c φ w ( ξ ) C q w ( ξ ) = − c β / c φ √ − c β / c φ w ( ξ ) − ( − c β / c φ ) w ( ξ ) C q w ( ξ ) = − w ( ξ ) . (4.1)The following easily obtained inner products will also be needed later. u ( ˜ q ) ⋅ u ( q ) = − ( β / ) cos φw ( ξ ) ⋅ u ( q ) = √ − cos ( β / ) cos φw ( ξ ) ⋅ u ( q ) = − cos ( β / ) cos φ. (4.2)Figure 7 shows (two copies of the fundamental domain of) the configuration manifoldof the no-slip Sinai billiard. Here the billiard table is the complement of a circularscatterer in a two-dimensional torus and M is the cartesian product of the latter witha one-dimensional torus. Notice that there is a whole one-parameter family of initialconditions giving period- orbits, parametrized by the angle φ . We obtain infinitely manysuch families by choosing different pairs of fundamental domains.Figure 7: The figure shows two fundamental domains of the no-slip Sinai billiard and an initialvelocity v of a periodic orbit with period . This trajectory lies in a one-parameterfamily of period- trajectories parametrized by the angle φ . We will return to the no-slip Sinai billiard in Section 9. 11
The differential of the no-slip billiard map
Mostly, in this section, we write ⟨ u, v ⟩ instead of u ⋅ v for the standard inner product of R .Let q ( s ) be a smooth curve in ∂M such that q ( ) = q and q ′ ( ) = X ∈ T q ( ∂M ) . Define ω q ( X ) ∶= dds ∣ s = σ ( q ( )) − σ ( q ( s )) ∈ so ( ) where so ( ) is the space of antisymmetric × matrices (the Lie algebra of the rotationgroup) and σ ( q ) ∶= σ q is the product frame. As e is a parallel field and ω q ( X ) isantisymmetric we have ω q ( X ) ij = except possibly for ( i, j ) = ( , ) and ( , ) . Denotingby D X directional derivative of vector fields along X at q , ω q ( X ) = (cid:15) ⋅ [ dds ∣ s = σ ( q ( )) − σ ( q ( s )) (cid:15) ] = ⟨ e ( q ) , D X e ⟩ = ⟨ e ( q ) , X ⟩ ⟨ e ( q ) , D e ( q ) e ⟩ since D e e = . The inner product κ ( q ) ∶= ⟨ e ( q ) , D e ( q ) e ⟩ is the geodesic curvature ofthe boundary of B at ¯ q , where ¯ q is the base point of q in ∂ B . Thus(5.1) ω q ( X ) = κ ( q )⟨ e ( q ) , X ⟩ A where A = ⎛⎜⎝ − ⎞⎟⎠ . Given vector fields µ, ν , we define µ ⊙ ν as the map(5.2) ( q, v ) ↦ ( µ ⊙ ν ) q v ∶= ⟨ µ q , v ⟩ ν q + ⟨ ν q , v ⟩ µ q . Lemma 5.1.
The directional derivative of C along X ∈ T q ( ∂M ) is D X C = κ ( q ) ⟨ e ( q ) , X ⟩ O q where O q = σ q O σ − q , O ∶= [ A , C ] = ( β / ) ⎛⎜⎝ ( β / ) − cos ( β / ) sin ( β / ) − cos ( β / ) ⎞⎟⎠ and C was defined above in (2.3). Furthermore, O q = ( β / )( u ⊙ e ) q and D X C = ( β / ) κ ( q )⟨ X, e ⟩ q ( u ⊙ e ) q . Proof.
Notice that = D X I = D X ( σ − σ ) = ( D X σ − ) σ + σ − D X σ. Thus D X σ − = − σ − ( D X σ ) σ − . Therefore, D X C = ( D X σ ) C σ − + σ C D X σ − = σ [ σ − D X σ ] C σ − − σ C [ σ − D X σ ] σ − = σ [ ω ( X ) , C ] σ − . D X C is now a consequence of Equation 5.1. A simplecomputation also gives, for any given v ∈ R ,(5.3) σ ( q ) O σ ( q − ) v = ( β / )( e ⊙ u ) q v yielding the second expression for D X C .It is convenient to define the following two projections. Let ξ = ( q, v ) ∈ N ± . The space T ξ N ± decomposes as a direct sum T ξ N ± = H ξ ⊕ V ξ where H ξ = T q N = { X ∈ R ∶ X ⋅ e ( q ) = } and V ξ = v ⊥ = { Y ∈ R ∶ Y ⋅ v = } . (Recall that N ∶= N + .) We refer to these as the horizontal and vertical subspaces of T ξ N ± .We use the same symbols to denote the projections H ξ ∶ R → T q ( ∂M ) and V ξ ∶ R → v ⊥ defined by H ξ Z ∶= Z − ⟨ Z, e ( q )⟩⟨ v, e ( q )⟩ v, V ξ Z ∶= Z − ⟨ Z, v ⟩ v. Notice that for ξ = ( q, v ) ∈ N ± and Z ∈ R ⟨ e ( q ) , H ξ Z ⟩ = ⟨ Z, e ( q )⟩⟨ v, e ( q )⟩ − ⟨ Z, e ( q )⟩⟨ v, e ( q )⟩⟨ v, e ( q )⟩ = ⟨ v × e ( q ) , Z ⟩⟨ v, e ( q )⟩ . Also observe that v × e = ∣ v ∣ w ( ξ ) , where w is the second vector in the wavefront frame(cf. Definition 3.1) and v is the orthogonal projection of v to the plane perpendicularto e . Thus, denoting by φ ( ξ ) the angle between v and e ( q ) (this is the same φ as inFigures 5, 6 and 7)(5.4) ⟨ e ( q ) , H ξ Z ⟩ = φ ( ξ ) ⟨ w ( ξ ) , Z ⟩ . Let q ∈ ∂M , v = v − ∈ N − q , v + ∶= C q v − ∈ N + q , ξ = ξ − = ( q, v − ) , ξ + = ( q, v + ) . Define(5.5) Λ ξ ∶= V ξ + H ξ − ∶ v ⊥− → v ⊥+ . Clearly Λ ξ is defined on all of R , not only on v ⊥− , but we are particularly interested in itsrestriction to the latter subspace.Let ξ = ( q, v ) be a point contained in a neighborhood of N where T is defined anddifferentiable. Set ˜ ξ = T ( ξ ) . We wish to describe dT ξ ∶ T ξ N → T ˜ ξ N . Let ξ ( s ) = ( q ( s ) , v ( s )) be a differentiable curve in N with ξ ( ) = ξ and define X ∶= q ′ ( ) ∈ T q N, Y ∶= v ′ ( ) ∈ v ⊥ . Then ˜ ξ ( s ) = T ( ξ ( s )) = ( ˜ q ( s ) , ˜ v ( s )) ∈ N where ˜ q ( s ) = q ( s ) + t ( s ) v ( s ) and ˜ v ( s ) = C ˜ q ( s ) v ( s ) .From the equality ⟨ ˜ q ′ ( ) , e ( ˜ q )⟩ = it follows that t ′ ( ) = − ⟨ X + tY, e ( ˜ q )⟩⟨ v, e ( ˜ q )⟩ . ˜ X ∶= ˜ q ′ ( ) ∈ T ˜ q N and ˜ Y ∶= ˜ v ′ ( ) ∈ ˜ v ⊥ satisfy ˜ X = X + tY − ⟨ X + tY, e ( ˜ q )⟩⟨ v, e ( ˜ q )⟩ v = H ˜ ξ − ( X + tY ) and ˜ Y = C ˜ q Y + [ dds ∣ s = C ˜ q ( s ) ] v = C ˜ q Y + κ ( ˜ q )⟨ e ( ˜ q ) , ˜ X ⟩ O ˜ q v where we have used Lemma 5.1. From the same lemma, σ ( ˜ q ) O σ ( ˜ q ) − v = − ( β / )( ν ⊙ u ) ˜ q v. Thus ˜ X = H ˜ ξ − ( X + tY ) ˜ Y = C ˜ q Y − ( β / ) κ ( ˜ q ) ⟨ e ( ˜ q ) , H ˜ ξ − ( X + tY )⟩ ( ν ⊙ u ) ˜ q v. (5.6)As already noted, T ξ N + = T q ( ∂M ) ⊕ v ⊥ . By using the projection V ξ ∶ T q ( ∂M ) → v ⊥ introduced earlier we may identify T ξ N + with the sum v ⊥ ⊕ v ⊥ . In this way dT ξ is regardedas a map from v ⊥ ⊕ v ⊥ to ˜ v ⊥ ⊕ ˜ v ⊥ . Proposition 5.2.
Let T ∶ N → N be the billiard map, ξ = ( q, v ) ∈ N and ( ˜ q, ˜ v ) = ˜ ξ = T ( ξ ) ,where ˜ q = q + tv , and ˜ v = C ˜ q v . Under the identification of the tangent space T ξ N with v ⊥ ⊕ v ⊥ as indicated just above, we may regard the differential dT ξ as a linear mapfrom v ⊥ ⊕ v ⊥ to ˜ v ⊥ ⊕ ˜ v ⊥ . Also recall from (5.4) the definition of Λ ˜ ξ ∶ v ⊥ → ˜ v ⊥ . Then dT ξ ∶ T ξ N → T ˜ ξ N is given by ( XY ) ↦ ⎛⎝ Λ ˜ ξ ( X + tY ) C ˜ q Y + ( β / ) κ ( ˜ q ) ( e ⊙ u ) ˜ q v cos φ ( ˜ q,v ) ⟨ w ( ξ ) , X + tY ⟩ ⎞⎠ where cos φ ( ˜ q, v ) = ⟨ v /∣ v ∣ , e ( ˜ q )⟩ and v is the orthogonal projection of v to e ⊥ .Proof. This is a consequence of the preceding remarks and definitions.
Corollary 5.1. If ξ = ( q, v ) is periodic of period , then C ˜ q v = − v , ⟨ v, u ( ˜ q )⟩ = , andthe map of Proposition 5.2 reduces to ( XY ) ↦ ( X + tYC ˜ q Y + ( β / ) κ ( ˜ q ) cos ψ ( ˜ q,v ) cos φ ( ˜ q,v ) ⟨ w ( ξ ) , X + tY ⟩ u ( ˜ q ) ) . where cos ψ ( ˜ q, v ) ∶= ⟨ v, e ( ˜ q )⟩ , cos φ ( ˜ q, v ) = ⟨ v /∣ v ∣ , e ( ˜ q )⟩ .Proof. Clearly, C ˜ q v = − v , whence ⟨ v, u ( ˜ q )⟩ = and ( e ⊙ u ) ˜ q v = ⟨ e ( ˜ q ) , v ⟩ u ( ˜ q ) . Alsonotice that Λ ˜ ξ Z = Z whenever ⟨ Z, v ⟩ = . The corollary follows. 14 Measure invariance and time reversibility
It will be seen below that the no-slip billiard map does not preserve the natural symplecticform on N , so these mechanical systems are not Hamiltonian. Nevertheless, the canonicalbilliard measure derived from the symplectic form (the Liouville measure) is invariant andthe system is time reversible, so some of the good features of Hamiltonian systems arestill present. (See, for example, [14, 15] where a KAM theory is developed for reversiblesystems.)Recall that the invertible map T is said to be reversible if there exists an involution R such that R ○ T ○ R = T − . In order to see that the no-slip billiard map T is reversible we first define the followingmaps: Φ ∶ ( q, v ) ↦ ( q + tv, v ) , where t is the time of free motion of the trajectory with initialstate ( q, v ) , so that q, q + tv ∈ ∂M ; the collision map C ∶ N → N given by C ( q, v ) = ( q, C q v ) ;and the flip map J ∶ ( q, v ) ↦ ( q, − v ) where q ∈ ∂M and v ∈ R . Recall that T = C ○ Φ .Now set R ∶= J ○ C = C ○ J . It is clear (since C q is an involution by Proposition 2.1) that R = I and that J ○ Φ ○ J = Φ − . Therefore, R ○ T ○ R = J ○ C ○ Φ ○ J ○ C = J ○ Φ ○ J ○ C = Φ − ○ C = ( C ○ Φ ) − = T − . Notice that if L ∶ V → V is a reversible isomorphism of a vector space V with timereversal map R ∶ V → V (so that R ○ L ○ R = L − ) then for any eigenvalue λ of L associatedto eigenvector u , / λ is also an eigenvalue for the eigenvector R u , as easily checked. Thesesimple observations have the following useful corollary. Proposition 6.1.
Let ξ ∈ N be a periodic point of period k of the no-slip billiard systemand let λ be an eigenvalue of the differential map dT kξ ∶ T ξ N → T ξ N corresponding toeigenvector u . Then / λ is also an eigenvalue of dT kξ corresponding to eigenvector R u ,where R is the composition of the collision map C and the flip map J . Furthermore, e (see Definition 3.1) is always an eigenvector of dT ξ and all its powers, corresponding tothe eigenvalue . We now turn to invariance of the canonical measure. The canonical -form θ on N isdefined by θ ξ ( U ) ∶= v ⋅ X for ξ = ( q, v ) ∈ N and U = ( X, Y ) ∈ T q N ⊕ v ⊥ = T ξ N . Its differential dθ is a symplecticform on N ∩ { v ∈ T q ( ∂M ) ∶ ∣ v ∣ = } c and Ω = dθ ∧ dθ is the canonical volume form on thissame set. In terms of horizontal and vertical components of vectors in T N , the symplecticform is expressed as dθ ( U , U ) = Y ⋅ X − Y ⋅ X where U i = ( X i , Y i ) . An elementary computation shows that the measure on N associatedto Ω is given by(6.1) ∣ Ω ξ ∣ = v ⋅ ν ( q ) dA ∂M ( q ) dA N ( v ) where ν ( q ) ∶= e ( q ) , dA ∂M ( q ) is the area measure on ∂M , and dA N ( v ) is the area measureon the hemisphere N q = { v ∈ R ∶ v ⋅ ν ( q ) > } . 15 roposition 6.2. The canonical -form Ω on N transforms under the no-slip billiardmap as T ∗ Ω = − Ω . In particular, the associated measure ∣ Ω ∣ , shown explicitly in Equation6.1, is invariant under T .Proof. Let u be a vector field on ∂M and introduce the one-form θ u on N given by θ uξ ( U ) ∶= ( v ⋅ u ( q ))( u ( q ) ⋅ X ) for ξ = ( q, v ) and U = ( X, Y ) . Taking u to be each of the vector fields u , u we obtainthe -forms θ u and θ u . As v = ( v ⋅ u ) u + ( v ⋅ u ) u + ( v ⋅ ν ) ν and X ⋅ ν = , we have θ = θ u + θ u . The no-slip collision map C acts on u = θ u i as follows: For U = ( X, Y ) ∈ T q ( ∂M ) ⊕ v ⊥ , ( C ∗ θ u ) ξ ( U ) = ( C q ( v ) ⋅ u ( q ))( u ( q ) ⋅ X ) = ( v ⋅ C q ( u ( q )))( u ( q ) ⋅ X ) . It follows that C ∗ θ u = θ u , C ∗ θ u = − θ u . We now compute the differentials dθ u for u = u , u . Observe that θ u = f u ( ξ )( π ∗ u ♭ ) ,where f ξ is the function on N defined by f u ( ξ ) ∶= v ⋅ u ( q ) and π ∗ u ♭ is the pull-back underthe projection map π ∶ N → ∂M of the -form u ♭ on ∂M given by u βq ( X ) = u ( q ) ⋅ X . Thus dθ u = df u ∧ ( π ∗ u ♭ ) + f u π ∗ du ♭ . A simple calculation gives df uξ ( X, Y ) = v ⋅ ( D X u ) + u ( q ) ⋅ Y. The vector field u = u i is parallel on ∂M . In fact, its derivative in direction X ∈ T q ( ∂M ) only has component in the normal direction, given by D X u = κ ( q )( X ⋅ e ( q ))( u ( q ) ⋅ e ( q )) ν ( q ) . Omitting the dependence on q , we have df uξ ( X, Y ) = κ ( q )( X ⋅ e )( u ⋅ e )( v ⋅ ν ) + u ⋅ Y. Another simple calculation gives du ♭ q ( X , X ) = ( D X u ) ⋅ X − ( D X u ) ⋅ X = so dθ u = df u ∧ π ∗ u ♭ . Explicitly, dθ u ( U , U ) = ( u ⋅ Y )( u ⋅ X ) − ( u ⋅ Y )( u ⋅ X ) − κ ( q )( v ⋅ ν )( u ⋅ e )( u ⋅ e ) ω ( X , X ) where ω ( X , X ) ∶= ( e ⋅ X )( e ⋅ X ) − ( e ⋅ X )( e ⋅ X ) . ω is the area form on ∂M . A convenient way to express dθ u is as follows.Define the -form ˜ u on N by ˜ u ξ ( U ) = u ( q ) ⋅ Y , where U = ( X, Y ) ∈ T ξ N , and the function g u ( ξ ) ∶= − κ ( q )( v ⋅ ν )( u ⋅ e )( u ⋅ e ) . These extra bits of notation now allow us to write dθ uξ = g u ( ξ )( π ∗ ω ) + ˜ u ∧ ( π ∗ u ♭ ) . The main conclusion we wish to derive from these observations is that dθ u ∧ dθ u = . Thisis the case because, as dim ( ∂M ) = , we must have ω = and ω ∧ u ♭ = . Therefore, Ω ∶= dθ ∧ dθ = ( dθ u + dθ u ) ∧ ( dθ u + dθ u ) = dθ u ∧ dθ u . Finally, C ∗ Ω = d ( C ∗ θ u ) ∧ d ( C ∗ θ u ) = − dθ u ∧ dθ u = − Ω . The forms dθ and Ω are invariant under the geodesic flow and under the map it induceson N . As T is the composition of this map and C , the proposition is established. We set the following conventions for a wedge table with corner angle φ . See Figure 8.(This is the same φ that has appeared before in previous figures.) The boundary planesof the configuration manifold are denoted P and P . The orthonormal vectors of theconstant product frame on plane P i are e ,i , e ,i , e ,i = ν i for i = , where e , = ⎛⎜⎝ ⎞⎟⎠ , e , = ⎛⎜⎝ cos φ − sin φ ⎞⎟⎠ , e , = ⎛⎜⎝ sin φ cos φ ⎞⎟⎠ ,e , = ⎛⎜⎝ ⎞⎟⎠ , e , = − ⎛⎜⎝ cos φ sin φ ⎞⎟⎠ , e , = ⎛⎜⎝ sin φ − cos φ ⎞⎟⎠ . Let σ i ∶ R → T q ⊕ R ν i be the constant orthogonal map such that σ i (cid:15) j = e j,i , where (cid:15) i , i = , , , is our notation for the standard basis vectors in R . Let u ,i = sin ( β / ) e ,i − cos ( β / ) e ,i , u ,i = cos ( β / ) e ,i + sin ( β / ) e ,i , u ,i = e ,i = ν i be the eigenvectors of the no-slip reflection map associated to the plane P i and set ζ i (cid:15) j ∶= u j,i . For easy reference we record their matrices here: ζ i = ⎛⎜⎝ (− ) i cos ( β / ) cos φ −(− ) i sin ( β / ) cos φ sin φ cos ( β / ) sin φ − sin ( β / ) sin φ −(− ) i cos φ sin ( β / ) cos ( β / ) ⎞⎟⎠ . The initial velocity v for the period- trajectory points in the direction of u , × u , andis given by v = √ − cos ( β / ) cos φ ⎛⎜⎝ ( β / ) cos ( β / ) sin φ ⎞⎟⎠ . q ∈ P and q ∈ P . Any such pair of pointscan be written as q = a ⎛⎜⎝ sin ( β / ) cos φ − sin ( β / ) sin φb − cos ( β / ) sin φ ⎞⎟⎠ , q = a ⎛⎜⎝ sin ( β / ) cos φ sin ( β / ) sin φb + cos ( β / ) sin φ ⎞⎟⎠ where a, b ∈ R , a > . In what follows we assume without loss of generality that a = and b = . Thus q i = ( sin ( β / ) cos φ, (− ) i sin ( β / ) sin φ, (− ) i cos ( β / ) sin φ ) t . Figure 8:
Some notation specific to the wedge billiard table. The P i are the half-plane compo-nents of the boundary of the configuration manifold. Let S ± i = { v ∈ R ∶ ∣ v ∣ = , ± v ⋅ ν i > } . The collision maps C i ∶ S − i → S + i , i = , are givenby the matrices C i = σ i C σ − i = ζ i ⎛⎜⎝ − − ⎞⎟⎠ ζ − i where C was defined in 2.3. We now introduce coordinates on P i × S + i as follows. Let S + = { z ∈ R ∶ ∣ z ∣ = , z > } and define Φ i ∶ R × S + → P i × S + i by Φ i ( x, y ) = ( q i + x u ,i + x u ,i , y u ,i + y u ,i + y u ,i ) . Regarding x ∈ R as ( x, ) ∈ R , we may then write Φ i ( x, y ) = ( q i + ζ i x, ζ i y ) . Clearly, the billiard map is not defined on all of ⋃ i P i × S + i since those initial velocitiesnot pointing towards the other plane will escape to infinity, but we are interested in thebehavior of the map on a neighborhood of the periodic point ξ i = ( q i , v i ) , v i = −(− ) i v .The question of interest here is whether some open neighborhood of ξ i remains invariantunder the billiard map. It is easily shown that the coordinates of the state ξ i (of theperiod- orbit at the plane P i ) are Φ − i ( ξ i ) = ( , y i ) ∈ R × S + where y i = √ − cos ( β / ) cos φ ( , (− ) i sin φ, sin ( β / ) cos φ ) t T i ∶ D i ⊂ R × S + → R × S + be the billiard map restricted to P i × S + i expressed in thecoordinate system defined by Φ i . Thus T = Φ − T Φ , T = Φ − T Φ on their domains D i . We now find the explicit form of T i . Define ¯ i = ⎧⎪⎪⎨⎪⎪⎩ if i = if i = andorthogonal matrices A i ∶= ζ − i ζ i and S = diag ( , − , − ) , both in SO ( ) . Also define α ∶= φ √ − cos ( β / ) cos φ. Observe that ζ − i C ¯ i ζ i = SA i . It is easily shown that q i − q ¯ i = − αv i , v i = ζ i y i , A i y i = − y ¯ i , SA i y i = y ¯ i . In particular, ζ − i ( q i − q ¯ i ) = − αy i . Let Q ∶ R × S + → R be defined by Q ( x, y ) ∶= x − x ⋅ (cid:15) y ⋅ (cid:15) y. Notice that Q ( x, y ) ⋅ (cid:15) = . We now have(7.1) T i ∶ ( x, y ) ↦ ( Q ( A i ( x − γy i ) , A i y ) , SA i y ) . Figure 9:
The velocity factor of orbits of the return billiard map T T in coordinate system Φ lie in concentric circles with axis y = ˆ (cid:15) . We use spherical coordinates ϕ and ψ relativeto the axis ˆ (cid:15) to represent the velocity y ∈ S + . With respect to these coordinates, thereturn map sends w ( ϕ ) to w ( ϕ + θ ) , where θ is a function of the wedge angle α andthe characteristic angle β of the no-slip reflection. For easy reference we record αy i = φ ⎛⎜⎝ (− ) i sin φ sin ( β / ) cos φ ⎞⎟⎠ , S = ⎛⎜⎝ − − ⎞⎟⎠ A = A t = ζ − ζ = ⎛⎜⎝ − ( β / ) cos φ − sin β cos φ cos ( β / ) sin ( φ )− sin β cos φ − ( β / ) cos φ sin ( β / ) sin ( φ )− cos ( β / ) sin ( φ ) − sin ( β / ) sin ( φ ) − cos ( φ ) ⎞⎟⎠ . Using the notation [ z ] ∶= z ⋅ (cid:15) and elementary computations based on the above gives: Proposition 7.1.
The return map in the coordinate system defined by Φ has the form T T ( x, y ) = ( x + [ A ( x − αy )] V ( y ) , SA t SA y ) where V ( y ) = [ y ] A t SA y − [ A t SA y ] y [ A y ] [ A t SA y ] . This vector satisfies: [ V ( y )] = and V ( y ) = . In particular, T T ( x, y ) = ( x, y ) whenever ( x, y ) is in the domain of T T . In order to study this return map in a neighborhood of ( x, y ) we use sphericalcoordinates about the axis y :(7.2) y = cos ψ y + sin ψ cos ϕ ˆ (cid:15) + sin ψ sin ϕ ˆ (cid:15) where ˆ (cid:15) ∶= (cid:15) , ˆ (cid:15) ∶= √ − cos ( β / ) cos φ ( sin ( β / ) cos φ (cid:15) + sin φ (cid:15) ) , ˆ (cid:15) ∶= y form an orthonormal frame. See Figure 9. (Notice the typographical distinction betweenthe corner angle φ of the wedge domain and the spherical coordinate ϕ .) Let ( X ( x, ϕ, ψ ) , Y ( x, ϕ, ψ )) ∶= T T ( x, cos ψ y + sin ψ cos ϕ ˆ (cid:15) + sin ψ sin ϕ ˆ (cid:15) ) and define w ∶= w ( ϕ ) ∶= cos ϕ ˆ (cid:15) + sin ϕ ˆ (cid:15) . Thus we may write y = cos ψ ( y + tan ψw ( ϕ )) . Since the rotation S ∶= SA t SA fixes y ,it acts on w as S w ( ϕ ) = w ( ϕ + θ ) for some constant angle θ . It follows that S y = cos ψ y + sin ψ w ( ϕ + θ ) . The following proposition summarizes these observations and notations.
Proposition 7.2.
For points y ∈ S + in a neighborhood of y we adopt spherical coordinatesrelative to the axis y = ˆ (cid:15) , so that y = cos ψ ( y + tan ψ w ( ϕ )) where w ∶= w ( ϕ ) ∶= cos ϕ ˆ (cid:15) + sin ϕ ˆ (cid:15) . See Figure 9. We also use the notations [ z ] ∶= z ⋅ (cid:15) , S ∶= A − SA , and S = SA − SA .Let R ∶= T T be the -step return map as defined above, whose domain contains aneighborhood of ( x, y ) for all x ∈ R . Then R ( x, y ) = ( x, y ) for all x and R ( x, y + tan ψ w ( ϕ )) = ( X, y + tan ψ S w ( ϕ )) = ( X, tan ψ w ( ϕ + θ )) or an angle θ , depending only on the wedge angle φ and the characteristic angle β ofthe no-slip reflection, such that cos θ = ( S ˆ (cid:15) ) ⋅ ˆ (cid:15) = − δ + δ sin θ = ( S ˆ (cid:15) ) ⋅ ˆ (cid:15) = δ ( − δ )√ − δ where δ ∶= cos ( β / ) cos φ . Writing ( X, Φ , Ψ ) = R ( x, ϕ, ψ ) we have (7.3) R ∶ ⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩ X = x + tan ψ [ A ( x − γy )] [ y ] ( I + S ) w − [( I + S ) w ] y [ y ] + tan ψ [ w ] S w −[ S w ] w [ y ] − tan ψ ( [( A + S ) w ] [ y ] − tan ψ [ A w ] [ S w ] [ y ] ) Φ = ϕ + θ Ψ = ψ Denoting µ ∶= ζ − (cid:15) ∈ R , we further have X ( x + sµ , ϕ, ψ ) = X ( x, ϕ, ψ ) + sµ . Since ψ remains constant under iterations of the return map R = T T , we regard ψ as a fixed parameter. We are interested in small values of r ∶= tan ψ . Notice that [ A z ] ∶= ( A z ) ⋅ (cid:15) = z ⋅ ( A t (cid:15) ) = µ ⋅ z , where µ ∶= A t (cid:15) = ⎛⎜⎝ cos ( β / ) sin ( φ ) sin ( β / ) sin ( φ )− cos ( φ ) ⎞⎟⎠ . Write x ∶= αy , so x = φ ⎛⎜⎝ − sin φ sin ( β / ) cos φ ⎞⎟⎠ . Then the proposition shows that R has the form(7.4) R ∶ ( x, ϕ ) ↦ ( X = x + µ ⋅ ( x − x ) V r ( ϕ ) , Φ = ϕ + θ ) where the vector V r ( ϕ ) can be made arbitrarily (uniformly) small by choosing ψ (or r = tan ψ ) sufficiently close to . Observe from the explicit form V r ( ϕ ) = [ y ] r (( I + S ) w − [( I + S ) w ] y [ y ] ) + r [ w ] S w −[ S w ] w [ y ] − r [( A + S ) w ] [ y ] + r [ A w ] [ S w ] [ y ] that V r ( ϕ ) ⋅ (cid:15) = so that X is indeed in R . Proposition 7.3.
The quantity + µ ⋅ V r ( ϕ ) satisfies the coboundary relation (7.5) + µ ⋅ V r ( ϕ ) = ρ ( ϕ ) ρ ( ϕ + θ ) where ρ ( ϕ ) = + r tan φ sin ( β / ) sin ϕ. n fact, the transformation R on the -dimensional space R × R /( π Z ) , obtained by fixinga value of ψ (hence of r = tan ψ ), leaves invariant the measure dµ = c ( + r tan φ sin ( β / ) sin ϕ ) dA dϕ where c is a positive constant (only dependent on the fixed parameters β, ψ, φ ) and A isthe standard area measure on R .Proof. The canonical invariant measure on R × S + has the form y ⋅ (cid:15) dA dA S , where A S is the area measure on S + . For a fixed value of ψ we obtain an invariant measure on R × S of the form y ⋅ (cid:15) dA dϕ . Using the form of y given by (7.2), one obtains y ⋅ (cid:15) = cos ψ cos φ sin ( β / )√ − cos ( β / ) cos φ ( + r tan φ sin ( β / ) sin ϕ ) . This shows that, up to a multiplicative constant, the invariant measure µ has the indicatedform. Equation (7.5) is an easy consequence of the invariance of µ with respect to R .Figure 10: The map R sends fibers x + R µ onto other such fibers preserving length. That is, dR ( x,ϕ ) µ = µ . The quotient is a measure preserving transformation on R × T . Thecoordinate on the first factor of the quotient is ¯ x = x ⋅ µ . The curve shown above istypical of the set to which orbits of R project in R . By using the coordinate system ( ¯ x, ¯ y ) ↦ ¯ xµ + ¯ yµ on R , the area measure is dA = d ¯ x d ¯ y and, as observed at the end of Proposition 7.2, the transformation R maps the fibersof the projection ( ¯ x, ¯ y ) ↦ ¯ x to fibers preserving the length measure on fibers. Thus weobtain a transformation ¯ R on R × S preserving the measure d ¯ µ ( ¯ x, ϕ ) = ρ ( ϕ ) d ¯ x dϕ where ρ ( ϕ ) has the stated expression. Using the quotient coordinates ¯ x = x ⋅ µ and φ andwriting V r ( ϕ ) ∶= V r ( ϕ ) ⋅ µ we obtain R ( ¯ x, φ ) = (( + V r ( ϕ )) ¯ x − ¯ x V r ( ϕ ) , φ + θ ) . In particular, X = ρ ( ϕ ) ρ ( ϕ + θ ) ¯ x + ( − ρ ( ϕ ) ρ ( ϕ + θ ) ) ¯ x . d ¯ µ ( ¯ x, ϕ ) = ρ ( ϕ ) d ¯ xdϕ where ρ ( ϕ ) is the density given in Proposition 7.3. It is now immediate that R n ( ¯ x, ϕ ) = ( ρ ( ϕ ) ρ ( ϕ + nθ ) ¯ x + ( − ρ ( ϕ ) ρ ( ϕ + nθ ) ) ¯ x , ϕ + nθ ) . This shows that all the iterates of ( ¯ x, ϕ ) remain uniformly close to the initial point forsmall values of ψ . Also notice that ( ζ µ ) ⋅ e , = ν ⋅ e , = sin ( φ ) > . This means thatif ¯ x remains bounded, the length coordinate along the base of P also must be similarlybounded. From this we conclude: Corollary 7.1.
Assume the notation introduced at the beginning of this section. For all q ∈ P i ∖ ( P ∩ P ) , i = , , and any neighborhood V of the period- state ( q, v i ) ∈ S + i ,there exists a small enough neighborhood U ⊂ V of ( q, v i ) the orbits of whose pointsremain in V . Because any (bounded) polygonal billiard shape must have a corner with angle lessthan π , the following corollary holds. Theorem 7.1.
Polygonal no-slip billiards cannot be ergodic for the canonical invariantmeasure.
The analysis of the previous section is based on the existence of period- orbits in wedge-shaped no-slip billiard tables. Existence of periodic orbits of higher periods is in generaldifficult to establish, although one such result for wedge domains will be indicated belowin this section. We first point out a generalization of Corollary 7.1 to perturbations ofperiodic orbits in general polygon-shaped domains.Figure 11 illustrates the type of stability implied by the following Theorem 8.1. Theorem 8.1.
Periodic orbits in no-slip polygon-shaped billiard domains are locallystable. That is, given an initial state ξ = ( q , v ) for a period- n orbit in such a billiardsystem, and for any neighborhood V of ξ , there exists a small enough neighborhood U ⊂ V of ξ the orbits of whose elements remain in V .Proof. The idea is essentially the same as used in the proof of Proposition 7.3 andCorollary 7.1. We only indicate the outline. By a choice of convenient coordinates aroundthe periodic point, it is possible to show that the n -th iterate of the billiard map T ,denoted R ∶= T n , can be regarded as a map from an open subset of R × S into this latterset, having the form R ( x, ϕ ) = ( x + A ( ϕ )( x − x ) , ϕ + θ ) for a certain angle θ , where A ( ϕ ) is a linear transformation independent of x . Rotation invariance implies that R mustsatisfy the invariance property R ( x + su, ϕ ) = R ( x, ϕ ) + su for a vector u ∈ R . From thiswe define a map R on (a subset of) the quotient R × S , R / R u . Furthermore, denotingby ( ¯ x, ϕ ) the coordinates in this quotient space, invariance of the canonical measure23igure 11: Projection to the plane of orbits in the neighborhood of a period- periodic orbit ofa triangular no-slip billiard domain (left) showing typical stable behavior, along withthe velocity phase portrait projections (right). implies invariance of a measure µ on this quotient having the form dµ ( ¯ x, ϕ ) = ρ ( ϕ ) d ¯ x dϕ. Invariance is with respect to the quotient map R ( ¯ x, ϕ ) = ( x + a ( ϕ )( ¯ x − ¯ x ) , ϕ + θ ) forsome function a ( ϕ ) . This function must then take the form a ( ϕ ) = ρ ( ϕ )/ ρ ( ϕ + θ ) . Iteratesof R will then behave like the corresponding map for the wedge domain, defined prior toTheorem 7.1.Figure 12: From left to right: projections to the plane of periodic orbits of periods , , , . (Bounded orbits in the same wedge domain are all periodic with the same period.)The rotation angle θ in each case is πp / q where p / q = / , / , / , / . Massdistribution of the disc particle is uniform. We turn now to the question of existence of periodic orbits of higher (necessarily even)periods for wedge shapes. Clearly, a necessary condition is that the angle θ introduced inProposition 7.2 (see also Figure 9) be rational. For orbits that do not eventually escape toinfinity, this is also a sufficient condition, as a simple application of Poincaré recurrence24hows. (See [8].) Moreover, as θ is only a function of δ ∶= cos ( β / ) cos φ , which is givenby (Proposition 7.2)(8.1) cos θ = − δ + δ where β is the characteristic angle of the system (a function of the mass distribution onthe disc) and φ is the corner angle of the wedge domain, if a higher order periodic orbitexists for a given δ , all bounded orbits have the same period.Figure 13: From left to right: projections of periodic orbits of periods , , , . The rotationangle θ in each case is πp / q where p / q is / , / , / , / , respectively. Massdistribution is uniform. We give a few examples for the uniform mass distribution, for which cos ( β / ) = √ / . Solving 8.1 for cos φ , for θ = πp / q , choosing first the negative square root, gives(8.2) cos φ p,q ∶= √ ¿``(cid:192) − √ + cos ( πp / q ) A few examples are shown in Figure 12.Notice that there are no restrictions on the values of p and q . The following propositionis a consequence of these remarks. Theorem 8.2.
For any positive even integer n there exists a wedge domain for whichthe no-slip billiard has period- n orbits. More specifically, all bounded orbits of the no-slipbilliard in a wedge domain with corner angle φ p,q satisfying Equation 8.2 are periodicwith period q . Figure 14:
All orbits of an equilateral triangle no-slip billiard system are periodic with (notnecessarily least) period equal to or . Solving 8.1 for cos φ , for θ = πp / q , but choosing now the positive square root, gives cos φ p,q ∶= √ ¿``(cid:192) + √ + cos ( πp / q ) . . ≈ arccos (− / )/ π ≤ p / q ≤ . , which greatly restrictsthe choices of p and q . A few examples in this case are shown in Figure 13.It is interesting to observe that all orbits of the equilateral triangle are periodic withperiod or . (See Figure 14 and [8] for the proof.) We do not know of any other no-slipbillard domain all of whose orbits are periodic.Figure 15: Velocity phase portrait of the no-slip billiard on a regular pentagon. Orbits all seemto lie in a stable neighborhood of some periodic orbit.
A final observation concerning polygonal no-slip billiards is suggested by plots of theirvelocity phase portraits. A typical such plot is shown in Figure 15. It is apparent thatthe orbits drawn all seem to lie on a stable neighborhood of some periodic orbit, andthis pattern is seen at all scales that we have explored, but we do not yet have a cleartopological dynamical description of this observation.
We now turn to the problem of characterizing stability of period- orbits for no-slipbilliard domains whose boundary may have non-zero geodesic curvature. Here we onlyaddress linear rather than local stability as we did before for polygonal billiards. Inother words, we limit ourselves to the problem of determining when the differential of thebilliard map dT ξ at a period- collision state ξ = ( q, v ) is elliptic or hyperbolic, and precisethresholds (where it is parabolic). To go from this information to local stability wouldrequire developing a KAM theory for no-slip billiards in the model of [14], something wedo not do in this paper.A simple but key observation is contained in the following lemma. 26 emma 9.1. Let ξ = ( q, v ) be periodic with period for the no-slip billiard map andconsider the differential T ∶= dT ξ ∶ v ⊥ ⊕ v ⊥ → v ⊥ ⊕ v ⊥ . Then either all the eigenvalues of T are real, of the form , , r, / r or, if not all real, they are , , λ, λ where ∣ λ ∣ = .Proof. This is a consequence of the following observations. First, we know that T ∗ Ω = − Ω ,where Ω is the canonical symplectic form (cf. Section 6). Therefore, the product of theeigenvalues of T counted with multiplicity is . The vector ( e , w ) , where e is thefirst vector in the product frame and w is the first vector in the wavefront frame, isan eigenvector for eigenvalue of dT ξ due to rotation symmetry, as already noted. Ifwe regard dT ξ as a self-map of v ⊥ ⊕ v ⊥ as in the corollary to Proposition 5.2 then weshould use instead the vector ( w , w ) . (Recall that w is collinear with the orthogonalprojection of e to v ⊥ .) In addition, by reversibility of T , if λ is an eigenvalue of T ,then / λ is one also, and since T is a real valued linear map, the complex conjugates λ and / λ are also eigenvalues. As the dimension of the linear space is , if one of theeigenvalues, λ , is not real, it must be the case that λ = / λ and we are reduced to thecase , , λ, λ with λλ = . If all eigenvalues are real, and r ≠ is one eigenvalue, then weare reduced to the case , , r, / r . Corollary 9.1.
The period- point ξ is elliptic for T = dT ξ if and only if ∣ Tr ( T ) − ∣ < . To proceed, it is useful to express the differential map of Corollary 5.1 in somewhatdifferent form. First observe, in the period- case (in which ˜ v = − v and v ⊥ = ˜ v ⊥ ), that w ( ξ ) = − w ( ˜ ξ ) and cos ψ ( ˜ q, v ) cos φ ( ˜ q, v ) = cos ψ ( ˜ ξ ) cos φ ( ˜ ξ ) = cos ψ ( ξ ) cos φ ( ξ ) . (See Section 4.) Now define the rank- operator Θ ˜ ξ ( Z ) ∶= ( β / ) cos ψ ( ˜ ξ ) cos φ ( ˜ ξ ) ⟨ w ( ˜ ξ ) , Z ⟩ u ( ˜ q ) . Then(9.1) dT ξ ( XY ) = ( I tI − κ ( ˜ q ) Θ ˜ ξ C ˜ q − tκ ( ˜ q ) Θ ˜ ξ ) ( XY ) . When the geodesic curvature satisfies κ ( q ) = κ ( ˜ q ) we obtain a simplification in thecriterion for ellipticity, as will be seen shortly. With this special case in mind we definethe linear map R ξ on v ⊥ by R ξ w i ( ξ ) = −(− ) i w i ( ξ ) , i = , . Notice that Ru ( q ) = u ( ˜ q ) .Then R ˜ ξ C ˜ q = C q R ξ , R ˜ x Θ ˜ ξ = Θ ξ R ξ . The same notation R ξ will be used for the map on v ⊥ ⊕ v ⊥ given by ( z , z ) ↦ ( R ξ z , R ξ z ) .Notice that R ∶= R ξ = R ˜ ξ since w i ( ˜ ξ ) = −(− ) i w i ( ξ ) . It follows that(9.2) RdT ξ R = ( I tI − κ ( ˜ q ) Θ ξ C q − tκ ( ˜ q ) Θ ξ ) .
27n particular, when κ ( q ) = κ ( ˜ q ) , we have RdT ξ R = dT ˜ ξ and dT ξ = ( RdT ξ ) . Therefore,rather than computing the trace of dT ξ , we need only consider the easier to computetrace of RdT ξ . A straightforward calculation gives the trace of these maps, which werecord in the next lemma. Lemma 9.2.
Let ξ = ( q, v ) have period and set ˜ ξ ∶= T ( ξ ) , C ∶= C q , Θ ∶= Θ q . Then Tr ( dT ξ ) = Tr { I + ( CR ) − t ( κ ( q ) + κ ( ˜ q )) [ Θ + ( CR )( Θ R )] + t κ ( q ) κ ( ˜ q )( Θ R ) } . When κ ∶= κ ( ˜ q ) = κ ( q ) , we have Tr ( RdT ξ ) = Tr ( CR + tκ Θ ) . Proof.
These expressions follow easily given the above definitions and notations.These traces can now be computed using Equations (4.1) and (4.2). The matricesexpressing
C, R, Θ in the wavefront basis of v ⊥ are given as follows. For convenience wewrite c ∶= cos ( β / ) , c φ ∶= cos φ, c ψ ∶= cos ψ, (cid:37) ∶= √ − cos ( β / ) cos φ, where φ = φ ( ξ ) and ψ = ψ ( ξ ) are defined in Corollary 5.1. [ C ] w = ( − c c φ − cc φ (cid:37) − cc φ (cid:37) − + c c φ ) , [ R ] = ( − ) , [ Θ ] w = c c ψ c φ ( (cid:37) − cc φ ) . Let ¯ d be the distance between the projections of q and ˜ q on plane the billiard table, v the projection of v on the same plane and t , as before, the time between consecutivecollisions. From cos ψ = sin ( β / ) cos φ /√ − cos ( β ) cos φ it follows that t cos ψ = cos φ ¯ d .We then obtain(9.3) Tr ( RdT ξ ) = Tr ( CR ) + tκ Tr ( Θ ) = [ − ( β / ) cos φ ] − κ ¯ d cos ( β / ) cos φ and Tr ( dT ξ ) = {[ − ( β / ) cos φ ] − ( κ ( q ) + κ ( ˜ q )) cos ( β / ) cos φ [ − ( β / ) cos φ ] ¯ d + κ ( q ) κ ( ˜ q ) cos ( β / ) cos φ ¯ d } (9.4)Observe that in the special case in which κ ( q ) = κ ( ˜ q ) we haveTr ( dT ξ ) = { [ − ( β / ) cos φ ] − κ cos ( β / ) cos φ ¯ d } Theorem 9.1.
Suppose that the billiard domain has a piecewise smooth boundary withat least one corner having inner angle less than π . Then, arbitrarily close to that cornerpoint, the no-slip billiard has (linearly) elliptic period- orbits.Proof. Period- orbits exist arbitrarily close to the corners of a piecewise smooth billiarddomain as Figure 16 makes clear. For period- orbits near a corner the above expressionfor Tr ( dT ξ ) gives for small ¯ d < Tr ( dT ξ ) = [ − ( β / ) cos φ ] + O ( ¯ d ) < . ∣ Tr ( dT ξ ) − ∣ < and the theorem follows from Corollary 9.1.Theorem 9.1 (and numerical experiments) strongly suggests that such no-slip billiardswill aways admit small invariant open sets and thus cannot be ergodic with respect tothe canonical billiard measure.Figure 16: For a billiard domain with piecewise smooth boundary, arbitrarily near any cornerwith inner angle less than π , there are linearly stable period- orbits. We illustrate numerically the transition between elliptic and hyperbolic in the specialcase of equal curvatures at q and ˜ q . Define ζ ∶= κ ¯ d . When ζ > (equivalently, thecurvature is positive), the critical value of ζ is ζ = − ( β / ) cos φ cos ( β / ) cos φ . The condition for the periodic point to be elliptic is ζ > ζ . When ζ < , the critical valueof ζ is ζ = − φ and the condition for ellipticity is ∣ ζ ∣ < ∣ ζ ∣ . Figure 17:
Velocity phase portraits of single orbits near the periodic orbit of the no-slip Sinaibilliard corresponding to φ = . The mass distribution is uniform. The numbers arethe radius of the circular scatterer. Consider the example of the no-slip Sinai billiard. (See Figures 4 and 7.) We examinesmall perturbations of the periodic orbit corresponding to the angle φ = . Figure 17suggests a transition from chaotic to more regular type of behavior for a radius between . and . . In reality the critical radius for the φ = periodic orbits is exactly29 / . So the observed numbers are smaller. We should bear in mind that the periodicpoints are not isolated, but are part of a family parametrized by φ . As φ increases, thecritical parameter ζ changes (for the uniform mass distribution, where cos ( β / ) = / )according to the expression ζ = ( − cos φ )/ cos φ . Given in terms of the radius ofcurvature, ζ = ( − R cos φ )/ R . Solving for the critical R yields R = cos φ . Thus for aperiod- trajectory having a small but non-zero φ , the critical radius is less than / . Itis then to be expected that the experimental critical value of R , for orbits closed to thathaving φ = will give numbers close to but less than / . Moreover, as R approaches when φ approaches π / , we obtain the following proposition which, together withexperimental evidence indicates that the no-slip Sinai billiard is not ergodic.Figure 18: On the left: velocity phase portrait of the no-slip Sinai billiard with scatterer radius R = . . Since this is greater than the transition value R = / , the period- orbitsparametrized by φ are all elliptic. On the right, R = . and ellipticity has beendestroyed for orbits with smaller values of φ . No matter how small R is, ellipticorbits always exist for φ sufficiently close to π / . Proposition 9.3.
The no-slip Sinai billiard, for any choice of scatterer curvature, willcontain (linearly) elliptic periodic trajectories of period . As another example, consider the family of billiard regions bounded by two symmetricarcs of circle depicted in Figure 19.In this case, the critical transition from hyperbolic to elliptic, for the horizontal periodicorbit at middle height shown in the figure, happens for the disc. A transition behaviorsimilar to that observed for the Sinai billiard seems to occur near the period- orbitsshown in Figure 19. The number indicated below each velocity phase portrait is the angleof each circular arc. Thus, for example, the disc corresponds to angle π ; smaller anglesgive shapes like that on the left in Figure 19. The cut-off angle at which the indicatedperiodic orbit becomes elliptic is π . Notice, however, that the experimental value for thisangle is greater than π . Just as in the Sinai billiard example, we should keep in mind30 lliptic parabolic hyperbolic Figure 19:
Family of focusing no-slip billiards. that the periodic orbits are not isolated; in this case, the bias would be towards greatervalues of the angle.Figure 20:
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