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ISSN 1560-3547, Regular and Chaotic Dynamics, 2011, Vol. 16, No. 6, pp. 653­662. c Pleiades Publishing, Ltd., 2011.

On the Mo del of Non-holonomic Billiard
Alexey V. Borisov* , Alexander A. Kilin** , and Ivan S. Mamaev*
Institute of Computer Science, Udmurt State University ul. Universitetskaya 1, Izhevsk 426034, Russia
Received Novemb er 6, 2010; accepted July 14, 2011
**

Abstract--In this paper we develop a new model of non-holonomic billiard that accounts for the intrinsic rotation of the billiard ball. This model is a limit case of the problem of rolling without slipping of a ball without slipping over a quadric surface. The billiards between two parallel walls and inside a circle are studied in detail. Using the three-dimensional-point-map technique, the non-integrability of the non-holonomic billiard within an ellipse is shown. MSC2010 numbers: 34D20, 70E40, 37J35 DOI: 10.1134/S1560354711060062 Keywords: billiard, impact, point map, nonintegrability, periodic solution, nonholonomic constraint, integral of motion

1. INTRODUCTION The game of billiards has b een known since high antiquity and still remains extraordinarily p opular today. It has long since attracted the attention of researchers, mathematicians and technicians who try to describ e the physics of b ouncing balls and their unusual b ehavior, in particular to develop winning techniques for the game. A great deal of effort has gone into developing the theory of the game of billiards that describ es the interaction b etween balls and b etween a ball and a wall. However, this problem is very complicated. Unfortunately, there exists no unified theory that would explain all asp ects of this game. One of the first researchers to study the theory of the game of billiards was G. Coriolis, whose results are presented in his famous b ook "Th´orie math´matique des effets du jeu billard" [3]. e e Among the classical works on the theory of billiards that are worth mentioning is the pap er of H. Resal [12], in which he enters into a controversy with Coriolis concerning a numb er of additional hyp otheses necessary to describ e collisions of balls or the impact of a ball against a wall, the work of P. App ell [1] on the motion of a billiard-ball with friction taken into consideration and a systematic b ook of G. W. Hemming [9]. The substantiation of the theory of the game of billiards and the investigation of numerous physical phenomena observed in this game entails analysis of the main features of general impact theory. Nowadays the impact theory (see [16, 18]) is a separate discipline and has several branches, such as stereomechanical and wave impact theory etc. However, a full-blown theoretical impact model does not exist. Therefore, when solving concrete problems some additional hyp otheses are needed. One of the classical hyp otheses most widely adopted today is the hyp othesis of the coefficient of restitution, which was first prop osed by Newton. Without pretending to give an exhaustive list, we refer to several theoretical studies on the impact theory [8, 10, 11, 21, 22] and mention a numb er of exp erimental investigations that describ e b oth the methodology and results of sp ecific exp eriments on the collision of various b odies [4­7, 24]. In addition, many results can also b e found in practical manuals on the game of billiards [23].
* ** ***

E-mail: borisov@rcd.ru E-mail: aka@rcd.ru E-mail: mamaev@rcd.ru

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In this article we develop a formal approach to deriving the laws of billiards by passage to the limit from some more general problem [20]. As is well known, there exists a mathematical model of the game of billiards (often referred to as Birkhoff 's bil liard) and there is quite an amount of literature on this model (for a review see, e.g., [14, 17]). Within this model the reflection of a material p oint from a flat curve ob eys the law "the angle of incidence is equal to the angle of reflection". Originally this model was prop osed by Birkhoff who used passage to the limit from the problem of motion of a material p oint on the surface of an ellipsoid. Although this model b ears the name "billiard", it does not capture many features of this game. In particular, it does not incorp orate the ball's self-rotation. One of the goals of this pap er is, therefore, to derive a law of reflection that takes the ball's self-rotation into account. We refer the readers to [19, 20], which develop the impact model based on the "impact on a constraint" hyp othesis, where the velocity of the p oint of contact reverses direction at the moment of the impact. Here we develop another model of the game that accounts for the ball's rotation. Following Birkhoff, we also resort to passage to the limit choosing a ball rolling on a surface without slipping as the "parent system". This "parent" problem was examined in detail in [2]. The resulting billiard model, which we call a nonholonomic bil liard, inherits the conservation laws of the parent problem. In particular, it is conservative and preserves the normal comp onent of the ball's velocity. On the one hand, we will prove a few theorems on passage to the limit and obtain a new model of mathematical billiard, on the other hand, we will describ e a class of billiards that are isomorphic to the new model and are in good agreement with the well-known physical exp eriments. Thereby we p ostulate a more complicated, but more adequate billiard model incorp orating the ball's self-rotation. We also note that while the study of the Birkhoff billiard is traditionally based on the analysis of an appropriate two-dimensional p oint map, the model we consider allows an adequate representation only via a three-dimensional map. In some cases (e.g., billiard in an ellipse) the existence of additional invariants makes it p ossible to reduce the order of the system and therefore deal with a two-dimensional p oint map on a surface. A more detailed discussion of methods for construction of three-dimensional maps and maps on surfaces, as applied to nonholonomic dynamical systems, can b e found in [2]. We conclude by p ointing out one more asp ect of our nonholonomic-billiard model: this model can also b e regarded as another discretization of the problem of a slipping-free rolling motion of a ball on a surface [13]. In contrast to most other p opular discretizations, which are formal and do not seem to have any reasonable physical grounds, the model we prop ose has a clear physical sense. 2. A NONHOLONOMIC BILLIARD IN A STRIP 2.1. Passage to the Limit Consider the problem of a ball rolling on a cylinder without slipping. The equations of motion for this problem can b e written in the following form [2]: v = -(v , ) + u â , (2.1) u = ( â , v ), x = v.

Here x and v are the coordinates and the velocities of the ball's center of mass, u = R( , ) is the pro jection of the angular velocity of the ball (hereinafter referred to as the spin) onto the normal to the surface which is given by the Gauss map =- F (x) , |F (x)| (2.2)

I where F (x) is the equation of the surface and = I +mR2 is a constant coefficient that dep ends on the mass distribution within the ball and assumes values from 0 to 2/5. We consider an elliptical

cylinder and therefore F (x) = and Eq. (2.2) b ecomes

x2 1 b2 1

+

x2 2 b2 2

- 1, where b1 and b2 are the semi-axes of the cylinder's base, A = diag(b
-2 1

=-

Ax , |Ax|

, b-2 , 0). 2
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Eqs. (2.1) admit the following first integrals of motion: (x, Ax) = 1, (v , Ax) = 0, 12 12 2 H2 = (v3 + u2 ), H1 = (v1 + v2 ), 2 2 (v , Av ) 2 b, where B = diag(b2 , b2 , 0). K= 12 ( , B ) 2

(2.4)

The first two integrals are geometric, the integrals H1 and H2 represent two indep endent parts of the energy integral H = H1 + H2 , and the integral K is an additional quadratic indep endent integral, the most general form of which was found in [2]. Now consider the passage to the limit in Eqs. (2.1) when one of the semi-axes of the cylinder's base tends to zero b2 0. As a result we obtain the problem of the motion of a ball rolling without slipping over b oth sides of the strip enclosed b etween two straight lines. It is straightforward to show that the motion of the ball inside the strip is uniform and rectilinear and that the spin u remains unchanged. While the ball is approaching the b oundary, it instantaneously rolls over to the reverse of the strip. When pro jected onto a plane this process can b e viewed as the ball's impact against a wall. Thus we obtain a new model of the nonholonomic billiard. We now consider in greater detail the process of passage to the limit and the resulting law of impact. It turns out that the following theorem holds. Theorem 1. Let the bal l rol l without slipping on an el liptic cylinder. Then if one of the semi-axes of the cylinder's base tends to zero (b2 0) we get the problem on nonholonomic bil liard inside a flat strip with the fol lowing law of reflection 0 0 -1 = 0 cos - sin 0 - sin - cos

V where V = (vn , v , ua ).

+

- V ,

(2.5)

Proof. First derive a law of reflection for the normal comp onent of velocity v . To do this, introduce the variable x1 = b1 cos , x2 = b2 sin . (2.6) Substituting (2.6) into the expression for the integral K and using the geometrical constraint (v , Ax) = 0, we obtain K = (b2 sin2 + b2 cos2 ) 1 2
v
2

2 v1 . b4 sin2 1

(2.7)

1 As b2 0 we get K = b2 . Thus, at the moment of impact, the conditions of conservation of the 1 integral K imply that the normal velocity of the ball remains unchanged, i. e. + - ± (2.8) v1 = -v1 , |v1 | = b1 K .

To establish the law of reflection for u and v3 we introduce the variable as follows v3 = 2H2 cos , u= 2H2 sin . (2.9)

Choosing as a new time, we write the evolution equation in : d b1 b2 =- 2 2 . d b1 sin + b2 cos2 2
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Putting x = b1 tg and using the representation of the -function b2 , (x) = lim 2 + b2 ) b2 0 (x 2 we obtain in the limit b2 0 Making an inverse change gives d = - (x). dx = -s (t - t0 ),

(2.11)

(2.12)

(2.13)

where V = (v1 , v3 , u). Notice that the law (2.15) is written in terms of pro jections onto the absolute axes x1 and x2 and the normal vector . However, at the impact against a wall the vector reverses direction. Therefore, instead of the pro jection u of the angular velocity onto the vector it is necessary to consider the pro jection ua of the angular velocity onto some constant vector a . The relation b etween ua and u is given by the relation ua = u, > 0, -u, < 0. (2.16)

where t0 is the moment of impact against a wall and s = sign( ) determines the direction of the ball's rolling motion on the cylinder; moreover s is constant along each tra jectory. From (2.13) it follows that the law of reflection for reads: (2.14) + = - - s . Inverting (2.9) and introducing the variable u = u, we arrive at the law of reflection 0 0 -1 + (2.15) V = 0 cos s sin V - , 0 -s sin cos

In addition, the law of impact is traditionally written in terms of pro jections onto local coordinate axes, which determine the normal and tangential comp onent of the velocity at the impact p oint. These pro jections are related to the pro jections on the axes of the absolute coordinate system in the following way: vn = v = v1 , = 0, -v1 , = , v3 , = 0, -v3 , = .

(2.17)

In terms of the ab ove-mentioned pro jections the law of reflection takes the form 0 0 -1 V + = 0 cos - sin V - , 0 - sin - cos where V = (vn , v , ua ).

(2.18)

2.2. The Analysis of Dynamics Let us consider the prop erties of the law (2.18). As an example we use the limit problem of a billiard in a strip. In this case it is more simple to present an analysis in pro jections onto the
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absolute axes v1 , v3 , therefore, we will consider the law of reflection in the form (2.15). It is not hard to notice that on the plane (v3 , u) th e law of reflection (2.15) represents a rotation by an angle = (Fig. 1). Thus, after successive reflections from the walls the corresp onding p oints on the plane (v3 , u) will b e evenly spaced in angle on the 2 circumference given by the integral H2 = 1 (v3 + u) and the angular 2 spacing b etween the neighb ors will b e = (Fig. 1). Using this prop erty one can prove a theorem that parallels the famous theorem which states that the average vertical departure of a ball rolling on a cylinder is zero (Stubler's problem). ¨ Theorem 2. The nonholonomic bil liard in the strip is bounded, and the maximum departure along the strip is zm
(n) ax

Fig. 1

=

2 2H2 . K sin 2

Proof. Let v3 b e the velocity along the strip on the n-th step. It is easy to obtain an explicit expression from the law of impact (n) (2.19) v3 = 2H2 cos(0 + n ), where 0 and H2 define the nonholonomic velocity v3 and the spin u via (2.9). The b oundedness of the ball's motion along the strip follows from the fact that in the Fourier (n) expansion of v3 the zero mode (the term indep endent of n) vanishes. Let us now estimate the maximum p ossible departure of the tra jectory along the strip. The time b etween two successive impacts against the opp osite walls is constant and given by the formula 2b1 2 t= (2.20) = . |v1 | K The total shift along the strip in N impacts can b e written as N -1 1 2 2H2 2H2 sin(0 + (N - 2 )) - sin(0 - zN = cos(0 + n ) = K K n =0 sin 2
2

)

.

(2.21)

From (2.21) it follows that the maximum departure of the ball along the strip is zm
ax

= zm

ax

- zmin =

2H2 2 . K sin 2

3. A NONHOLONOMIC BILLIARD IN A CIRCLE Consider an analogous passage to the limit from the system of a ball rolling without slipping on an ellipsoid of revolution. In the limit we obtain a billiard inside a circle. The original equations of motion for the ball on the ellipsoid coincide with Eq. (2.1), however, the equation of surface now reads x2 x2 x2 3 1 + 2 + 2 - 1. (3.1) R2 R2 b Note that the expression (2.3) for the normal vector and the first two integrals (2.4) remain unchanged but now A = diag(R-2 , R-2 , b-2 ) F (x) =
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By virtue of the cylindrical symmetry of the limit problem, it is useful to choose as new variables the pro jections of the ball's velocity onto the orts of the cylindrical coordinate system v1 x2 - v2 x1 v1 x1 + v2 x2 , v = , v3 (3.2) vn = 2 + x2 x1 x2 + x2 2 1 2 and the spherical coordinates x = (R cos cos , R cos sin , b sin ). (3.3) In these variables the first of the integrals (2.4) looks the same and the second integral simplifies to v3 = - ctg vn , (3.4)
b where = R . The equations of motion in vn , v , u, and are 2 2 sin2 v sin v u 2 cos vn vn = + - , R cos (sin2 + 2 cos2 ) R(sin2 + 2 cos2 ) R sin2 (sin2 + 2 cos2 ) uvn vn v - , v = - R cos R sin (sin2 + 2 cos2 ) 2 2 vn v u = (1 - ) cos vn v , =- , =- . 2 2 cos2 ) R sin R cos R sin (sin +

(3.5)

Equations (3.5) admit two first integrals of motion 12 2 2 H = (vn + v + v3 + u2 ), 2 2 2 K = (1 + 2 ctg2 )(vn + v sin2 ).
+ - vn = -vn .

(3.6)

As ab ove, since K is preserved, we obtain in the limit (3.7) To determine the law of reflection for the other variables we choose as a new time and write the equations of motion for v and u v = tg v + u, 2 sin + 2 cos2 (3.8) (1 - 2 ) cos2 u = - v , sin2 + 2 cos2 where the prime denotes differentiation with resp ect to . Equations (3.8) admit an integral of motion of the form 2H - K 2 (3.9) J2 = = v cos2 + u2 . 1 - 2 1 - 2 Using this integral, we reduce the system (3.8) to one differential equation by means of the following change of variables: J 1 - 2 J cos , u= sin . (3.10) u = cos The differential equation in reads 1 - 2 cos =- 2 . sin + 2 cos2




(3.11)

Putting z = sin , = 0)

1-

2

and using the representation of the -function (2.11), we obtain (as d = - (z ) dz
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or, after the inverse change, Here s = sign( ) and s = -1 when passing from the upp er half-sphere ( > 0) to the lower one ( < 0) and s = 1 in the case of the inverse passage. Integrating Eq. (3.13) and making a transformation inverse to (3.10), we get a law of impact that agrees with (2.15). As in the case of a billiard in a strip, we introduce the pro jection ua of the angular velocity onto a constant vector and thus obtain a law of impact that is in full agreement with (2.5). Thus, we have proved the following theorem. Theorem 3. Assume the bal l is rol ling without slipping on an el lipsoid of revolution which is governed by Eq. (3.1), then, as the third semi-axis tends to zero (b 0), we obtain a nonholonomic bil liard with the law of reflection (2.5) inside a circle. 3.1. Construction of a Three-dimensional Map The classical problem of a mathematical billiard is traditionally investigated by means of a twodimensional p oint map. However, we need three-dimensional maps for a prop er treatment of the nonholonomic billiards with the law (2.18) b ecause with the spin taken into consideration the numb er of dimensions is increased by one. As an example we construct such a map for nonholonomic billiard in an ellipse. We use the following variables: [0, ) "-- the angular coordinate of the impact p oint, [- , ] "-- the angle b etween the incident ball velocity and the normal to the 22 b oundary of of the ellipse at the impact p oint (Fig. 2) and the spin ua . = - s(t - t0 ). (3.13)

Fig. 2

The energy value H and the comp onents of the incident velocity are related as follows:
- vn = - v =

2H - u2 cos , a 2H - u2 sin . a

(3.14)

Thus, the coordinates (, , ua ) and H uniquely define the initial conditions for a tra jectory. For a billiard in an ellipse there exists an additional integral K= (b2 sin2 + b2 cos2 ) 2 2 1 vn , b2 b2 12 (3.15)

such that all p oints are mapp ed onto the surface
2 b2 sin2 + b2 cos2 1 (2H - u2 ) cos2 = K. a 2 b2 b1 2

(3.16)

An example of such a map is given in Fig. 3. As can b e seen from the figure, the map contains a chaotic layer, which is indicative of nonintegrability of the nonholonomic billiard in an ellipse. In the case of a billiard in a circle the situation is simplified. The integral K no longer dep ends on . Furthermore, it is straightforward to show that the identity Vk+ = Vk+ 2 is satisfied, where -
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Fig. 3

Fig. 4

k is the iteration numb er. Consequently, the p oints of each tra jectory are alternately made to lie on two parallel straight lines ua = const, = const, with the values of u and b eing related by This prop erty is illustrated with the three-dimensional map depicted in Fig. 4. 4. THE AXIOMATIC APPROACH The model of nonholonomic billiard (2.5) fails to describ e prop erly some effects encountered in real billiards. Therefore, we present a general nonholonomic impact model which is consistent with the conservation laws and captures the effects that the model (2.5) overlooks. We assume the nonholonomic law of impact to b e an arbitrary transformation that does not change the value of the integrals 12 2 H = (vn + v + u2 ), 2 (4.1) 2 K = vn . The most general form of such a law reads -1 0 0 - + V = 0 (4.2) V , A 0 where V = (vn , v , u) and A is a matrix that includes an arbitrary turn and various reflections. The angle of rotation in the nonholonomic setting is determined from the passage to the limit(see, e.g., 2) and is equal to . Thus, the most general form of the matrix A is s1 s3 cos s2 s3 sin A= , (4.3) -s1 s4 sin s2 s4 cos (2H - u2 ) cos2 = R2 K. a

where s1 , . . . , s4 assume the values ±1. It is natural to ask the question: for which values of si the impact model gives the b est approximation to real billiards? To answer this question consider several model examples of a ball b ouncing off of a wall. 1. Effe Stroke (Fig. 5) The initial and final values of the velocities at this impact satisfy the conditions
- v = 0,

u- > 0,

+ v > 0,

u+ > 0.

For the model (4.3) to prop erly describ e this impact in the case of a homogeneous ball I = 2 2 2 5 me ( = 7 ), it is necessary that the equalities s2 · s4 = -1 and s2 · s3 = 1 b e satisfied.
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661

Fig. 5

Fig. 6

2. Stroke without spinning (Fig. 6) The initial and final velocities satisfy the conditions u- = 0,
- v > 0,

u+ > 0,

+ v > 0.

For our model to apply, the equalities s1 s4 = -1 and s1 s3 = -1 must hold. It is straightforward to show that these four conditions cannot b e satisfied simultaneously. Thus, the model we have constructed can describ e only a part of the real effects. We present here the law of a nonholonomic impact, which prop erly describ es the spin stroke and gives a correct direction of the ball's spin after the stroke without spinning -1 0 0 V + = 0 cos sin V -. (4.4) 0 sin - cos 1 Note that the law (4.4) applies only if the distribution of mass is such that cos( ) < 0, i. e. > 4 (e.g. a homogeneous ball). For distributions with < 1 the corresp onding law takes the form 4 0 0 -1 - + V = 0 - cos sin V . (4.5) 0 sin cos The laws (4.4), (4.5) and (2.5) coincide up axial symmetries; therefore, the results on nonintegrability obtained in Section 3 hold also for (4.4) and (4.5). To conclude, we note that it would b e interesting to investigate the p eriodic solutions and their stability for an elliptic nonholonomic billiard and in particular the prop erty of integrability of nonholonomic billiards in p olygons [15, 20]. There are several intriguing allied problems which do not seem to have a direct relationship to the billiard model describ ed ab ove but advance further the model of mathematical billiard. One such a problem in which three-dimensional maps naturally arise is concerned with b ounces of a rigid b ody off a smooth (or absolutely rough) plane. It would b e interesting to study its integrability and existence and stability of p eriodic solutions. We would like to thank A. P. Ivanov and A. P. Markeev for useful comments and discussions during the workshops at the ICS in Izhevsk. Our thanks are due also to V. Dragovi´ for c useful discussions. The work has b een carried out within the framework of the Federal target program "Scientific and scientific-p edagogical p ersonnel of innovative Russia" for 2009­2013. (pro ject code 2009-1.1-111-048-011). The work of A. A. Kilin has b een supp orted by a grant of the President of the Russian Federation for supp ort of young Russian scientists with a Candidate of Science degree (pro ject code MK-8428.2010.1).
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