Numerical evaluation of the escape time of a classical point particle from an annular billiard

Estimação numérica do tempo de escape de uma partícula puntual clássica em um bilhar anular

ARTIGOS GERAIS

Numerical evaluation of the escape time of a classical point particle from an annular billiard

Estimação numérica do tempo de escape de uma partícula puntual clássica em um bilhar anular

R. De Luca¹

Dipartimento di Matematica e Informatica, Università degli Studi di Salerno, Fisciano, SA, Italy

ABSTRACT

A two-dimensional annular billiard consisting of a region confined within two concentric circumferences, the outer of radius R and the inner of radius r, is considered. The escape time of a point particle projected at a given angle within this billiard is numerically evaluated in terms of the size of the opening in the billiard. The problem is solved by means of classical mechanics and can be of interest for advanced high school physics students or undergraduate college physics students.

Keywords: classical mechanics, chaos, computational physics.

RESUMO

Consideramos neste trabalho um bilhar bidimensional que consiste em uma região circunscrita por duas circunferências concêntricas de raio externo R e interno r, respectivamente. O tempo de escape de uma partícula projetada por um certo ângulo dentro deste sistema é calculada numericamente em termos do tamanho da abertura no bilhar. O problema é resolvido usando mecânica clássica em um nível que pode ser apresentado a estudantes do ensino médio e graduandos em física.

Palavras-chave: mecânica clássica, caos, física computacional.

1. Defining the problem

The study of reflections of point particles within a billiard of a given shape with perfectly reflecting walls is a well known topic in the literature [1-3]. Here we consider a billiard whose walls are two reflecting concentric circumferences; i.e., an annular billiard. Our interest in this particular shape is given by the possibility of tracing a parallel between the escape time of the point particle moving at a constant speed V within the billiard and the analogous problem of a light ray "trapped" within systems of similar shape [4].

The main question we would like to address in the present work is the following. In case we project an incoming point particle from an aperture in the billiard a given angle θ with respect to the horizontal, after what interval of time will the same particle emerge out of the billiard, assuming that the aperture has a finite size? This question is well posed, since, as we shall see, one could consider two cases. In the first case, for angles θ greater than some critical value θ_c, the system behaves as if the inner circumference were not present, since the particle never hits its wall. The problem becomes thus similar to the well-known topic of circular billiards. In the second case, on the other hand, when collisions with the inner wall are allowed (0 < θ < θ_c), either a periodic or non-periodic trajectory is followed by the point particle in case the aperture is point-like. Of course, in case the aperture has finite dimensions, there is a finite escape time and the trajectories in the configuration space cannot in general be classified in the same way.

In the case the point particle goes through a closed trajectory, it comes to the starting point after a given number of reflections, and the escape time is easily calculated even in the presence of a point-like opening. In the other case, when a non-periodic trajectory is followed, the particle will eventually come close enough to the starting point in such a way that it will escape from the area enclosed within the concentric circumferences if the intake has finite dimensions. In the latter case, a computer algorithm can allow numerical solution of the problem.

2. Some graphical and analytic notes

Let us start by defining the radius of the inner and outer circumferences as r and R, respectively. Let also assume that the point particle enters the outer circumference of center O at P₀, as shown in Fig. 1, at an angle θ with respect to the horizontal. If we were just to consider the particle's trajectory inside a single circumference, as shown in Fig. 1, then the path followed is obtained in the following simple way: First draw the segment P₀P₁; then draw segments P_n-1P_n all tangent to the circumference of radius d equal to the distance between the point O and the segment P₀ P₁. This procedure gives the starred pattern in Fig. 1.

We would now like to discuss the above construction from an analytic point of view. A representation of segment P₀ P₁ can be given in terms of the difference of two complex numbers, ρ₀ and ρ₁ (see Fig. 1), in such a way that

where ø₀ and ø₁ are the angles the vectors ρ₀ e ρ₁, represented on the complex plane, make with the horizontal. If we now set ø = ø₁ - ø₀, we can write

where k is an integer. The segment P_kP_k-1 will thus coincide with P₀P₁ if the following condition is satisfied after k iterations of the procedure described above

s being an integer. Eq. (3) can be written as follows

by setting k - 1= m and s = n - 2, with n > 2 and m >1 integers. Notice that for m = 2 and n = 3 we have φ = π, so that the particle enters at ø₀ = 0, bounces once at ø1 = π, and then leaves the circumference at ø₂ =2π. In this case, the quantity t_E = , V being the particle's speed, is the minimum escape time from a circumference of radius R. In the case n = 3 and m = 3, 4, 5, . . . we obtain regular polygons of perimeter pm with 3, 4, 5 . . . sides, respectively, all inscribed in the circumference of radius R. In this cases, the escape time is t_E = .

In the case of non-periodic orbits, on the other hand, condition (3) is not satisfied and the point particle is seen to wander within the annular region of outer radius R and inner radius d, never coming back to the starting point P₀.

3. Escape time

The purpose of the present work is to find the escape time for a point particle confined within two concentric circumferences, whose trajectory is schematically shown in Fig. 2. The complex representation of the trajectory, as done in the case of a single circumference, is useful also in this case. Let us then start by considering a particle entering the region of interest for our analysis from P₀ as shown in Fig. 2. The partial trajectory P₀Q₀P₁ is a broken line, whose characteristics are shown in Fig. 2, if the angle θ (0 < θ < ) at which the particle enters the annular region is such that θ < tan ^-1 = θ_c. In the case tan^-1< θ < , the particle moves as if the inner circumference were not present. Notice that the critical angle α_c is complementary to θ_c, so that α_c = tan^-1 . The relation between the angles α and θ can be found by trigonometry to be

where x = .

We shall not consider cases similar to those treated in the previous section, and will thus assume θ < θ_c. We therefore indicate with P_k (k = 0, 1, 2,...) and Q_k (k = g, 1, 2,...) the points on which the particle impinges on the outer and inner circumference, respectively. Therefore, we may represent segments Q₀ P₀ and P₁ Q ₀ by means of the three following vectors ρ₀ = R, σ0 = re^iα and ρ1 = Re^2iσ, where α is the angle between ρ₀ and σ₀, as follows

In general, we may thus write

for k = g, 1, 2,. . . In Eqs. (7a-b), of course, we have set ρ_k = Re^i2kα and σ_k = r e^i(2k+1)α. By now imposing that Q_kP_k = Q₀P₀, or, equivalently, P_k+1Q_k = P₁Q₀, in order to have closure of the trajectory after exactly k mechanical reflections on the inner circumference, we have e^i2kα = 1, so that

where N is an integer. Notice that Eq. (10) is only formally similar to Eq. (4). Here, in fact, one should kheep in mind that the angle α must lie in the range [0, tan^-1) and that it is different, in definition, from the angle φ in Eq. (4). The dependence of the critical angle α_c = tan^-1 on the positive ratio x = < 1 is given in Fig. 3, where it can be seen that α_c is a monotonously decreasing function of x. In this way, not all possible choices of k and N in Eq. (10) represents a path defined in Eqs. (7a-b). In summary, we may state that if the indices k and N satisfy the following condition

then the escape time can be calculated in terms of the angle α as follows

where V is the constant speed of the particle. Eq. (5) can be used to relate the escape time t_E to θ. The above relation can be used, more generally, as we shall see in the following section, even when α ≠ .

4. General behavior of the system

Up to this point we have not considered the possibility of having a point particle in P₀ with an initial velocity making, with respect to the horizontal, an angle α being an irrational multiple of π. Therefore, in the present section we shall consider also cases in which α ≠ π, N and k being positive integers and the angle α is such that 0 < tan α < . in this case the point particle will suffer reflection on both circular walls and will never go back to its starting point, even though it can, after a certain time, get close enough to it. The particle will thus be able to exit from the annular region in which it is temporarily confined if the opening has finite height ε, as shown in Fig. 4. In this way, a finite escape time t_E will still exist for the system. Of course, in this case the quantity t_E will go to infinity as the dimension ε of the opening goes to zero and, in the absence of dissipation, the system will realize a non-periodic perpetual motion within the annular region. In what follows we shall give a numerical estimate of the escape time in the case the particle wanders within the region in between the two concentric circumferences and, at the same time, the dimension of the opening is assumed to be finite.

By considering the inset in Fig. 4, where we set α_k = 2kα, we see that the condition for which the light ray might escape after k reflections is simply the following

provided cos (2kα) > 0. The above relation can readily be used in a numerical algorithm, so that the escape time may be computed for varying values of θ, recalling Eq. (5), in the interval 0 < θ < θ_c. In Fig. 5 we report a numerical evaluation of an averaged escape time T. In order to find T, we start from a list of values of t_E as given by Eq. (12), found by a numerical algorithm based on condition (13) for various values of the dimensions of the opening , choosing = 0.75 and V = 10 . We then average the outcome of the escape times calculated for specific values of the angle θ over an angular range of about Δθ = 1.21 degrees.

This averaging process is necessary in order to show, in an experimentally significant fashion, the outcome of our analysis, given the highly scattered nature of plots obtained when evaluation of t_E is performed. A large amount (10.000 in our case) of t_E values in the angular range 0 < θ < θ_c are then grouped in such a way to form intervals of amplitude Δθ =1.21degrees. Considering the outcome of this averaging process, in Fig. 5 we notice that the quantity T lies below the value of 200 s for most projection angles θ for the choice of parameters given. However, as the angle θ increases, the average escape time rises in the vicinity of θ_c. Naturally, the obvious result that for increasing values of the dimension of the opening the escape times are, on the average, smaller for a given angular range, is confirmed by the present simple analysis.

5. Conclusions

We have considered the problem of the motion of a classical point particle confined within an annular region with perfectly reflecting walls. Assuming finite dimensions of the hole from which the point particle is projected within the billiard with speed V at an angle θ with respect to the horizontal, we evaluate, numerically, by means of a rather simple algorithm, the escape time from the billiard. The escape time is seen to increase, on the average, with increasing values of the projecting angle θ. The analysis of the problem can be proposed to advanced high school students or undergraduate college students. Future work will involve light propagation within a region enclosed between two coaxial cylinders, the external one presenting a circular opening from which a light ray may enter or exit.

Acknowledgements

The author gratefully acknowledges pleasant and fruitful discussions with A. Fedullo and F. Romeo.

Recebido em 28/10/2009; Aceito em 5/1/2010; Publicado em 15/2/2011

• [1] M.V. Berry, Eur. J. Phys. 2, 91 (1981).
• [2] R.W. Robinett, Am. J. Phys. 67, 67 (1999).
• [3] M.A. Bandres and J.C. Gutiérrez-Vega, Am. J. Phys. 72, 810 (2004).
• [4] R. De Luca, F. Romeo and P. Zozzaro, Eur. J. Phys. 27, 347 (2006).

Publication Dates

• Publication in this collection
  15 Mar 2011
• Date of issue
  Sept 2010

History

• Received
  28 Oct 2009
• Reviewed
  05 Jan 2010
• Accepted
  15 Feb 2011