Billiards & Ballyards

In (mathematical) billiards, the ball travels in a straight line between impacts with the boundary, when it changes suddenly and discontinuously We can approximate the hard-edged, flat-bedded billiard by a smooth sloping surface, that we call a “ballyard”. Then the continuous dynamics of the ballyard approach the motions on a billiard.

Elliptical tray in the form of a Ballyard.

Elliptical Billiards

We idealize the game of billiards, assuming the ball is a point mass moving at constant velocity between elastic impacts with the boundary, or cushion, of the billiard table. The energy is taken to be constant.

Each bounce is a specular reflection: the angle between the normal to the cushion and the incoming trajectory is equal to the corresponding angle for the outgoing trajectory.

We assume that the table is elliptical, with the boundary is described by the equation

$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

The foci are at ${(f,0)}$ and ${(-f,0)}$ where ${f^2 = a^2 - b^2}$ . it is well-known that the motion on an elliptic billiard is completely integrable. Each linear segment of the trajectory is tangent to another conic confocal with the boundary, called the caustic of the orbit.

Box Orbits & Loop Orbits

There are two generic types of orbit. The first arises when the ball crosses the major axis between the foci, the second when it crosses outside (see figure).

Left: Box orbit, crossing between the foci. Right: Loop orbit, crossing outside the foci.

The two generic families of orbits are separated by the homoclinic orbit, for which the ball passes through a focus. This orbit rapidly approaches horizontal motion back and forth along the major axis [See earlier post on Boxes and Loops].

Since the system has no dissipation and since energy is conserved at boundary impacts, the kinetic energy is a constant of the motion.

For an elliptical table, the angular momentum about the centre is not conserved. However, there is another conserved quantity. It is convenient to use elliptic coordinates ${(\xi,\eta)}$ :

Elliptic coordinates ${(\xi,\eta)}$ .

$\displaystyle x = f \cosh\xi \cos\eta \,,\qquad y = f \sinh\xi \sin\eta \,. \nonumber \ \ \ \ \$

Then it is easy to show that

$\displaystyle L_1 L_2 = f^4(\cosh^2\xi-\cos^2\eta)\bigl[(-\sin^2\eta)\dot\xi^2+(\sinh^2\xi)\dot\eta^2\bigr] \,.$

where ${L_1}$ and ${L_2}$ are the angular momenta about the foci. It turns out that ${L_1 L_2}$ is a constant of the motion, which also serves as a discriminant for the nature of the orbit:

$\displaystyle \begin{cases} \mbox{Box type} &\mbox{if\ \ } L_1 L_2 < 0 \\ \mbox{Homoclinic} &\mbox{if\ \ } L_1 L_2 = 0 \\ \mbox{Loop type } &\mbox{if\ \ } L_1 L_2 > 0. \end{cases}$

Ballyards

In 1848 Joseph Liouville identified a broad class of dynamical systems that can be integrated (see Whittaker, 1937, §43). If the kinetic and potential energies can be expressed in the form

$\displaystyle T = \frac{1}{2}({\cal U}_1(q_1)+{\cal U}_2(q_2))[{\cal V}_1(q_1)\dot q_1^2 + {\cal V}_2(q_2)\dot q_2^2] \qquad V = \displaystyle{ \frac{{\cal W}_1(q_1)+{\cal W}_2(q_2)}{{\cal U}_1(q_1)+{\cal U}_2(q_2)} } \,.$

then the solution can be solved in quadratures.

The kinetic energy term is already of the required form with

$\displaystyle {\cal U}_1(\xi) = f^2\cosh^2\xi \, \qquad {\cal U}_2(\eta) = -f^2\cos^2\eta \, \qquad {\cal V}_1 \equiv 1 \qquad {\cal V}_2 \equiv 1 \,.$

We seek a potential surface that is close to constant within the elliptical region defined by ${(x/a)^2+(y/b)^2=1}$ and that rises rapidly in a boundary zone. We define the potential surfaces by setting

$\displaystyle {\cal W}_1(\xi) = V_N f^2 \cosh^N\xi \qquad {\cal W}_2(\eta) = - V_N f^2 \cos^N\eta$

where ${V_N = V_0 /(\cosh^{N}\xi_B) }$ with ${V_0}$ a constant, ${\xi_B}$ the value of ${\xi}$ defining the reference ellipse ( ${\cosh\xi_B=a/f=1/e}$ ) and ${N}$ an even integer.

The potential energy function is then

$\displaystyle V(\xi,\eta) = \frac{{\cal W}_1(\xi)+{\cal W}_2(\eta)}{{\cal U}_1(\xi)+{\cal U}_2(\eta)} = V_N \left[ \frac{\cosh^{N}\xi - \cos^{N}\eta}{\cosh^2\xi-\cos^2\eta} \right] \,.$

which is of the form required for the system to be of Liouville type and thus integrable. Two examples of the potential energy surface are shown in the figure.

Potential energy surfaces for N=6 and N=32.

For the case ${N=6}$ :

$\displaystyle {\cal W} = V_6 f^2 (\cosh^6\xi-\cos^6\eta) \\ \quad = V_6 f^2 (\cosh^2\xi-\cos^2\eta) (\cosh^4\xi+\cosh^2\xi\cos^2\eta+\cos^4\eta) \,.$

The Angular Momentum Integral

For the billiard dynamics, the product of the angular momenta about the foci, ${L_1 L_2}$ , is constant. We have found a corresponding integral for the ballyard (Lynch 2019). If we define the quantity

$\displaystyle \Lambda[\xi,\eta] = \frac{2f^2(\sinh^2\xi\ {\cal W}_2 - \sin^2\eta\ {\cal W}_1)}{\cal U}$

then we get

$\displaystyle \mathbb{L} \equiv L_1 L_2 + \Lambda = 2f^2 E + 2\gamma \,$

and $\mathbb{L}$ is an integral of the motion.

Numerical solutions for N = 6. Left: Box orbit. Right: Loop orbit.

Examples of box and loop orbits on a ballyard are shown above.

The function ${\Lambda(\xi,\eta)}$ is constant on the major axis ( ${y=0}$ ). Therefore, ${L_1 L_2}$ is also constant there. This means that the orbits fall into boxes and loops. The integral $\mathbb{L}$ corresponds in the limit ${N\rightarrow\infty}$ to ${L_1 L_2}$ , the quantity conserved for motion on a billiard.

Sources

Lynch, Peter, 2019: Integrable Elliptical Billiards & Ballyards. Submitted to Eur. J. Phys. PDF on arXiv.

Lynch, Peter, 2019: Integrable Elliptical Billiards & Ballyards. Seminar at Imperial College London, 24 July 2019. PDF of presentation.

Whittaker, E.T., 1937 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. 4th Edn., Cambridge Univ. Press, 456pp.

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