Franz J. Vesely > Notes> Billiards with Gravity 
 
 
F VESELY COMP PHYS U VIENNA






 
Billiard on a Tilted Table

Franz J. Vesely, CompPhys Vienna U
Nov 2015


   






In Dynamical Billiards the minimum requirement for attaining chaotic behavior is the presence of a dispersive element such as a curved section of the table wall. The basic step leading to chaos is the intersection of the straight particle track with the curved wall portion. Alternatively, the track may be curved, with the table contour a simple polygon. In the simplest case the table could be an equilateral triangle tilted along its base side, such that the action of gravity will render the tracks parabolic.
[Note: Classical billiards in magnetic fields have been investigated early on; see e.g. Robnik & Berry, Phys. A: Math. Gen. 18 (1985) 1361, or M. Aichinger, S. Janecek, and E. Räsänen, Phys. Rev. E 81, 016703. Thermostats were not considered.]

By considering such a system we gain the advantage that physical quantitites such as speed, temperature, kinetic and potential energy can be defined and used to couple the system to heat reservoirs - i.e. to apply a thermostat. This promotes the system from a purely mathematical toy to a physical model.

For those who have activated Java in their browser, here is an Applet that simulates a triangular billiard with gravity and thermostats. A gravitational acceleration $g$ is assumed to act in the $-y$ direction. A heat reservoir below amplifies the modulus of a particle reflected by the bottom wall by the factor $f_{v}$, while a particle hitting any of the top sides is decelerated by the same factor. The starting position $q_{0}$ along the contour is counted from the base line center, and is normalised by the total circumference length $3$, such that $q_{0} \epsilon [0,1]$. By $\phi_{0}$ we denote the starting angle between the local normal vector (pointing inside) at $q_{0}$ and the starting velocity vector $\vec{v}_{0}$.


Applet: Triangular billiard, tilted around the base side

If your browser will not let you play my Applet, here are some selected Poincaré plots. Abscissa: $q$; Ordinate: $\phi$; start at $q=0.15$; $\phi=0.13 \pi$ in all cases:


$g=0.1$, $f_{v}=1.00$
 
$g=0.1$, $f_{v}=1.001$
 
$g=0.5$, $f_{v}=1.00$
 
$g=0.5$, $f_{v}=1.001$



To do:
  • Determine Ljapunov exponents for various parameters
  • extend the Applet to a square table tilted around a diagonal (otherwise the velocity components cannot mix), and to a hexagonal table
  • Ask Harald and Bill what it all means...
vesely nov-2015