Dynamic billiards

The Bunimowitsch Stadium, named after Leonid Abramowitsch Bunimowitsch , is a chaotic billiard table with two degrees of freedom in the form of a stadium. Two trajectories of a particle are drawn in red and yellow . A property of chaotic systems is that the trajectories diverge greatly with almost identical starting conditions.

Sinai billiards, named after the mathematician and theoretical physicist Jakow Grigoryevich Sinai

Dynamic billiards is a dynamic system that describes the movement of a mass point that moves without force in an area with a piecewise smooth edge and is elastically reflected at the edges of the area. In the case of an elastic reflection or an elastic impact on a solid object, the energy , momentum and speed of the particle are retained and the angle of incidence is equal to the angle of reflection . It is therefore a Hamiltonian system .

Depending on the choice of the area under consideration, dynamic billiards can show all behaviors, from integrable to chaotic . One advantage of a billiard over other Hamiltonian models is that the behavior can be reduced to a billiard image without having to integrate the equations of motion. A billiards map is a special Poincaré map that maps the coordinates and angles of one reflection to the coordinates and angles of the next reflection.

properties

The Hamiltonian that a particle with coordinates and mass and momentum in a potential is describes ${\displaystyle q}$ ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle V(q)}$

${\displaystyle H(p,q)={\frac {p^{2}}{2m}}+V(q)}$

The mass as the proportionality constant between the momentum and the time derivative of the coordinates can be set equal to one without loss of generality. The coordinates and momentum components indicate the state of a particle in phase space . A dynamic billiard is therefore completely defined by the choice of the area . The potential is zero inside the area and infinite outside: ${\displaystyle (p,q)}$ ${\displaystyle \Omega }$

${\displaystyle V(q)={\begin{cases}0&q\in \Omega \\\infty &q\notin \Omega \end{cases}}}$

The dynamics receive the phase space volume from trajectories. Trajectories that have a phase space volume of zero can mostly be neglected. This applies in particular to trajectories that meet a singularity at the edge of the area, such as the transition from the straight line to the curve at the Bunimowitsch Stadium.

generalization

In a generalized billiard table for a particle that moves in a non-Euclidean manifold , the scalar product has to be formed taking the metric tensor into account . ${\displaystyle p^{2}=p^{i}p^{j}g_{ij}(q)}$ ${\displaystyle g}$

For a quantum mechanical billiard the momentum is given by the momentum operator and one gets the eigenvalue problem ${\displaystyle p=-\mathrm {i} \hbar \nabla }$

${\displaystyle -{\frac {\hbar ^{2}}{2m}}\nabla ^{2}\psi _{n}(q)=E_{n}\psi _{n}(q)}$

with the energy eigenvalues and the wave functions . The elastic scattering at the edge of the area becomes the Dirichlet boundary condition ${\displaystyle E_{n}}$ ${\displaystyle \psi _{n}}$

${\displaystyle \psi _{n}(q)=0\quad {\text{für}}\quad q\notin \Omega }$

Web links

• Dynamical Billards at Scholarpedia
• Billiards at Wolfram Math-World
• Quantum chaos in the Bunimovich Stadium

Individual evidence

1. ^ Bunimovich Stadium . In: Lexicon of Physics . Spectrum Akademischer Verlag ( Spektrum.de [accessed on August 5, 2016]).
2. ^ Sinai billiards . In: Lexicon of Physics . Spectrum Akademischer Verlag ( Spektrum.de [accessed on August 5, 2016]).
3. ↑ ^{a b} Leonid Bunimovich: Dynamical billiards . In: Scholarpedia . tape 2 , no. 8 , 2007, p. 1813 , doi : 10.4249 / scholarpedia.1813 .