Classical and quantum chaos of the wedge billiard. I. Classical mechanics

In this first of two papers on the classical-quantum correspondence of the wedge billiard, the classical mechanics for wedge angles giving hard chaos is described. Attention is focused on the periodic orbits of the system, all of which are known to be unstable. Each primitive periodic orbit has been found to correspond uniquely to a sequence of two symbols, except for certain orbits of the 60° wedge that go directly into the vertex. However, not every binary sequence has a corresponding periodic orbit. A total of 1048 primitive periodic orbits of the 49° wedge and 1920 primitive periodic orbits of the 60° wedge have been found and their actions, Maslov indices, and stability exponents determined. The primitive periodic orbits of word length n have been used to calculate the mean action S¯(n), the mean Maslov index ν¯(n), and the mean stability exponent u¯(n). To a good approximation, each of these quantities increases linearly with n. It is also shown that there exist families consisting of many or possibly an infinite number of primitive periodic orbits with nearly the same action.