By Simon Fischer
Simon Fischer - fundamentals (Violin Method)(Peters, 1997)(ISBN 1901507009)
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Additional resources for Basics: 300 exercises and practice routines for the violin, Peters Edition
25. Two periodic orbits which are members of the same family of the quarter 2D SB. The two periodic orbits have the same =, hence they re#ect o! the same discs. erent. For simplicity, the illustration is made for the 2D SB, but the same principle applies also to the 3D Sinai billiard. Left: unfolded representation, right: standard representation. Fig. 26. Two families of periodic orbits of the 3D SB, represented in the unfolded representation of the SB. The faint spheres do not participate in the code.
This partition is the proposed source of the correlations that were observed in the form factor. This concept is illustrated in Fig. 25, and graphic representations of two families are displayed in Fig. 26. The most outstanding feature which emerges from Fig. 26 is that the orbits occupy a very narrow volume of phase-space throughout most of their length, and they fan out appreciably only at a single sphere. The above arguments suggest that the main source of correlations are the similarities of orbits within each family (=).
The term d consists of the contributions of the non-generic bouncing-ball manifolds. erent prefactors (which are possibly 0) due to partial or complete shadowing of the bouncing-ball family by the sphere. The last term is the contribution of the set of generic and unstable periodic orbits, where ¸ denote their lengths and A are semiclassical amplitudes. erence between its two sides. This cannot be done by a straightforward substitution, since three obstacles must be removed: E The spectrum of wavenumbers k was computed for the fully desymmetrized Sinai billiard.
Basics: 300 exercises and practice routines for the violin, Peters Edition by Simon Fischer