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# Scaling potentials and the billiard case

A scaling system [126] is created when the potential obeys

 (D.11)

The potential contours all have the same shape, and the radial dependence is a -power-law in any direction ( is even for bounded motion). This gives the property everywhere in space, so that the constant of the motion can be written . This last expression is composed of integrand terms in (D.8), in the case of isotropic , corresponding to dilation. Thus in this case the integrand is constant [and equal to , in order to have zero time-average of ]; there is no diffusive growth and dilation is special'. The limit of hard-walled billiards corresponds to , in which case the constant of the motion is simply the kinetic term .

To summarize, in dimensions in a scaling potential (of which the billiard is a special case), counting the special' degrees of freedom gives: for translations (vector ), for rotations (antisymmetric part of ), and 1 for dilation (isotropic part of ). The total is .

I have strong numerical evidence that dilation is the only new special deformation which always arises when a hard-walled limit is taken of a general potential. Certainly the above arguments are sufficient to exclude simple cases, such as shear-type deformations. However I cannot exclude the possibility that a which is not differentiable everywhere in space allows new special functions to arise, which are not expressible as the Taylor series of (D.6). Also worthy of study is the general Hamiltonian system , no longer restricted to a constant mass tensor . This restriction played a key role in the above arguments.

Next: Appendix E: Cross correlations Up: Appendix D: How Many Previous: General potential case
Alex Barnett 2001-10-03