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