Scaling potentials and the billiard case

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

$$U(\alpha {\mathbf r}) \; = \; \alpha^c U{({\mathbf r})}, \hspace{0.5in} \mbox{all $\alpha$}.$$ (D.11)

The potential contours all have the same shape, and the radial dependence is a $c^{th}$-power-law in any direction ($c$ is even for bounded motion). This gives the property ${\mathbf r}\cdot\nabla U = cU$ everywhere in space, so that the constant of the motion $\mathcal{H}$ can be written $\mbox{\small$\frac{1}{2}$}m\dot{r}^2 - (m/c){\mathbf r}\cdot \ddot{{\mathbf r}}$. This last expression is composed of integrand terms in (D.8), in the case of isotropic $B_{ij} \propto \delta_{ij}$, corresponding to dilation. Thus in this case the integrand is constant [and equal to $F(x)$, in order to have zero time-average of ${\mathcal{F}}(t)$]; there is no diffusive growth and dilation is `special'. The limit of hard-walled billiards corresponds to $c\rightarrow\infty$, in which case the constant of the motion is simply the kinetic term $p^2$.

To summarize, in $d$ dimensions in a scaling potential (of which the billiard is a special case), counting the `special' degrees of freedom gives: $d$ for translations (vector ${\mathbf a}$), $\mbox{\small$\frac{1}{2}$}d(d-1)$ for rotations (antisymmetric part of $B$), and 1 for dilation (isotropic part of $B$). The total is $\mbox{\small$\frac{1}{2}$}d(d+1) + 1$.

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 $U{({\mathbf r})}$ which is not differentiable everywhere in space allows new special ${\mathbf D}{({\mathbf r})}$ functions to arise, which are not expressible as the Taylor series of (D.6). Also worthy of study is the general Hamiltonian system ${\mathcal{H}}({\mathbf r},{\mathbf p};x)$, no longer restricted to a constant mass tensor ${\mathbf p} = M\dot{{\mathbf r}}$. This restriction played a key role in the above arguments.