  We study a long-recognised but under-appreciated symmetry called "dynamical
similarity" and illustrate its relevance to many important conceptual problems
in fundamental physics. Dynamical similarities are general transformations of a
system where the unit of Hamilton's principal function is rescaled, and
therefore represent a kind of dynamical scaling symmetry with formal properties
that differ from many standard symmetries. To study this symmetry, we develop a
general framework for symmetries that distinguishes the observable and surplus
structures of a theory by using the minimal freely specifiable initial data for
the theory that is necessary to achieve empirical adequacy. This framework is
then applied to well-studied examples including Galilean invariance and the
symmetries of the Kepler problem. We find that our framework gives a precise
dynamical criterion for identifying the observables of those systems, and that
those observables agree with epistemic expectations. We then apply our
framework to dynamical similarity. First we give a general definition of
dynamical similarity. Then we show, with the help of some previous results, how
the dynamics of our observables leads to singularity resolution and the
emergence of an arrow of time in cosmology.
