  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 symmetry. To study this symmetry, we introduce a
novel set of principles for identifying the surplus structure of a theory
designed to accommodate the distinct features of dynamical similarities. We
test our framework on a variety of examples and find that it sidesteps tensions
between dynamic and epistemic aspects of symmetry while reproducing orthodox
intuitions. When applied to cosmology, our framework prescribes that dynamical
similarities should be associated with surplus structure. We then review
previous technical results showing that an elimination of this surplus
structure leads to a resolution of a large class of singularities in general
relativity and suggests a novel explanation for the arrow of time. We conclude
by highlighting the potential relevance of dynamical similarity to
thermodynamic and quantum systems.
