  A new and universal method for implementing scale invariance, called best
matching, is presented. It extends to scaling the method introduced by Bertotti
and the author to create a fully relational dynamics that satisfies Mach's
principle. The method is illustrated here in the context of non-relativistic
gravitational particle dynamics.. It leads to far stronger predictions than
general Newtonian dynamics. The energy and angular momentum of an `island
universe' must be exactly zero and its size, measured by its moment of inertia,
cannot change. This constancy is enforced because the scale invariance requires
all potentials to be homogeneous of degree -2. It is remarkable that one can
nevertheless exactly recover the standard observed Newtonian laws and forces,
which are merely accompanied by an extremally weak universal force like the one
due to Einstein's cosmological constant. In contrast to Newtonian and
Einsteinian dynamics, both the gravitational constant G and the strength of the
cosmological force are uniquely determined by the matter distribution of the
universe. Estimates of their values in agreement with observations are
obtained. Best matching implements a dynamics of pure shape for which the
action is a dimensionless number. If the universe obeys such scale invariant
law, steadily increasing inhomogeneity, not expansion of the universe, causes
the Hubble red shift. The application of best matching to geometrodynamics is
treated in a companion paper.
