  Absolute space is eliminated from the body of mechanics by gauging
translations and rotations in the Lagrangian of a classical system. The
procedure implies the addition of compensating terms to the kinetic energy, in
such a way that the resulting equations of motion are valid in any frame. The
compensating terms provide inertial forces depending on the total momentum P,
intrinsic angular momentum J and intrinsic inertia tensor I. Therefore, the
privileged frames where Newton's equations are valid (Newtonian frames) are
completely determined by the matter distribution of the universe
(Machianization). At the Hamiltonian level, the gauge invariance leads to first
class constraints that remove those degrees of freedom that make no sense once
the absolute space has been eliminated. This reformulation of classical
mechanics is entirely relational, since it is a dynamics for the distances
between particles. It is also Machian, since the rotation of the rest of the
universe produces centrifugal effects. It then provides a new perspective to
consider the foundational ideas of general relativity, like Mach's principle
and the weak equivalence principle. With regard to the concept of time, the
absence of an absolute time is known to be a characteristic of parametrized
systems. Furthermore, the scale invariance of those parametrized systems whose
potentials are inversely proportional to the squared distances can be also
gauged by introducing another compensating term associated with the intrinsic
virial G (shape-dynamics).
