The equivalence class of absolute configurations of a system under the group
    of similarity transformations $Sim(3)$ is called the shape of the system. It is
    explained in this paper how the direct application of the Principle of
    Relationalism leads to a new theory whose Lagrangian is $Sim(3)$-invariant,
    which in turn ensures the existence of the theory’s law of motion on shape
    space. To find out the equations of motion for a system’s shape degrees of
    freedom, the Boltzman-Hammel equations of motion in an anholonomic frame on the
    tangent space $T(Q)$ to the system’s absolute configuration space $Q$, is
    adapted to the $Sim(3)$-fiber-bundle structure of the configuration space. The
    derived equations of motion on shape space enable us, among others, to predict
    the evolution of the shape of a classical system governed by this new theory
    without any reference to its absolute position, orientation, or size in space.
    We will explain by treating the measuring instruments as part of the matter in
    the theory, how the mass metric $\textbf{M}$ on the configuration space $Q$
    uniquely defines a metric on the reduced tangent bundle $\frac{T(Q)}{Sim(3)}$,
    and how the unique metric structure on shape space $S$ can be derived. After
    treating the general $N$-body system, the shape equations of motion of a
    three-body system are derived explicitly as an illustration of the general
    method. Some cosmological implications of this theory are also worked out. In
    particular, we explain how the observed universe’s accelerated expansion
    follows from the conservation of the dilational momentum in the modified
    Newtonian theory. Finally, we compare the present work with two other
    approaches to relational physics and discuss their essential differences.

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