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.
