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For $N$ labelled particles with collisions excluded, Newtonian gravity hasSimultaneously translating every position or applying the same time-independent spatial rotation preserves both kinetic energy and the distance-dependent potential. The conserved momentum and angular momentum obtained from Noether's theorem areTime translations give the conserved Hamiltonian. Galilean boosts change the Lagrangian by a total derivative, rather than leaving it strictly unchanged; they give the conserved centre-of-mass quantity $M\mathbf{R}-t\mathbf{P}$, where $M={\sum }_i{m}_i$ and $\mathbf{R}=M^{-1}{\sum }_i{m}_i{\mathbf{q}}_i$. Thus absolute uniform velocity is not singled out by the isolated dynamics.

Translations can be separated explicitly. Put ${\mathbf{r}}_i={\mathbf{q}}_i-\mathbf{R}$, so ${\sum }_i{m}_i{\mathbf{r}}_i=0$. Thenwith the cross-term vanishing. The potential depends only on ${\mathbf{r}}_i-{\mathbf{r}}_j$. The centre-of-mass motion is therefore a decoupled free motion; at fixed $\mathbf{P}$ its energy is ${\mathbf{P}}^2/(2M)$ and it can be eliminated from the internal equations. Choosing a centre-of-mass rest frame sets $\mathbf{P}=0$.

One can then quotient internal configurations by simultaneous rotations. On generic noncollinear configurations with $N\ge 3$ the resulting relational shape space has dimension $3N-6$; collinear configurations have stabilizers and belong to singular strata. For two particles only the separation remains, so that generic dimension formula is inapplicable. This configuration quotient identifies absolute placement and orientation, but it does not by itself specify all the reduced dynamics.

At the phase-space level, symplectic reduction is performed at a fixed moment-map value: $J^{-1}(\mu )/G_{\mu }$, where $G_{\mu }$ is the coadjoint stabilizer. In particular, at fixed nonzero angular momentum one does not simply divide the momentum level by every rotation. The reduction retains the appropriate angular-momentum data and centrifugal effects. Even for two gravitating particles, specifying $r$ and $\dot{r}$ alone is insufficient unless the angular-momentum parameter is supplied:Systems with identical instantaneous separation and radial velocity but different $\ell$ have different radial futures. Eliminating orientation while discarding this parameter would not be a faithful reduction.

These results support a relational treatment of overall position and orientation: isolated configurations differing only by one fixed translation or rotation have the same internal predictions. However, a symmetry and a declared gauge equivalence are conceptually different, and the quotient alone does not settle the ontology of space. Arbitrary time-dependent rotations are not symmetries of the original inertial equations; a rotating coordinate description brings Coriolis and centrifugal terms. Newtonian inertial structure and absolute acceleration therefore cannot be eliminated merely by citing rotational invariance. A relational theory may encode that structure through additional dynamical or connection data, but it must reproduce these effects rather than erase them.

The symmetries license mathematically controlled elimination of collective variables, with momentum and regularity qualifications; they do not alone prove either absolute space or complete relationalism. This distinguishes redundant descriptions of a solution from genuinely different possible motions.

The relational shape space of a particle configuration is obtained by removing collective translations and rotations; here scale is retained. For generic collision-free, noncollinear configurations of $N\ge 3$ labelled particles in three dimensions, the resulting configuration dimension is $3N-6$. Collinear configurations have nontrivial rotational stabilizers and require singular strata. Removing position and orientation does not automatically remove conserved angular momentum or its centrifugal effect; a dynamically faithful reduction must retain the appropriate momentum-level data.