Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 59 / 3

For a finite-dimensional mechanical system let $Q$ be its configuration space, a smooth manifold incorporating the holonomic constraints. Lagrangian mechanics starts with $L:TQ\to \mathbb{R}$; a curve $q(t)$ makes the action stationary under fixed-endpoint variations. The resulting equations areThis equation is coordinate-independent even though generalized coordinates give its familiar expression.

The fibre derivative $\mathbb{F}L(q,v)=(q,p)$, with $p_i=L_{v^i}$, defines the Legendre transform in mechanics. The Cartan form of a regular mechanical Lagrangian isFor a regular Lagrangian, the velocity Hessian is invertible, ${\omega }_L$ is symplectic, and ${\iota }_{{\Gamma }_L}{\omega }_L=d{E}_L$ determines a second-order vector field on $TQ$. In coordinates this recovers the displayed Euler–Lagrange equations: the velocity components of the vector field are $v^i$, and its acceleration components are fixed by the invertible Hessian.

Hamiltonian mechanics takes place on $T^{\ast }Q$. Its tautological one-form is $\theta =p_{i}d{q}^i$, and its canonical symplectic form is $\omega =-d\theta =d{q}^i\wedge d{p}_i$. With ${\iota }_{X_H}\omega =dH$,The associated Poisson bracket is $\{f,g\}=f_{q^i}{g}_{p_i}-f_{p_i}{g}_{q^i}$, so $X_{H}f=\{f,H\}$. For a hyperregular Lagrangian, $\mathbb{F}L$ is a global diffeomorphism, $H=E_L\circ (\mathbb{F}L{)}^{-1}$ and $(\mathbb{F}L{)}^{\ast }\omega ={\omega }_L$. The two formulations then describe the same trajectories. Regularity gives only local equivalence; degenerate Lagrangians require constraint analysis instead. These geometric structures distinguish equations of motion from arbitrary coordinate formulae and make symmetry reduction precise.

In Lagrangian mechanics, let a one-parameter group on $Q$ have generator ${\xi }_Q={\xi }^i(q){\partial }_{q^i}$. Its tangent lift acts on $TQ$ asIf $L$ is invariant, ${\xi }_{TQ}L=0$. Along an Euler–Lagrange trajectory, the Noether conserved quantity for a mechanical point symmetry isbecauseThus the conserved quantity is explicitly derived, not merely attached to the name of a symmetry. If the change of $L$ is the total derivative $dB/dt$, then $p_i{\xi }^i-B$ is conserved. For a general point transformation with time component $\tau$, the conserved expression is $p_i{\xi }^i-E_L\tau -B$, provided the variation of $Ldt$ is $dB$. Time-translation invariance of an autonomous Lagrangian therefore yields conservation of energy.

In Hamiltonian mechanics, a cotangent-lifted action preserves $\theta$ and hence $\omega$. For its infinitesimal generator ${\xi }_{T^{\ast }Q}$, Cartan's magic formula giveswhere $J_{\xi }=p_i{\xi }^i$. These components define the canonical moment map $J:T^{\ast }Q\to {\mathfrak{g}}^{\ast }$, with $⟨J,\xi ⟩=J_{\xi }$. If $H$ is invariant, $\{H,J_{\xi }\}=0$, so ${\dot{J}}_{\xi }=\{J_{\xi },H\}=0$. This is the Hamiltonian form of Noether's theorem, and the Legendre transform in mechanics identifies its charges with the Lagrangian ones.

More generally, a Hamiltonian group action is an action whose infinitesimal symplectic generators have globally defined Hamiltonian functions. The existence of those functions is an assumption, not a consequence of every symplectic action. For example, translation on a symplectic two-torus preserves $dq\wedge dp$ but contraction with ${\partial }_q$ gives $dp$, which is closed and not globally exact. Such a symmetry need not have a globally single-valued moment-map component. On $T^{\ast }Q$, the cotangent lift does have the canonical components just derived. A Hamiltonian symmetry preserving $H$ gives a conserved generator; the global momentum-map hypotheses must be stated.

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.