Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S7 / 2

A constant of motion has zero total derivative on every solution, and may explicitly depend on time. For an infinitesimal transformation $\delta q_i=ϵ{\xi }_i$ with $\delta L=ϵdF/dt$, the on-shell first variation is $\delta L=ϵd({\sum }_i{p}_i{\xi }_i)/dt$. Therefore $\boxed{I=\sum _i{p}_i{\xi }_i-F}$ is conserved. Including a time shift $\delta t=ϵ\tau$ gives the corresponding $I={\sum }_i{p}_i{\xi }_i-H\tau -F$. This proves Noether's theorem in its usual point-transformation form. A symmetry of the action up to a boundary term is sufficient; exact invariance of the displayed Lagrangian is stronger than necessary. Discrete transformations by themselves do not give this continuous-charge construction.

The separability of the printed Lagrangian is important. Every $x_i$ is an independent oscillator of the same frequency, irrespective of mass. Define $X_i=\sqrt{m_i}{x}_i$, $P_i=p_{x_i}/\sqrt{m_i}$. Then the complete constant matrix $Q_{ij}=P_i{P}_j+{\omega }^2{X}_i{X}_j$, and angular quantities $J_{ij}=X_i{P}_j-X_j{P}_i$, include each oscillator energy and the rotational symmetries of this mass-weighted oscillator space. Alternatively its explicitly time-dependent constants are $X_i\cos \omega t-P_i\sin \omega t/\omega$ and $X_i\sin \omega t+P_i\cos \omega t/\omega$.

Translations in all $y_i$ give conserved $P_y=\sum p_{y_i}$. Translation of $z_1,z_2$ together gives $P_{z12}=p_{z_1}+p_{z_2}$; $z_3$ is free and gives $p_{z_3}$. Their center-of-mass positions minus the associated constant velocities times $t$ are boost constants. The decoupled relative $z_1-z_2$ motion conserves $E_{z,\mathrm{rel}}=(p_{z_1}/m_1-p_{z_2}/m_2{)}^2{\mu }_{12}/2+V(z_1-z_2)$, where ${\mu }_{12}=m_1{m}_2/(m_1+m_2)$. In mass-weighted versions, the three free center coordinates also have conserved mutual angular momenta. These families are not all functionally independent.

The $y$ relative subsystem alone is driven through $\cos [\lambda (m_1-m_2)t]$. For generic nonzero $\lambda (m_1-m_2)$ its energy, and hence the total energy, is not conserved. Omitting the mass-weighted oscillator symmetries simply because the physical masses differ would miss genuine symmetries of this particular model.

When $m_1=m_2$, the cosine becomes one and the previously driven $y$ relative subsystem becomes autonomous. Its energy is one new generic independent time-independent integral; equivalently total energy becomes conserved because the remaining subsystem energies already were. Making $m_3$ equal as well introduces no further generic continuous symmetry in the printed model. The arbitrary $U_{13}$ and $V_{12}$ distinguish particle labels and directions, so equal masses alone do not create a permutation or physical-rotation invariance.

A finite “number of conserved quantities” must mean independent generators/integrals: arbitrary functions of any integral give infinitely many. Mass-weighted rotations and free-coordinate symmetries identified in part a already exist at unequal masses. Thus the justified generic increment is one at $m_1=m_2$, and zero further at complete equality. Special potentials or $\lambda =0$ can change the count, but were not stipulated.

For equal masses $m$, the newly conserved relative $y$ energy isIt is kinetic energy relative to the free $y$ center of mass plus the two interactions. Adding its constant center kinetic energy and the already conserved $x,z$ energies gives the total Hamiltonian. The equal-mass oscillator rotations simplify to $J_{ij}=x_i{p}_{x_j}-x_j{p}_{x_i}$, but are not new: for unequal masses their conserved predecessors contain the mass-weight factors. As a concrete check, independent transformations of $X_i=\sqrt{m_i}{x}_i$ rotate an isotropic oscillator at any positive masses. Complete mass equality does not magically make generic $U_{12}(y_1-y_2)+U_{13}(y_1-y_3)+V(z_1-z_2)$ invariant under rotations of physical space.