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 / b

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