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 / 3

Vary $S=\int L(q,\dot{q},t)dt$ with fixed endpoints. Integration by parts gives $\delta S=\int {\sum }_i[L_{q_i}-d{L}_{{\dot{q}}_i}/dt]\delta q_{i}dt$, hence the Euler-Lagrange equations. Define $p_i=L_{{\dot{q}}_i}$ and, assuming an invertible velocity-momentum relation, $H=p\cdot \dot{q}-L$. The equivalent action is $\int (p\cdot \dot{q}-H)dt$. Independent variations of $p$ and $q$ yieldBoundary terms vanish for fixed endpoint coordinates; constraining momenta independently at the same endpoints is unnecessary. Singular Lagrangians require constraint analysis instead of this unconstrained Legendre transform.

Phase space combines all independent canonical coordinates and momenta. Its Hamiltonian velocity has divergence ${\sum }_i[{\partial }_{q_i}{H}_{p_i}-{\partial }_{p_i}{H}_{q_i}]=0$ by equality of mixed derivatives. Thus an infinitesimal material phase volume has zero fractional rate of change. This is Liouville's theorem, valid also for explicitly time-dependent Hamiltonians. It preserves fine-grained volume, not necessarily the area of any projected coordinate pair.

The oscillator flow isIts determinant is ${\cos }^2\omega t+{\sin }^2\omega t=1$, explicitly preserving area. The orbits are energy ellipses; stretching one coordinate during transport is compensated by the conjugate direction.

Lossless Hamiltonian focusing can reduce a beam's transverse position width at the expense of momentum spread, but cannot reduce its occupied canonical phase volume. Coupling to the longitudinal direction can redistribute projected emittance without reducing total emittance. True cooling requires exporting entropy to additional degrees of freedom or discarding particles: stochastic feedback, electron cooling or radiative damping coupled to an external environment provide possible mechanisms, with very different efficiency for high-energy protons. Collimation shrinks the accepted distribution at the cost of intensity. Acceleration also reduces angular divergence at fixed normalized emittance, an apparent geometrical cooling rather than a violation of Liouville's theorem. The apparatus and radiation cannot be omitted from the accounting when the bunch is no longer an isolated Hamiltonian subsystem.