  We derive the John-Sclavounos equations describing the motion of a fluid
particle on the sea surface from first principles using Lagrangian and
Hamiltonian formalisms applied to the motion of a frictionless particle
constrained on an unsteady surface. The main result is that vorticity generated
on a stress-free surface vanishes at a wave crest when the horizontal particle
velocity equals the crest propagation speed, which is the kinematic criterion
for wave breaking. If this holds for the largest crest, then the symplectic
two-form associated with the Hamiltonian dynamics reduces instantaneously to
that associated with the motion of a particle in free flight, as if the surface
did not exist. Further, exploiting the conservation of the Hamiltonian function
for steady surfaces and traveling waves we show that particle velocities remain
bounded at all times, ruling out the possibility of the finite-time blowup of
solutions.
