  The stochastic dynamics of a rigid inclusion constrained to move on a curved
surface has many applications in biological and soft matter physics, ranging
from the diffusion of passive or active membrane proteins to the motion of
phoretic particles on liquid-liquid interfaces. Here we construct intrinsic
Langevin equations for an oriented rigid inclusion on a curved surface using
Cartan's method of moving frames. We first derive the Hamiltonian equations of
motion for the translational and rotational momenta in the body frame.
Surprisingly, surface curvature couples the linear and angular momenta of the
inclusion. We then add to the Hamiltonian equations linear friction, white
noise and arbitrary configuration-dependent forces and torques to obtain
intrinsic Langevin equations of motion in phase space. We provide the
integrability conditions, made non-trivial by surface curvature, for the forces
and torques to admit a potential, thus distinguishing between passive and
active stochastic motion. We derive the corresponding Fokker-Planck equation in
geometric form and obtain fluctuation-dissipation relations that ensure
Gibbsian equilibrium. We extract the overdamped equations of motion by
adiabatically eliminating the momenta from the Fokker-Planck equation, showing
how a peculiar cancellation leads to the naively expected Smoluchowski limit.
The overdamped equations can be used for accurate and efficient intrinsic
Brownian dynamics simulations of passive, driven and active diffusion processes
on curved surfaces. Our work generalises to the collective dynamics of many
inclusions on curved surfaces.
