  The reduced-particle model is the central element for the systematic
derivation of the gyrokinetic Vlasov-Maxwell equations from first principles.
Coupled to the fields inside the gyrokinetic field-particle Lagrangian, the
reduced-particle model defines polarization and magnetization effects appearing
in the gyrokinetic Maxwell equations. It is also used for the reconstruction of
the gyrokinetic Vlasov equation from the particle characteristics. Various
representations of reduced-particle models are available according to the
choice of the gyrokinetic phase space coordinates. In this paper, the
Hamiltonian representation of the reduced particle dynamics at an order
suitable for the implementation in particle-in-cell simulations is explicitly
derived from the general reduction procedure. The second-order (with respect to
the fluctuating electromagnetic fields), full Finite Larmor Radius (FLR)
Hamiltonian gyrokinetic particle model as well as the second-order model
suitable specifically for the long-wavelength approximation (i.e., containing
up to the second-order FLR corrections), are derived and compared to the model
recently implemented in the particle-in-cell code ORB5. We show that the same
long-wavelength approximate equations can also be derived by taking the proper
limit of the full FLR model.
