  In this paper, we study the weak gravitational deflection of relativistic
massive particles for a receiver and source at finite distance from the lens in
stationary, axisymmetric and asymptotically flat spacetimes. For this purpose,
we extend the generalized optical metric method to the generalized Jacobi
metric method by using the Jacobi-Maupertuis Randers-Finsler metric. More
specifically,we apply the Gauss-Bonnet theorem to the generalized Jacobi metric
space and then obtain an expression for calculating the deflection angle, which
is related to Gaussian curvature of generalized optical metric and geodesic
curvature of particles orbit. In particular, the finite-distance correction to
the deflection angle of signal with general velocity in the the Kerr black hole
and Teo wormhole spacetimes are considered. Our results cover the previous work
of the deflection angle of light, as well as the deflection angle of massive
particles in the limit for the receive and source at infinite distance from the
lens object. In Kerr black hole spacetime, we compared the effects due to the
black hole spin, the finite-distance of source or receiver, and the
relativistic velocity in microlensings and lensing by galaxies. It is found in
these cases, the effect of BH spin is usually a few orders larger than that of
the finite-distance and relativistic velocity, while the relative size of the
latter two could vary according to the particle velocity, source or observer
distance and other lensing parameters.
