Extra force in f ( R ) modified theories of gravity

The equation of motion for massive particles in $f(R)$ modified theories of gravity is derived. By considering an explicit coupling between an arbitrary function of the scalar curvature, $R$, and the Lagrangian density of matter, it is shown that an extra force arises. This extra force is orthogonal to the four-velocity and the corresponding acceleration law is obtained in the weak-field limit. Connections with MOND and with the Pioneer anomaly are further discussed.