  We investigate the possibility of extending Newton's second law to the
general framework of theories in which special relativity is locally valid, and
in which gravitation changes the flat Galilean space-time metric into a curved
metric. This framework is first recalled, underlining the possibility to
uniquely define a space metric and a local time in any given reference frame,
hence to define velocity and momentum in terms of the local space and time
standards. It is shown that a unique consistent definition can be given for the
derivative of a vector (the momentum) along a trajectory. Then the possible
form of the gravitation force is investigated. It is shown that, if the motion
of free particles has to follow space-time geodesics, then the expression for
the gravity acceleration is determined uniquely. It depends on the variation of
the metric with space and time, and it involves the velocity of the particle.

General Relativity and Quantum Cosmology