  In classical theory, the trajectory of a particle is entirely predetermined
by the complete set of initial conditions via dynamical laws. Based on this, we
formulate a no-go theorem for the dynamics of classical particles, i.e., a
Bell's inequality for trajectories, and discuss its possible violation in a
quantum scenario. A trajectory, however, is not an outcome of a quantum
measurement, in the sense that there is no observable associated with it, and
thus there is no "direct" experimental test of the Bell's inequality for
trajectories. Nevertheless, we show how to overcome this problem by considering
a special case of our generic inequality that can be experimentally tested
point-by-point in time. Such inequality is indeed violated by quantum
mechanics, and the violation persists during an entire interval of time and not
just at a particular singular instant. We interpret the violation to imply that
trajectories (or at least pieces thereof) cannot exist predetermined, within a
local-realistic theory.
