  The authors describe the general motion of radiation-pushed sails accelerated
near the speed of light with directed energy propulsion. Practical applications
of the model are also given, including the interstellar flyby mission to the
Alpha Centauri star system envisioned by the Breakthrough Starshot program. Any
misalignment between the driving light beam and the direction of the sail's
motion is naturally swept away during acceleration toward relativistic speed,
yet leads to a deviation of about 80 A.U. in the case of an initial
misalignment of 1 arc sec for a sail accelerated up to 0.2c toward Alpha
Centauri. Then, the huge proper acceleration felt by the probes (of order 2500
g), the tremendous energy cost (of about 13 kt per probe) for poor efficiency
(of about 3 \%), the trip duration (between 22 and 33 years), the temperature
at thermodynamic equilibrium (about 1500 K), and the time dilation aboard
(about 160-days difference) are all presented and their variation with the
sail's reflectivity is discussed. We also present an application to single
trips within the Solar System using double-stage light sails. A spaceship of
mass 24 tons can start from Earth and stop at Mars in about seven months with a
peak velocity of 30 km/s but at the price of a huge energy cost of about
$5.3\times 10^4$ GWh due to extremely low efficiency of the directed energy
system, around $10^{-4}$ in this low-velocity case.
