  The character of radiation of relativistic charged particles in strong
magnetic fields largely depends on the disposition of particle trajectories
relative to the field lines. The motion of particles with trajectories close to
the curved magnetic lines is usually referred to the so-called curvature
radiation. The latter is treated within the formalism of synchrotron radiation
by replacing the particle Larmor radius with the curvature radius of the field
lines. However, even at small pitch angles, the curvatures of the particle
trajectory and the field line may differ significantly. Moreover, as we show in
this paper the trajectory curvature varies with time, i.e. the process has a
stochastic character. Therefore for calculations of observable characteristics
of radiation by an ensemble of particles, the radiation intensities should be
averaged over time. In this paper, for determination of particle trajectories
we use the Hamiltonian formalism, and show that that close to curved magnetic
lines, for the given configuration of the magnetic field, the initial point and
particle energy, always exist a smooth trajectory without fast oscillations of
the curvature radius. This is the trajectory which is responsible for the
curvature radiation. The realization of this regime requires the initial
particle velocity to be directed strictly along the smooth trajectory. This
result might have direct relation to the recent spectral measurements of
gamma-radiation of pulsars by the Fermi Gamma-ray Space Telescope.
