  The equations of motion for a Lagrangian ${\cal L}(k_1)$, depending on the
curvature $k_1$ of the particle worldline, embedded in a space--time of
constant curvature, are considered and reformulated in terms of the principal
curvatures. It is shown that for arbitrary Lagrangian function ${\cal L}(k_1)$
the general solution of the motion equations can be obtained by integrals. By
analogy with the flat space--time case, the constants of integration are
interpreted as the particle mass and its spin. As examples, we completely
investigate Lagrangians linear and quadratic in $(k_1)$ and the model of
relativistic particle with maximal proper acceleration, in a space--time with
constant curvature.
