  Models of relativistic particle with Lagrangian ${\cal L}(k_1)$, depending on
the curvature of the worldline $k_1$, are considered. By making use of the
Frenet basis, the equations of motion are reformulated in terms of the
principal curvatures of the worldline. It is shown that for arbitrary
Lagrangian function ${\cal L}(k_1)$ these equations are completely integrable,
i.e., the principal curvatures are defined by integrals. The constants of
integration are the particle mass and its spin. The developed method is applied
to the study of a model of relativistic particle with maximal proper
acceleration, whose Lagrangian is uniquely determined by a modified form of the
invariant relativistic interval. This model gives us an example of a consistent
relativistic dynamics obeying the principle of a superiorly limited value of
the acceleration, advanced recently.
