An elementary particle is defined as a mechanical system whose kinematical space is a homogeneous space of the Poincaré group. Lagrangians for describing these systems depend on higher-order derivatives and some of them are analyzed. For bradyons the Lagrangian depends on the acceleration and angular velocity of the particle and is characterized by two parameters m and s, the rest mass and absolute value of spin, respectively. In general the spin is of kinematical nature, related to the rotation and internal orbital motion of the system. Two different kinds of bradyons appear according to the spin structure. One has a spin related to the generalized acceleration. Photons are massless particles with spin lying along the direction of motion and energy hν, where ν is the frequency of its rotational motion. Particles moving in circles with velocity c in their center of mass frame are also predicted, showing a Dirac-type Hamiltonian. There also appear particles with tachyonic orbital motion whose center of mass is bradyonic motion. Transformation properties under space and time reversal are also analyzed.