Generalizes Lagrangians and spinning particles (Q2784874)

The author studies the problem of classical description of spin. After introducing the notions of kinematical variables and kinematical space, he defines a classical elementary particle as a Lagrangian system whose dynamical space is a homogeneous space of the kinematical group. As a consequence, the usual requirement that Lagrangians must depend only on first-order derivatives is revised. Moreover, it is shown that the spin structure of elementary particles can be naturally described by Lagrangians depending on higher-order derivatives. The author mentions the Ostrogradskij's contribution into the development of variational methods in mechanics and, in particular, Ostrogradskij's results concerning Lagrangians depending on higher-order derivatives.NEWLINENEWLINENEWLINEAs an example, the author considers an elementary particle whose kinematical space is the quotient space \({\mathcal G}/SO(3)\), where \({\mathcal G}\) is the Galileo group. In this example, the Lagrangian depends on the acceleration. The author shows how in this situation the internal orbital motion appears, known as the zitterbewegung. Such a motion gives rise to the spin structure and also to the magnetic properties of the particle.