Lagrangian for the Frenkel electron

We found Lagrangian action which describes spinning particle on the base of non-Grassmann vector and involves only one auxiliary variable. It provides the right number of physical degrees of freedom and yields generalization of the Frenkel and BMT equations to the case of an arbitrary electromagnetic field. For a particle with anomalous magnetic moment, singularity in the relativistic equations generally occurs at the speed different from the speed of light. Detailed discussion of the ultra-relativistic motion is presented in the work: A. A. Deriglazov and W. G. Ramirez, World-line geometry probed by fast spinning particle, arXiv:1409.4756.Comment: 8 pages, close to published version: paragraph about ultra-relativistic motion is extended, the detailed discussion of this point is presented in arXiv:1409.475

From Noncommutative Sphere to Nonrelativistic Spin

Reparametrization invariant dynamics on a sphere, being parameterized by angular momentum coordinates, represents an example of noncommutative theory. It can be quantized according to Berezin-Marinov prescription, replacing the coordinates by Pauli matrices. Following the scheme, we present two semiclassical models for description of spin without use of Grassman variables. The first model implies Pauli equation upon the canonical quantization. The second model produces nonrelativistic limit of the Dirac equation implying correct value for the electron spin magnetic moment

Lagrangian and Hamiltonian formulations of asymmetric rigid body, considered as a constrained system

We discuss the dynamics of a rigid body, taking its Lagrangian action with kinematic constraints as the only starting point. Several equivalent forms for the equations of motion of rotational degrees of freedom are deduced and discussed on this basis. Using the resulting formulation, we revise some cases of integrability, and discuss a number of features, that are not always taken into account when formulating the laws of motion of a rigid body.Comment: 30 pages, 8 figures, discussions added, new section added: Chetaev bracket is the Dirac bracke

Dynamics on a submanifold: intermediate formalism versus Hamiltonian reduction of Dirac bracket, and integrability

We consider the Lagrangian dynamical system forced to move on a submanifold $G_\alpha(q^A)=0$. If on some reasons we are interested to know dynamics of all original variables $q^A(t)$, the most economical would be Hamiltonian formulation on the intermediate phase-space submanifold spanned by reducible variables $q^A$ and an irreducible set of momenta $p_i$, $[i]=[A]-[\alpha]$. We describe and compare two different possibilities to establish the Poisson structure and Hamiltonian dynamics on intermediate submanifold. They are Hamiltonian reduction of Dirac bracket and intermediate formalism. As an example of the application of the intermediate formalism, we deduce on this base the Euler-Poisson equations of a spinning body, establishing the underlying Poisson structure, and write their general solution in terms of exponential of Hamiltonian vector field.Comment: 15 pages, typos correcte

Lagrange top: integrability according to Liouville and examples of analytic solutions

The Euler-Poisson equations for the Lagrange top are derived on the basis of a variational problem with kinematic constraints. The Hamiltonian structure of these equations is established using the intermediate formalism presented in the recent work arXiv:2302.12423. General solution to the equations of motion is reduced to the calculation of four elliptic integrals. Several solutions in terms of elementary functions are presented. The case of precession without nutation has a surprisingly rich relationship between the rotation and precession rates, and is discussed in detail.Comment: 15 pages. arXiv admin note: text overlap with arXiv:2304.1037