Equations of motion for a classical color particle in background non-Abelian fermionic and bosonic fields: Inclusion of pseudoclassical spin

A generalization of the Lagrangian introduced earlier in [2011 {\it J. Phys. G} ${\bf 37}$ 105001] for a classical color spinning particle interacting with background non-Abelian gauge and fermion fields for purpose of considering a change in time of the spin particle degree of freedom, is suggested. In the case under consideration the spin degree of freedom is described by a commuting $c$-number Dirac spinor $\psi_{\alpha}$. A mapping of this spinor into new variables: anticommuting pseudovector $\xi_{\mu}$ and pseudoscalar $\xi_5$ commonly used in a description of the spin degree of freedom of a massive spin-1/2 particle, is constructed. An analysis of one-to-one correspondence of this mapping is given. It is shown that for the one-to-one correspondence it is necessary to extend a class of real tensor quantities including besides $\xi_{\mu}$ and $\xi_5$, also odd vector $\hat{\xi}_{\mu}$, scalar $\hat{\xi}_5$, and (dual) pseudotensor $^{\ast}\zeta_{\mu \nu}$. In addition, it is shown that it is necessary either to restrict a class of the initial spinor $\psi_{\alpha}$ to Majorana one or to double the number of variables in the tensor aggregate $(\xi_{\mu},\,\xi_5,, ^{\ast}\zeta_{\mu \nu},\,\hat{\xi}_{\mu},\,\hat{\xi}_5)$. Various special cases of the desired mapping are considered. In particular, a connection with the Lagrangian suggested by A.M. Polyakov, is studied. It is also offered the way of obtaining the supersymmetric Lagrangian in terms of the even $\psi_{\alpha}$ and odd $\theta_{\alpha}$ spinors. The map of the Lagrandian leads to local SUSY Lagrangian in terms of $\xi_{\mu}$ and $\xi_5$.