Resumo:
In this work, we use anticommutative variables from Grassmann algebra to generalize the Lagrangian and Hamiltonian formalisms, aiming to describe massive particles with spin 1/2, both relativistic and non-relativistic, within a framework called pseudomechanics. Initially, we present the fundamental concepts of Grassmann algebra, emphasizing the anticommutative property of its elements. The incorporation of these variables into the aforementioned formalisms resulted in new Euler-Lagrange and Hamilton equations, as well as extensions of Noether's theorem and Poisson brackets. Given that pseudomechanical systems are intrinsically singular, we introduced Dirac's formalism to properly handle such systems. In the non-relativistic regime, we developed models describing particles interacting with external magnetic fields, spin-orbit coupling, and interactions between the spins of two particles, further demonstrating the conservation of total angular momentum in systems with rotational symmetry. In the relativistic regime, we verified the invariance of the system under supersymmetry and showed that the canonical quantization of this formalism leads to the Dirac equation, linking it to the symmetries associated with supersymmetric transformations.
