Generalized Velocities Degenerate Lagrangian Field Theory System And Its Local Issues

In this article we start from the Lagrangian density of Dirac field, to elicit a kind of system to which can not give the Hamiltonian expression through the nomal Legendre transformation. The Lagrangian function of this kind of system is linearly depending on the velocity. We call them linear velocity degenerate Lagrangian system. It is found that this kind of system is very common. To study the property of this kind of degenerate Lagrangian system systematically is our job. From its Lagrangian equation, we derive the Poisson bracket and Hamiltonian structure. We prove the symmetry properties of the Poisson bracket by studying the corresponding super matrix. We study the Bosonic system and the Bosonic-Fermionic system seperately, and put emphasis on the characters of the Poisson bracket and especially on the Jacobi identity. This kind of Lagrangian system and its Hamiltonian structure was used in literature. However, we did not yet find a systematic detailed derivation. Our next main point is that we find there is a localization problem when we extend the method to the two-dimension field system of such degenerate Lagrangian function. We give the sufficient condition which can assure the system to be local. And then we finish the work about extending our approach to the two-dimension field system with such degenerate Lagrangian function.