Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userConstructing a class of solutions for the Hamilton-Jacobi equation in field theory
237
0
0.0
(
0
)
Authors
Danilo Bruno
Ask ChatGPT about the research
No Arabic abstract
A new approach leading to the formulation of the Hamilton-Jacobi equation for field theories is investigated within the framework of jet-bundles and multi-symplectic manifolds. An algorithm associating classes of solutions to given sets of boundary conditions of the field equations is provided. The paper also puts into evidence the intrinsic limits of the Hamilton-Jacobi method as an algorithm to determine families of solutions of the field equations, showing how the choice of the boundary data is often limited by compatibility conditions.
rate research
Read More
The paper is devoted to prove the existence of a local solution of the Hamilton-Jacobi equation in field theory, whence the general solution of the field equations can be obtained. The solution is adapted to the choice of the submanifold where the initial data of the field equations are assigned. Finally, a technique to obtain the general solution of the field equations, starting from the given initial manifold, is deduced.
By using the Hamilton-Jacobi [HJ] framework the three dimensional Palatini theory plus a Chern-Simons term [PCS] is analyzed. We report the complete set of $HJ$ Hamiltonians and a generalized $HJ$ differential from which all symmetries of the theory are identified. Moreover, we show that in spite of PCS Lagrangian produces Einsteins equations, the generalized $HJ$ brackets depend on a Barbero-Immirzi like parameter. In addition we complete our study by performing a canonical covariant analysis, and we construct a closed and gauge invariant two form that encodes the symplectic geometry of the covariant phase space.
Dirac equation is solved for some exponential potentials, hypergeometric-type potential, generalized Morse potential and Poschl-Teller potential with any spin-orbit quantum number $kappa$ in the case of spin and pseudospin symmetry, respectively. We have approximated for non s-waves the centrifugal term by an exponential form. The energy eigenvalue equations, and the corresponding wave functions are obtained by using the generalization of the Nikiforov-Uvarov method.
A systematic construction of a Lax pair and an infinite set of conservation laws for the Ernst equation is described. The matrix form of this equation is rewritten as a differential ideal of gl(2,R)-valued differential forms, and its symmetry condition is expressed as an exterior equation which is linear in the symmetry characteristic and has the form of a conservation law. By means of a recursive process, an infinite collection of such laws is then obtained, and the conserved charges are used to derive a linear exterior equation whose components constitute a Lax pair.
By using the Hamilton-Jacobi [HJ] framework the topological theories associated with Euler and Pontryagin classes are analyzed. We report the construction of a fundamental $HJ$ differential where the characteristic equations and the symmetries of the theory are identified. Moreover, we work in both theories with the same phase space variables and we show that in spite of Pontryagin and Euler classes share the same equations of motion their symmetries are different. In addition, we report all HJ Hamiltonians and we compare our results with other formulations reported in the literature.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
