n-symplectic Hamilton-Jacobi Theory
نویسنده
چکیده
In previous work n-symplectic geometry on the adapted frame bundle λ : LπE → E of an n = (m + k)-dimensional fiber bundle π : E → M has been used to forumulate covariant Lagrangian field theory that is standardly formulated on the bundle J1π of 1 jets of sections of π. In this paper we set up an n-symplectic Hamilton-Jacobi equation in order to identify the analogue of a polarization that plays an important role in geometric quantization theory. We find that a local solution of the n-symplectic H-J equation yields a locally defined H = GL(m) × GL(k) subbundle of LπE. This suggests that the gobal structure on LπE that is generated by local solutions of the Hamilton-Jacobi equations is a foliation of LπE by H subbundles. Such a foliation of LπE is shown to exist for the k-tuple of scalar fields on Minkowski spacetime.
منابع مشابه
Analysis of the Geometric Structure of the Hamilton-Jacobi Equation via Generating Functions of Symplectic Transforms
The geometric structure of the Hamilton-Jacobi equation is analyzed by using symplectic geometry. Generating function of symplectic transform plays an important role. It will be shown that the Hamilton-Jacobi equation possesses important geometric properties such as existence condition and maximality of the stabilizing solution, which are well-known in the Riccati equation.
متن کاملGenerating functions of multi-symplectic RK methods via DW Hamilton-Jacobi equations
In this paper we investigate Donder-Weyl (DW) Hamilton-Jacobi equations and establish the connection between DW Hamilton-Jacobi equations and multi-symplectic Hamiltonian systems. Based on the study of DW Hamilton-Jacobi equations, we present the generating functions for multi-symplectic partitioned Runge-Kutta (PRK) methods.
متن کاملConstructing a class of solutions for the Hamilton–Jacobi equation in field theory
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...
متن کاملCosmological perturbations in FRW model with scalar field within Hamilton-Jacobi formalism and symplectic projector method
The Hamilton-Jacobi analysis is applied to the dynamics of the scalar fluctuations about the Friedmann-Robertson-Walker (FRW). The gauge conditions are found from the consistency conditions. The physical degrees of freedom of the model are obtain by symplectic projector method. The role of the linearly dependent Hamiltonians and the gauge variables in Hamilton-Jacobi formalism is discussed. ∗E-...
متن کاملExtended Applicability of the Symplectic Pontryagin Method
Abstract. The Symplectic Pontryagin method was introduced in a previous paper. This work shows that this method is applicable under less restrictive assumptions. Existence of solutions to the Symplectic Pontryagin scheme are shown to exist without the previous assumption on a bounded gradient of the discrete dual variable. The convergence proof uses the representation of solutions to a Hamilton...
متن کامل