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Register a new userThe Hamilton-Jacobi characteristic equations for topological invariants: Pontryagin and Euler classes
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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.
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The symplectic analysis for the four dimensional Pontryagin and Euler invariants is performed within the Faddeev-Jackiw context. The Faddeev-Jackiw constraints and the generalized Faddeev-Jackiw brackets are reported; we show that in spite of the Pontryagin and Euler classes give rise the same equations of motion, its respective symplectic structures are different to each other. In addition, a quantum state that solves the Faddeev-Jackiw constraints is found, and we show that the quantum states for these invariants are different to each other. Finally, we present some remarks and conclusions.
The Hamilton-Jacobi analysis of three dimensional gravity defined in terms of Ashtekar-like variables is performed. We report a detailed analysis where the complete set of Hamilton-Jacobi constraints, the characteristic equations and the gauge transformations of the theory are found. We find from integrability conditions on the Hamilton-Jacobi Hamiltonians that the theory is reduced to a $BF$ field theory defined only in terms of self-dual (or anti-self-dual) variables; we identify the dynamical variables and the counting of physical degrees of freedom is performed. In addition, we compare our results with those reported by using the canonical formalism.
The Euler characteristic $chi =|V|-|E|$ and the total length $mathcal{L}$ are the most important topological and geometrical characteristics of a metric graph. Here, $|V|$ and $|E|$ denote the number of vertices and edges of a graph. The Euler characteristic determines the number $beta$ of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via the Weyls law. We show theoretically and confirm experimentally that the Euler characteristic can be determined (heard) from a finite sequence of the lowest eigenenergies $lambda_1, ldots, lambda_N$ of a simple quantum graph, without any need to inspect the system visually. In the experiment quantum graphs are simulated by microwave networks. We demonstrate that the sequence of the lowest resonances of microwave networks with $beta leq 3$ can be directly used in determining whether a network is planar, i.e., can be embedded in the plane. Moreover, we show that the measured Euler characteristic $chi$ can be used as a sensitive revealer of the fully connected graphs.
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.
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.
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