Boundary values as Hamiltonian variables. I. New Poisson brackets

Functional Analysis and Its Applications, 2003

2007

Abstract iopas120323 2/36 q The Hamiltonian for interacting classical fields with quite general theories of dynamic geometry generates the evolution of a spatial region along a time-like vector field. q It includes a boundary term which determines the value of the Hamiltonian. From this value one obtains the quasi-local quantities: energy-momentum, angular-momentum/center-of-mass. q The Hamiltonian boundary term also directly controls the boundary term in the variation of the Hamiltonian.

1995

In this report it is proposed to generalize the definition of Poisson brackets in order to treat spatial integrals of divergences as Hamiltonians which generate a kind of Hamiltonian equations on the boundary. Nonlinear Schrodinger equation is used as an illustrative example.

2012

We give an exposition of some recent crucial achievements in the theory of multidimensional Poisson brackets of hydrodynamic type. In particular, we solve the well-known Dubrovin-Novikov problem posed as long ago as 1984 in connection with the Hamiltonian theory of systems of hydrodynamic type, namely, the classification problem for multidimensional Poisson brackets of hydrodynamic type. In contrast to the one-dimensional case, in the general case, a nondegenerate multidimensional Poisson bracket of hydrodynamic type cannot be reduced to a constant form by a local change of coordinates. We obtain the classification of all nonsingular nondegenerate multidimensional Poisson brackets of hydrodynamic type for any number N of components and for any dimension n by differentialgeometric methods. This problem is equivalent to the classification of a special class of flat pencils of metrics. A key role in the solution of this problem was played by the theory of compatible metrics that had been earlier constructed by the present author.

Annals of Physics, 1986

Poisson brackets that are spacetime covariant are presented for a variety of relativistic field theories. These theories include electromagnetism, general relativity, and general relativistic fluids and plasmas in Eulerian representation. The examples presented suggest the development of a general theory; the beginnings of such a theory are presented. Our covariant bracket formalism provides a general setting for, amongst other things, clarifying the transition from the covariant formalism to the dynamical 3 + 1 Hamiltonian formalism of Dirac and Arnowitt, Deser, and Misner. We illustrate the relevant procedures with electromagnetism.

2002

1981

SciPost Physics Proceedings, 2021

In this paper, we are going to construct the classical field theory on the boundary of the embedding of \mathbb{R} \times S^{1}ℝ×S1 into the manifold MM by the Jacobi sigma model. By applying the poissonization procedure and by generalizing the known method for Poisson sigma models, we express the fields of the model as perturbative expansions in terms of the reduced phase space of the boundary. We calculate these fields up to the second order and illustrate the procedure for contact manifolds.

Physica D: Nonlinear Phenomena, 1986

Hamiltonian structures for 2-or 3-dimensional incompressible flows with a free boundary are determined which generalize a previous structure of Zakharov for irrotational flow. Our Poisson bracket is determined using the method of Arnold, namely reduction from canonical variables in the Lagrangian (material) description. Using this bracket, the Hamiltonian form for the equations of a liquid drop with a free boundary having surface tension is demonstrated. The structure of the bracket in terms of a reduced cotangent bundle of a principal bundle is explained. In the case of two-dimensional flows, the vorticity bracket is determined and the generalized enstrophy is shown to be a Casimir function, This investigation also clears up some confusion in the literature concerning the vorticity bracket, even for fixed boundary flows.

Journal of Nonlinear Mathematical Physics, 2021

A general differential-algebraic approach is devised for constructing multi-component Hamiltonian operators as differentiations on suitably constructed loop Lie algebras. The related Novikov-Leibniz algebraic structures are presented and a new non-associative "Riemann" algebra is constructed, which is closely related to the infinite multi-component Riemann integrable hierarchies. A close relationship to the standard symplectic analysis techniques is also discussed.