On Field Theoretic Generalizations of a Poisson Algebra

A few generalizations of a Poisson algebra to field theory canonically formulated in terms of the polymomentum variables are discussed. A graded Poisson bracket on differential forms and an $(n+1)$-ary bracket on functions are considered. The Poisson bracket on differential forms gives rise to various generalizations of a Gerstenhaber algebra: the noncommutative (in the sense of Loday) and the higher-order (in the sense of the higher order graded Leibniz rule). The $(n+1)$-ary bracket fulfills the properties of the Nambu bracket including the ``fundamental identity'', thus leading to the Nambu-Poisson algebra. We point out that in the field theory context the Nambu bracket with a properly defined covariant analogue of Hamilton's function determines a joint evolution of several dynamical variables.Comment: 10 pages, LaTeX2e. Missprint in Ref. 1 is corrected (hep-th/9709229 instead of ...029

Canonical Structure of Classical Field Theory in the Polymomentum Phase Space

Canonical structure of the space-time symmetric analogue of the Hamiltonian formalism in field theory based on the De Donder-Weyl (DW) theory is studied. In $n$ space-time dimensions the set of $n$ polymomenta is associated to the space-time derivatives of field variables. The polysymplectic $(n+1)$-form generalizes the simplectic form and gives rise to a map between horizontal forms playing the role of dynamical variables and vertical multivectors generalizing Hamiltonian vector fields. Graded Poisson bracket is defined on forms and leads to the structure of a Z-graded Lie algebra on the subspace of the so-called Hamiltonian forms for which the map above exists. A generalized Poisson structure arises in the form of what we call a ``higher-order'' and a right Gerstenhaber algebra. Field euations and the equations of motion of forms are formulated in terms of the graded Poisson bracket with the DW Hamiltonian $n$-form H\vol (\vol is the space-time volume form and $H$ is the DW Hamiltonian function). A few applications to scalar fields, electrodynamics and the Nambu-Goto string, and a relation to the standard Hamiltonian formalism in field theory are briefly discussed. This is a detailed and improved account of our earlier concise communications (hep-th/9312162, hep-th/9410238, and hep-th/9511039).Comment: 45 pages, LaTeX2e, to appear in Reports on Mathematical Physics v. 41 No. 1 (1998

Extensive sampling and high-throughput sequencing reveal Posidoniomyces atricolor gen. et sp. nov. (Aigialaceae, Pleosporales) as the dominant root mycobiont of the dominant Mediterranean seagrass Posidonia oceanica

Volume: 55Start Page: 59End Page: 8

Oofem.org - Project status, challenges and needs

The paper describes the object-oriented design of OOFEM, a general, open-source finite element solver. The design combines several design approaches, that contribute to highly modular and extensible structure and that allowed sustainable development over the last two decades. The current capabilities are briefly presented. The aim of the paper is also to discuss some challenges and needs not only for the future development of the code, but also for the open-source modelling community