Partial differential Hamiltonian systems (Q2871246)

The author presents his own approach to higher-order field theories in a general framework of the so-called partial differential Hamiltonian systems. As is well known, Lagrangian mechanics is developed on the tangent bundle of the configuration manifold, whereas the Hamiltonian mechanics has a dual picture on its cotangent bundle. In the case of first-order Lagrangian field theories the dynamics occurs on the space of 1-jets of sections of the configuration bundle, and the arena for the corresponding Hamiltonian description is the so-called multimomentum space, which is obtained using the space of basic forms on the configuration bundle. In both cases, one can connect the two sides (Lagrangian and Hamiltonian) using a Legendre transform.NEWLINENEWLINEThe situation becomes more involved for higher-order theories, due to the ambiguities on the definition of regularity and the identification of proper generalized momenta. A recent approach to this problem should be noticed that uses an extension of the Skinner and Rusk formalism for mechanics by \textit{C. M. Campos} et al. [J. Phys. A, Math. Theor. 43, No. 45, Article ID 455206, 26 p. (2010; Zbl 1248.58010); ibid. 42, No. 47, Article ID 475207, 24 p. (2009; Zbl 1231.58005)]. To avoid these problems, the author introduces the notion of partial differential Hamiltonian systems on a fiber bundle \(\alpha : P \longrightarrow M\) as a \(\delta\)-closed \(2\)-form on \(P\) satisfying some properties with \(\delta\) a (restricted) de Rham differential. The author also discusses symmetries and Poisson brackets in this framework. The connection with other approaches (multisymplectic geometry and similar structures) is also studied. Some examples are considered.