On a class of polynomial Lagrangians

In the framework of finite order variational sequences a new class of Lagrangians arises, namely, \emph{special} Lagrangians. These Lagrangians are the horizontalization of forms on a jet space of lower order. We describe their properties together with properties of related objects, such as Poincar\'e--Cartan and Euler--Lagrange forms, momenta and momenta of generating forms, a new geometric object arising in variational sequences. Finally, we provide a simple but important example of special Lagrangian, namely the Hilbert--Einstein Lagrangian.Comment: LaTeX2e, amsmath, diagrams, hyperref; 15 page

On the formalism of local variational differential operators

The calculus of local variational differential operators introduced by B. L. Voronov, I. V. Tyutin, and Sh. S. Shakhverdiev is studied in the context of jet super space geometry. In a coordinate-free way, we relate these operators to variational multivectors, for which we introduce and compute the variational Poisson and Schouten brackets by means of a unifying algebraic scheme. We give a geometric definition of the algebra of multilocal functionals and prove that local variational differential operators are well defined on this algebra. To achieve this, we obtain some analytical results on the calculus of variations in smooth vector bundles, which may be of independent interest. In addition, our results give a new a new efficient method for finding Hamiltonian structures of differential equations

On the bi-Hamiltonian Geometry of WDVV Equations

We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and third order, respectively).Comment: 21 pages, revised published version; exposition substantially improve

Computing with Hamiltonian operators

Hamiltonian operators are used in the theory of integrable partial differential equations to prove the existence of infinite sequences of commuting symmetries or integrals. In this paper it is illustrated the new Reduce package \cde for computations on Hamiltonian operators. \cde can compute the Hamiltonian properties of skew-adjointness and vanishing Schouten bracket for a differential operator, as well as the compatibility property of two Hamiltonian operators and the Lie derivative of a Hamiltonian operator with respect to a vector field. It can also make computations on (variational) multivectors, or functions on supermanifolds. This can open the way to applications in other fields of Mathematical Physics.Comment: 35 pages, published version; software is available on the web page of the author http://poincare.unisalento.it/vitol

Systems of conservation laws with third-order Hamiltonian structures

We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $\mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^{\alpha} \in \Lambda^2(W)$ such that $$\phi_{\beta \gamma}A^{\beta}\wedge A^{\gamma}=0,$$ for some non-degenerate symmetric $\phi$.Comment: 31 page