920 research outputs found
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 -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 satisfying additional
geometric constraints. Algebraically, the problem can be reformulated as
follows: for a vector space of dimension , classify -tuples of
skew-symmetric 2-forms such that for some non-degenerate symmetric
.Comment: 31 page
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