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Search Results: 1 - 10 of 208502 matches for " L. Vitagliano "
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On Higher Derivatives as Constraints in Field Theory: a Geometric Perspective
L. Vitagliano
Mathematics , 2010, DOI: 10.1142/S0219887811005968
Abstract: We formalize geometrically the idea that the (de Donder) Hamiltonian formulation of a higher derivative Lagrangian field theory can be constructed understanding the latter as a first derivative theory subjected to constraints.
Partial Differential Hamiltonian Systems
L. Vitagliano
Mathematics , 2009, DOI: 10.4153/CJM-2012-055-0
Abstract: We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field theory, in our formalism, the geometric structure (kinematics) and the dynamical information on the "phase space" appear as just different components of one single geometric object.
Secondary Calculus and the Covariant Phase Space
L. Vitagliano
Mathematics , 2008, DOI: 10.1016/j.geomphys.2008.12.001
Abstract: The covariant phase space of a Lagrangian field theory is the solution space of the associated Euler-Lagrange equations. It is, in principle, a nice environment for covariant quantization of a Lagrangian field theory. Indeed, it is manifestly covariant and possesses a canonical (functional) "presymplectic structure" w (as first noticed by Zuckerman in 1986) whose degeneracy (functional) distribution is naturally interpreted as the Lie algebra of gauge transformations. We propose a fully rigorous approach to the covariant phase space in the framework of secondary calculus. In particular we describe the degeneracy distribution of w. As a byproduct we rederive the existence of a Lie bracket among gauge invariant functions on the covariant phase space.
The Lagrangian-Hamiltonian Formalism for Higher Order Field Theories
L. Vitagliano
Mathematics , 2009, DOI: 10.1016/j.geomphys.2010.02.003
Abstract: We generalize the Lagrangian-Hamiltonian formalism of Skinner and Rusk to higher order field theories on fiber bundles. As a byproduct we solve the long standing problem of defining, in a coordinate free manner, a Hamiltonian formalism for higher order Lagrangian field theories. Namely, our formalism does only depend on the action functional and, therefore, unlike previously proposed ones, is free from any relevant ambiguity.
The Hamilton-Jacobi Formalism for Higher Order Field Theories
L. Vitagliano
Mathematics , 2010, DOI: 10.1142/S0219887810004889
Abstract: We extend the geometric Hamilton-Jacobi formalism for hamiltonian mechanics to higher order field theories with regular lagrangian density. We also investigate the dependence of the formalism on the lagrangian density in the class of those yelding the same Euler-Lagrange equations.
Hamilton-Jacobi Diffieties
L. Vitagliano
Mathematics , 2011, DOI: 10.1016/j.geomphys.2011.05.003
Abstract: Diffieties formalize geometrically the concept of differential equations. We introduce and study Hamilton-Jacobi diffieties. They are finite dimensional subdiffieties of a given diffiety and appear to play a special role in the field theoretic version of the geometric Hamilton-Jacobi theory.
Iterated Differential Forms II: Riemannian Geometry Revisited
A. M. Vinogradov,L. Vitagliano
Mathematics , 2006, DOI: 10.1134/S1064562406020074
Abstract: A natural extension of Riemannian geometry to a much wider context is presented on the basis of the iterated differential form formalism developed in math.DG/0605113 and an application to general relativity is given.
Iterated Differential Forms I: Tensors
A. M. Vinogradov,L. Vitagliano
Mathematics , 2006, DOI: 10.1134/S1064562406020037
Abstract: We interpret tensors on a smooth manifold M as differential forms over a graded commutative algebra called the algebra of iterated differential forms over M. This allows us to put standard tensor calculus in a new differentially closed context and, in particular, enriches it with new natural operations. Applications will be considered in subsequent notes.
Iterated Differential Forms VI: Differential Equations
A. M. Vinogradov,L. Vitagliano
Mathematics , 2007, DOI: 10.1134/S1064562407050146
Abstract: We describe the first term of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence (see math.DG/0610917) of the diffiety (E,C), E being the infinite prolongation of an l-normal system of partial differential equations, and C the Cartan distribution on it.
Iterated Differential Forms V: C-Spectral Sequence on Infinite Jet Spaces
A. M. Vinogradov,L. Vitagliano
Mathematics , 2007, DOI: 10.1134/S1064562407050092
Abstract: In the preceding note math.DG/0610917 the $\Lambda_{k-1}\mathcal{C}$--spectral sequence, whose first term is composed of \emph{secondary iterated differential forms}, was constructed for a generic diffiety. In this note the zero and first terms of this spectral sequence are explicitly computed for infinite jet spaces. In particular, this gives an explicit description of secondary covariant tensors on these spaces and some basic operations with them. On the basis of these results a description of the $\Lambda_{k-1}\mathcal{C}$--spectral sequence for infinitely prolonged PDE's will be given in the subsequent note.
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