Partial Differential Hamiltonian Systems

@article{Vitagliano2013PartialDH,
title={Partial Differential Hamiltonian Systems},
author={Luca Vitagliano},
year={2013},
volume={65},
pages={1164 - 1200}
}
• L. Vitagliano
• Published 2013
• Mathematics, Physics
Abstract We define partial differential ( $\text{PD}$ in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, $\text{PD}$ Hamilton equations, $\text{PD}$ Noether theorem, $\text{PD}$ Poisson bracket, etc. Unlike the 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… Expand
11 Citations
GEOMETRY OF LAGRANGIAN AND HAMILTONIAN FORMALISMS IN THE DYNAMICS OF STRINGS
• Physics, Mathematics
• 2014
The Lagrangian description of mechanical systems and the Legendre Transformation (considered as a passage from the Lagrangian to the Hamiltonian formulation of the dynamics) for point-like objects,Expand
Tulczyjew Triples: From Statics to Field Theory
• Mathematics, Physics
• 2013
A geometric approach to dynamical equations of physics, based on the idea of the Tulczyjew triple, is presented. We show the evolution of these concepts, starting with the roots lying in theExpand
Multisymplectic formalism and the covariant phase
The formulation of a relativistic dynamical problem as a system of Hamilton equations by respecting the principles of Relativity is a delicate task, because in their classical form the HamiltonExpand
ON HIGHER DERIVATIVES AS CONSTRAINTS IN FIELD THEORY: A GEOMETRIC PERSPECTIVE
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 derivativeExpand
Hamilton–Jacobi diffieties
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 andExpand
Multisymplectic formulation of Yang--Mills equations and Ehresmann connections
We present a multisymplectic formulation of the Yang--Mills equations. The connections are represented by normalized equivariant 1-forms on the total space of a principal bundle, with values in a LieExpand
Variational principles for multisymplectic second-order classical field theories
• Mathematics, Physics
• 2014
We state a unified geometrical version of the variational principles for second-order classical field theories. The standard Lagrangian and Hamiltonian variational principles and the correspondingExpand
VARIATIONAL PRINCIPLES FOR MULTISYMPLECTIC SECOND-ORDER CLASSICAL FIELD THEORIES
We state a unified geometrical version of the variational principles for second-order classical field theories. The standard Lagrangian and Hamiltonian variational principles and the correspondingExpand
GEOMETRIC HAMILTON–JACOBI FIELD THEORY
I review my proposal about how to extend the geometric Hamilton–Jacobi theory to higher derivative field theories on fiber bundles.
A Complete Bibliography of Publications in Canadian Journal of Mathematics = Journal canadien de mathématiques for the decade 1940–1949
33]. algebraic [5]. Angular [24]. Applications [30]. arbitrary [8]. associated [18]. Axiomatic [31]. between [27]. bounded [11, 14]. Boundedness [16]. Cayleyan [4]. certain [9]. characteristic [35].Expand

References

SHOWING 1-10 OF 82 REFERENCES
Symplectic and Poisson Geometry on Loop Spaces of Manifolds and Nonlinear Equations
We consider some differential geometric classes of local and nonlocal Poisson and symplectic structures on loop spaces of smooth manifolds which give natural Hamiltonian and multihamiltonianExpand
AV-differential geometry: Euler–Lagrange equations
• Mathematics, Physics
• 2007
Abstract A general, consistent and complete framework for geometrical formulation of mechanical systems is proposed, based on certain structures on affine bundles (affgebroids) that generalize LieExpand
Singular Lagrangian Systems on Jet Bundles
• Mathematics, Physics
• 2002
The jet bundle description of time-dependent mechanics is revisited. The constraint algorithm for singular Lagrangians is discussed and an exhaustive description of the constraint functions is given.Expand
A finite-dimensional canonical formalism in the classical field theory
AbstractA canonical formalism based on the geometrical approach to the calculus of variations is given. The notion of multi-phase space is introduced which enables to define whole the canonicalExpand
Geometry of Hamiltonian n-vector fields in multisymplectic field theory
• Mathematics, Physics
• 2001
Abstract Multisymplectic geometry—which originates from the well known De Donder–Weyl (DW) theory—is a natural framework for the study of classical field theories. Recently, two algebraic structuresExpand
Presymplectic manifolds and the Dirac-Bergmann theory of constraints
• Mathematics
• 1978
We present an algorithm which enables us to state necessary and sufficient conditions for the solvability of generalized Hamilton‐type equations of the form ι (X) ω=α on a presymplectic manifoldExpand
ON POISSON BRACKETS OF HYDRODYNAMIC TYPE
• Mathematics
• 2008
I. Riemannian geometry of multidimensional Poisson brackets of hydrodynamic type. In [1] we developed the Hamiltonian formalism of general onedimensional systems of hydrodynamic type. Now supposeExpand
The Hamiltonian formulation of regular rth-order Lagrangian field theories
A Hamiltonian formulation of regular rth-order Lagrangian field theories over an m-dimensional manifold is presented in terms of the Hamilton-Cartan formalism. It is demonstrated that a uniquelyExpand
Constraint algorithm for k-presymplectic Hamiltonian systems. Application to singular field theories
• Mathematics, Physics
• 2009
The k-symplectic formulation of field theories is especially simple, since only tangent and cotangent bundles are needed in its description. Its defining elements show a close relationship with thoseExpand
Geometry of Lagrangian First-order Classical Field Theories
• Mathematics, Physics
• 1995
We construct a lagrangian geometric formulation for first-order field theories using the canonical structures of first-order jet bundles, which are taken as the phase spaces of the systems inExpand