Related papers: Remarks on Hamiltonian Structures in G_2-Geometry
We study the Euler-Lagrange cohomology and explore the symplectic or multisymplectic geometry and their preserving properties in classical mechanism and classical field theory in Lagrangian and Hamiltonian formalism in each case…
Some aspects of the connection between differential geometry and multidimensional soliton equations are discussed.
We introduce the notions of generalised (bi-)Hamiltonian structures which generalise naturally the (bi-)Hamiltonian structures of evolutionary partial differential equations. In the hydrodynamic case, these structures are characterised in…
Is is known that the loop space associated to a Riemannian manifold admits a quasi-symplectic structure. This article shows that this structure is not likely to recover the underlying Riemannian metric by proving a result that is a strong…
The mathematical theory underlying Hamiltonian mechanics is called symplectic geometry. So symplectic geometry arose from the roots of mechanics and is seen as one of the most valuable links between physics and mathematics today. Symplectic…
Certain dissipative physical systems closely resemble Hamiltonian systems in $\mathbb{R}^{2n}$, but with the canonical equation for one of the variables in each conjugate pair rescaled by a real parameter. To generalise these dynamical…
We prove that an integrable system over a symplectic manifold, whose symplectic form is covariantly constant w.r.t. the Gauss-Manin connection, carries a natural hyper-symplectic structure. Moreover, a special Kaehler structure is induced…
This paper provides a geometric description for Lie--Hamilton systems on $\mathbb{R}^2$ with locally transitive Vessiot--Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of…
In this note, we extend to the singular case some results on the birational geometry of irreducible holomorphic symplectic manifolds.
This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…
The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds…
Several geometric flows on symplectic manifolds are introduced which are potentially of interest in symplectic geometry and topology. They are motivated by the Type IIA flow and T-duality between flows in symplectic geometry and flows in…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
We present a general classification of Hamiltonian multivector fields and of Poisson forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories. This is a prerequisite for computing…
Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a…
This paper defines a symplectic form on the infinite dimensional Fr\'echet manifold of framed curves of fixed length over a simply connected Riemannian manifold of constant curvature. The paper then considers Hamiltonian systems generated…
In symplectic topology one uses elliptic methods to prove rigidity results about symplectic manifolds and solutions of Hamiltonian equations on them, where the most basic example is given by geodesics on Riemannian manifolds. Harmonic maps…
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the…
In order to describe the impact of different geometric structures and constraints for the dynamics of a Hamiltonian system, in this paper, for a magnetic Hamiltonian system defined by a magnetic symplectic form, we first drive precisely the…
We propose a new definition of so called Hamiltonian forms in n-plectic geometry and show that they have a non-trivial Lie infinity-algebra structure.