Related papers: q-Cosymplectic Geometry, Integrability and Reducti…
I construct a global version of the local polysymplectic approach to covariant Hamiltonian field theory pioneered by C. Gunther. Beginning with the geometric framework of the theory, I specialize to vertical vector fields to construct the…
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…
This paper is devoted to the problem of classification, up to smooth isomorphisms or up to orbital equivalence, of smooth integrable vector fields on 2-dimensional surfaces, under some nondegeneracy conditions. The main continuous…
Just like decent classical difference-difference systems define symplectic maps on suitable phase spaces, their counterparts with properly ordered noncommutative entries come as Heisenberg equations of motion for corresponding quantum…
We study local analytical integrability in a neighborhood of $p:-q$ resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the non-smooth involution $…
It is shown that the geometry of locally homogeneous multisymplectic manifolds (that is, smooth manifolds equipped with a closed nondegenerate form of degree > 1, which is locally homogeneous of degree k with respect to a local Euler field)…
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 the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We…
Multisymplectic geometry - which originates from the well known de Donder-Weyl theory - is a natural framework for the study of classical field theories. Recently, two algebraic structures have been put forward to encode a given theory…
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…
Over the past five years, there has been significant progress on the problem of quantization of diffeomorphism covariant field theories with {\it local} degrees of freedom. The absence of a background space-time metric in these theories…
We derive a complete asymptotic expansion of generalized Witten integrals for Hamiltonian circle actions on arbitrary symplectic manifolds, characterizing the coefficients in the expansion as integrals over the symplectic strata of the…
Following the idea of a field quantization of gravity as realized in group field theory, we construct a minisuperspace model where the wavefunction of canonical quantum cosmology (either Wheeler-DeWitt or loop quantum cosmology) is promoted…
We introduce an algorithm for computing geodesics on sampled manifolds that relies on simulation of quantum dynamics on a graph embedding of the sampled data. Our approach exploits classic results in semiclassical analysis and the…
We study here some aspects of the topology of the space of smooth, stable, genus 0 curves in a Riemannian manifold $X$, i.e. the Kontsevich stable curves, which are not necessarily holomorphic. We use the Hofer-Wysocki-Zehnder polyfold…
We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main…
The higher dimensional Quantum General Relativity of a Riemannian manifold being an embedded space in a space-time being a Lorentzian manifold is investigated. The model of quantum geometrodynamics, based on the Wheeler-DeWitt equation…
We first give a pedagogical introduction to the differential calculus on q-groups and analize the relation between differential calculus and q-Lie algebra. Equivalent definitions of bicovariant differential calculus are studied and their…
Motivated by the recent connection between nonholonomic integrable systems and twisted Poisson manifolds made in \cite{balseiro_garcia_naranjo}, this paper investigates the global theory of integrable Hamiltonian systems on almost…
Dynamists have been studying Hamiltonian systems for a long time. However, many physical systems are dissipative and do not preserve a symplectic form. This is the case, for example, with systems involving friction, which multiply the…