Related papers: Discrete Dynamics and Supergeometry
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
This paper presents a geometric-variational approach to continuous and discrete mechanics and field theories. Using multisymplectic geometry, we show that the existence of the fundamental geometric structures as well as their preservation…
In this article, we generalize the theory of discrete Lagrangian mechanics and variational integrators in two principal directions. First, we show that Lagrangian submanifolds of symplectic groupoids give rise to discrete dynamical systems,…
A new geometric procedure to construct symplectic methods for constrained mechanical systems is developed in this paper. The definition of a map coming from the notion of retraction maps allows to adapt the continuous problem to the…
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…
We study deterministic and quantum dynamics from a constructive "finite" point of view, since the introduction of a continuum, or other actual infinities in physics poses serious conceptual and technical difficulties, without any need for…
This article reviews the role of hidden symmetries of dynamics in the study of physical systems, from the basic concepts of symmetries in phase space to the forefront of current research. Such symmetries emerge naturally in the description…
Classical Hamiltonian spin systems are continuous dynamical systems on the symplectic phase space $(S^2)^n$. In this paper we investigate the underlying geometry of a time discretization scheme for classical Hamiltonian spin systems called…
Symplectic and Poisson geometry emerged as a tool to understand the mathematical structure behind classical mechanics. However, due to its huge development over the past century, it has become an independent field of research in…
We investigate the tension between symplecticity and gauge covariance in classical Hamiltonian mechanics. The pursuit of manifest covariance over manifest symplecticity results in a unique geometric formulation. Firstly, covariant yet…
A classic approach in dynamical systems is to use particular geometric structures to deduce statistical properties, for example the existence of invariant measures with stochastic-like behaviour such as large deviations or decay of…
This article explores some geometric and algebraic properties of the dynamical system which is represented by matrix differential equations arising from inertial navigation problems, such as the symplecticity and the orthogonality.…
We introduce the Euler-Lagrange cohomology to study the symplectic and multisymplectic structures and their preserving properties in finite and infinite dimensional Lagrangian systems respectively. We also explore their certain difference…
We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of…
This lecture is devoted to review some of the main properties of multisymplectic geometry. In particular, after reminding the standard definition of multisymplectic manifold, we introduce its characteristic submanifolds, the canonical…
We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This…
Many interesting physical systems have mathematical descriptions as finite-dimensional or infinite-dimensional Hamiltonian systems. Poincare who started the modern theory of dynamical systems and symplectic geometry developed a particular…
Hybrid dynamical systems are systems which undergo both continuous and discrete transitions. The Bolza problem from optimal control theory was applied to these systems and a hybrid version of Pontryagin's maximum principle was presented.…
This paper is devoted to the study of symplectic manifolds and their connection with Hamiltonian dynamical systems. We review some properties and operations on these manifolds and see how they intervene when studying the complete…
We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the…