Related papers: Geometry of optimal control problems and Hamiltoni…
Cosymplectic geometry has been proven to be a very useful geometric background to describe time-dependent Hamiltonian dynamics. In this work, we address the globalization problem of locally cosymplectic Hamiltonian dynamics that failed to…
We develop a discrete analogue of Hamilton-Jacobi theory in the framework of discrete Hamiltonian mechanics. The resulting discrete Hamilton-Jacobi equation is discrete only in time. We describe a discrete analogue of Jacobi's solution and…
We study the geometry of a weak Riemannian metric on the infinite dimensional manifold of compact spacelike Cauchy hypersurfaces in a globally hyperbolic spacetime. We show that the geodesic distance (i.e. the infimum of lengths of paths…
Over the past six years, a detailed framework has been constructed to unravel the quantum nature of the Riemannian geometry of physical space. A review of these developments is presented at a level which should be accessible to graduate…
Lectures given at International School of Physics ``Enrico Fermi'', Varenna, Villa Monastero, June 28-July 7 1994
We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton-Jacobi equations (HJ), prove the comparison principle and that…
These notes form an extended version of a minicourse delivered in Universite de Montreal (June 2002) within the framework of a NATO workshop ``Normal Forms, Bifurcations and Finiteness Problems in Differential Equations''. The focus is on…
This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special…
A survey of some recent and important results which have to do with integrable equations and their relationship with the theory of surfaces is given. Some new results are also presented. The concept of the moving frame is examined, and it…
Contact Hamiltonian dynamics is a subject that has still a short history, but with relevant applications in many areas: thermodynamics, cosmology, control theory, and neurogeometry, among others. In recent years there has been a great…
In any dimension $n+1\ge 4$ we construct a sequence of closed $(n+1)$-dimensional Riemannian manifolds with positive Ricci curvature admitting embedded two-sided minimal hypersurfaces such that the following hold: (i) any such hypersurface…
We overview a new mechanism whereby classical Riemannian geometry emerges out of the differential structure on quantum spacetime, as extension data for the classical algebra of differential forms. Outcomes for physics include a new formula…
These notes, based on a graduate course I gave at Hamburg University in 2003, are intended to students having basic knowledges of differential geometry. Their main purpose is to provide a quick and accessible introduction to different…
The paper discusses various aspects of time-optimal control of quantum spin systems, modelled as right-invariant systems on a compact Lie group G. The main results are the reduction of such a system to an equivalent system on a homogeneous…
This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 -…
We describe the relation between block Jacobi matrices and minimization problems for discrete time optimal control problems. Using techniques developed for the continuous case, we provide new algorithms to compute spectral invariants of…
This is the note for the four lectures given by the author in the ``International Short-School/Conference on Affine Algebraic Geometry and the Jacobian Conjecture" at Chern Institute of Mathematics, Nankai University, Tianjin, China. July…
This is the text of a series of five lectures given by the author at the "Second Annual Spring Institute on Noncommutative Geometry and Operator Algebras" held at Vanderbilt University in May 2004. It is meant as an overview of recent…
This paper formulates an optimal control problem for a system of rigid bodies that are connected by ball joints and immersed in an irrotational and incompressible fluid. The rigid bodies can translate and rotate in three-dimensional space,…
This work is devoted to review the modern geometric description of the Lagrangian and Hamiltonian formalisms of the Hamilton--Jacobi theory. The relation with the "classical" Hamiltonian approach using canonical transformations is also…