Related papers: Geometry of Mechanics
We propose a geometrical approach to the investigation of Hamiltonian systems on (Pseudo) Riemannian manifolds. A new geometrical criterion of instability and chaos is proposed. This approach is more generic than well known reduction to the…
This work is a conceptual analysis of certain recent developments in the mathematical foundations of Classical and Quantum Mechanics which have allowed to formulate both theories in a common language. From the algebraic point of view, the…
Standandard Hamiltonian mechanics in its homogeneous formulation is applied to the study of discontinuities representing rapid changes of Hamiltonians. Different formulations of Hamiltonian mechanics are reviewed. An original representation…
In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we…
After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to…
The optimal control of a mechanical system is of crucial importance in many realms. Typical examples are the determination of a time-minimal path in vehicle dynamics, a minimal energy trajectory in space mission design, or optimal motion…
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…
In this paper we review a proposed geometrical formulation of quantum mechanics. We argue that this geometrization makes available mathematical methods from classical mechanics to the quantum frame work. We apply this formulation to the…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
In this article, we continue the program started in our previous article of exploring an important class of thermodynamic systems from a geometric point of view. In order to model the time evolution of systems verifying the two laws of…
The Lagrangian-Hamiltonian unified formalism of R. Skinner and R. Rusk was originally stated for autonomous dynamical systems in classical mechanics. It has been generalized for non-autonomous first-order mechanical systems, as well as for…
We study geometry of the phase space for finite-dimensional dynamical systems with degenerate Lagrangians. The Lagrangian and Hamiltonian constraint formalisms are treated as different local-coordinate pictures of the same invariant…
The geometrical description of Quantum Mechanics is reviewed and proposed as an alternative picture to the standard ones. The basic notions of observables, states, evolution and composition of systems are analised from this perspective, the…
We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean)…
Using a geometric formalism of elasticity theory we develop a systematic theoretical method for controlling and manipulating the mechanical response of slender solids to external loads. We formally express global mechanical properties…
In order to derive a large set of Hamiltonian dynamical systems, but with only first order Lagrangian, we resort to the formulation in terms of Lagrange-Souriau 2-form formalism. A wide class of systems derived in different phenomenological…
The treatment of exact conservation laws in Lagrangian gauge theories constitutes the main axis of the first part of the thesis. The formalism is developed as a self-consistent theory but is inspired by earlier works, mainly by…
This paper focuses on polynomial dynamical systems over finite fields. These systems appear in a variety of contexts, in computer science, engineering, and computational biology, for instance as models of intracellular biochemical networks.…
We study thermodynamical formalism of a discrete nonautonomous dynamical system determined by a sequence of continuous self-maps of a compact metric space. Using the methods of Convex Analysis we get variational principles for pressure…
This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of…