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Jiménez Alburquerque, Fernando

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fjimenez@ind.uned.es
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Jiménez Alburquerque
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Fernando
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2014 - 201962020 - 20257
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Mostrando 1 - 10 de 13
  • Miniatura
    Publicación
    De la manzana de Newton a los drones de Amazon
    (Prensa Científica, 2021-08-01) Jiménez Alburquerque, Fernando; Sardón, Cristina
    La mecánica clásica suele concebirse como una disciplina limitada a las leyes de Newton y al cálculo vectorial que aprendemos en la secundaria. Sin embargo, su tratamiento matemático abarca formalismos mucho más ricos y potentes. Una de sus encarnaciones actuales se basa en el empleo de la geometría diferencial, el área de las matemáticas que estudia las propiedades de los espacios curvos. Esta formulación se conoce como mecánica geométrica. Esta nueva disciplina ha contribuido a desarrollar varios ámbitos de la geometría moderna, como la geometría simpléctica. Al mismo tiempo, ha encontrado numerosas aplicaciones en campos como la robótica o el control de drones.
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    Fractional variational integrators based on convolution quadrature
    (Springer, 2025-02-05) Hariz Belgacem, Khaled; Jiménez Alburquerque, Fernando; Ober-Blöbaum, Sina
    Fractional dissipation is a powerful tool to study nonlocal physical phenomena such as damping models. The design of geometric, in particular, variational integrators for the numerical simulation of such systems relies on a variational formulation of the model. In Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021), a new approach is proposed to deal with dissipative systems including fractionally damped systems in a variational way for both, the continuous and discrete setting. It is based on the doubling of variables and their fractional derivatives. The aim of this work is to derive higher-order fractional variational integrators by means of convolution quadrature (CQ) based on backward difference formulas. We then provide numerical methods that are of order 2 improving a previous result in Jiménez and Ober-Blöbaum (J Nonlinear Sci 31:46, 2021). The convergence properties of the fractional variational integrators and saturation effects due to the approximation of the fractional derivatives by CQ are studied numerically.
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    Feedback integrators
    (Springer, 2016-06-23) Eui Chang, Dong; Jiménez Alburquerque, Fernando; Perlmutter, Matthew
    A new method is proposed to numerically integrate a dynamical system on a manifold such that the trajectory stably remains on the manifold and preserves the first integrals of the system. The idea is that given an initial point in the manifold we extend the dynamics from the manifold to its ambient Euclidean space and then modify the dynamics outside the intersection of the manifold and the level sets of the first integrals containing the initial point such that the intersection becomes a unique local attractor of the resultant dynamics. While the modified dynamics theoretically produces the same trajectory as the original dynamics, it yields a numerical trajectory that stably remains on the manifold and preserves the first integrals. The big merit of our method is that the modified dynamics can be integrated with any ordinary numerical integrator such as Euler or Runge–Kutta. We illustrate this method by applying it to three famous problems: the free rigid body, the Kepler problem and a perturbed Kepler problem with rotational symmetry. We also carry out simulation studies to demonstrate the excellence of our method and make comparisons with the standard projection method, a splitting method and Störmer–Verlet schemes.
  • Miniatura
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    Fractional damping through restricted calculus of variations
    (Springer, 2021-04-05) Jiménez Alburquerque, Fernando; Ober-Blöbaum, Sina; Engineering and Physical Sciences Research Council
    We deliver a novel approach towards the variational description of Lagrangian mechanical systems subject to fractional damping by establishing a restricted Hamilton’s principle. Fractional damping is a particular instance of non-local (in time) damping, which is ubiquitous in mechanical engineering applications. The restricted Hamilton’s principle relies on including fractional derivatives to the state space, the doubling of curves (which implies an extra mirror system) and the restriction of the class of varied curves. We will obtain the correct dynamics and will show rigorously that the extra mirror dynamics is nothing but the principal one in reversed time; thus, the restricted Hamilton’s principle is not adding extra physics to the original system. The price to pay, on the other hand, is that the fractional damped dynamics is only a sufficient condition for the extremals of the action. In addition, we proceed to discretise the new principle. This discretisation provides a set of numerical integrators for the continuous dynamics that we denote Fractional Variational Integrators (FVIs). The discrete dynamics is obtained upon the same ingredients, say doubling of discrete curves and restriction of the discrete variations. We display the performance of the FVIs, which have local truncation order 1, in two examples. As other integrators with variational origin, for instance those generated by the discrete Lagrange–d’Alembert principle, they show a superior performance tracking the dissipative energy, in opposition to direct (order 1) discretisations of the dissipative equations, such as explicit and implicit Euler schemes.
  • Miniatura
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    Geometry preserving numerical methods for physical systems with finite-dimensional Lie algebras
    (Springer, 2023-12-23) Blanco, L.; Jiménez Alburquerque, Fernando; Lucas, J. de; Sardón, Cristina
    We propose a geometric integrator to numerically approximate the flow of Lie systems. The key is a novel procedure that integrates the Lie system on a Lie group intrinsically associated with a Lie system on a general manifold via a Lie group action and then generates the discrete solution of the Lie system on the manifold via a solution of the Lie system on the Lie group. One major result from the integration of a Lie system on a Lie group is that one is able to solve all associated Lie systems on manifolds at the same time, and that Lie systems on Lie groups can be described through first-order systems of linear homogeneous ordinary differential equations (ODEs) in normal form. This brings a lot of advantages, since solving a linear system of ODEs involves less numerical cost. Specifically, we use two families of numerical schemes on the Lie group, which are designed to preserve its geometrical structure: the first one is based on the Magnus expansion, whereas the second is based on Runge–Kutta–Munthe–Kaas (RKMK) methods. Moreover, since the aforementioned action relates the Lie group and the manifold where the Lie system evolves, the resulting integrator preserves any geometric structure of the latter. We compare both methods for Lie systems with geometric invariants, particularly a class on Lie systems on curved spaces. We also illustrate the superiority of our method for describing long-term behavior and for differential equations admitting solutions whose geometric features depends heavily on initial conditions. As already mentioned, our milestone is to show that the method we propose preserves all the geometric invariants very faithfully, in comparison with non-geometric numerical methods.
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    Variational integrators for underactuated mechanical control systems with symmetries
    (American Institute of Mathematical Sciences, 2016-05-01) Colombo, Leonardo; Jiménez Alburquerque, Fernando; Martín de Diego, David
    Optimal control problems for underactuated mechanical systems can be seen as a higher-order variational problem subject to higher-order constraints (that is, when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces). In this paper we discuss the variational formalism for the class of underactuated mechanical control systems when the configuration space is a trivial principal bundle and the construction of variational integrators for such mechanical control systems. An interesting family of geometric integrators can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators, being one of their main properties the preservation of geometric features as the symplecticity, momentum preservation and good behavior of the energy. We construct variational integrators for higher-order mechanical systems on trivial principal bundles and their extension for higher-order constrained systems, paying particular attention to the case of underactuated mechanical systems.
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    Dirac Structures in Vakonomic Mechanics
    (Elsevier, 2014-11-11) Jiménez Alburquerque, Fernando; Yoshimura, Hiroaki
    In this paper, we explore dynamics of the nonholonomic system called vakonomic mechanics in the context of Lagrange-Dirac dynamical systems using a Dirac structure and its associated Hamilton-Pontryagin variational principle. We first show the link between vakonomic mechanics and nonholonomic mechanics from the viewpoints of Dirac structures as well as Lagrangian submanifolds. Namely, we clarify that Lagrangian submanifold theory cannot represent nonholonomic mechanics properly, but vakonomic mechanics instead. Second, in order to represent vakonomic mechanics, we employ the space T Q×V∗, where a vakonomic Lagrangian is defined from a given Lagrangian (possibly degenerate) subject to nonholonomic constraints. Then, we show how implicit vakonomic Euler-Lagrange equations can be formulated by the Hamilton-Pontryagin variational principle for the vakonomic Lagrangian on the extended Pontryagin bundle (T Q ⊕ T∗Q) × V∗. Associated with this variational principle, we establish a Dirac structure on (T Q ⊕ T∗Q) × V∗ to define an intrinsic vakonomic Lagrange-Dirac system. Furthermore, we establish another construction for the vakonomic Lagrange-Dirac system using a Dirac structure on T∗Q × V∗, where we introduce a vakonomic Dirac differential. Lastly, we illustrate our theory of vakonomic Lagrange-Dirac systems by some examples such as the vakonomic skate and the vertical rolling coin.
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    Geometric numerical methods for Lie systems and their application in optimal control
    (MDPI, 2023-06-19) Blanco Díaz, Luis; Sardón, Cristina; Jiménez Alburquerque, Fernando; Lucas, Javier de
    A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL(𝑛,ℝ) , which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given.
  • Miniatura
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    Una introducción a la integración geométrica
    (Instituto de Matemática de Bahía Blanca (INMABB), 2023-01-01) Jiménez Alburquerque, Fernando
    Dedicamos estas notas al estudio de ciertos integradores geométricos para sistemas hamiltonianos. Con integradores geométricos nos referimos a métodos numéricos de un paso y orden bajo (no consideraremos métodos Runge-Kutta) que preservan en el tiempo discreto algunas (o todas) de las propiedades cualitativas de los sistemas continuos que aproximan. En particular, nos centraremos en dos de los principales rasgos de los sistemas hamiltonianos: la simplecticidad y la preservación de la función hamiltoniana. Presentaremos integradores diseñados para preservar cada uno de ellos y en ocasiones ambos; puntualizaremos en qué circunstancias ocurre esta coincidencia y en cuáles no, que es el caso más habitual. Además, daremos una explicación razonada del buen comportamiento de los integradores simplécticos cuando no presernvan exactamente la función hamiltoniana, explicación basada en el Backward Error Analysis. Estas notas suponen el material en el que se basa el curso homónimo impartido en el Congreso Monteiro 2021.
  • Miniatura
    Publicación
    The variational discretization of the constrained higher-order Lagrange-Poincaré equations
    (American Institute of Mathematical Sciences, 2019-01-01) Bloch, Anthony; Colombo, Leonardo; Jiménez Alburquerque, Fernando
    In this paper we investigate a variational discretization for the class of mechanical systems in presence of symmetries described by the action of a Lie group which reduces the phase space to a (non-trivial) principal bundle. By introducing a discrete connection we are able to obtain the discrete constrained higher-order Lagrange-Poincaré equations. These equations describe the dynamics of a constrained Lagrangian system when the Lagrangian function and the constraints depend on higher-order derivatives such as the acceleration, jerk or jounces. The equations, under some mild regularity conditions, determine a well defined (local) flow which can be used to define a numerical scheme to integrate the constrained higher-order Lagrange-Poincaré equations. Optimal control problems for underactuated mechanical systems can be viewed as higher-order constrained variational problems. We study how a variational discretization can be used in the construction of variational integrators for optimal control of underactuated mechanical systems where control inputs act soley on the base manifold of a principal bundle (the shape space). Examples include the energy minimum control of an electron in a magnetic field and two coupled rigid bodies attached at a common center of mass.