Unified formalism for higher-order non-autonomous dynamical systems

This work is devoted to giving a geometric framework for describing higher-order non-autonomous mechanical systems. The starting point is to extend the Lagrangian-Hamiltonian unified formalism of Skinner and Rusk for these kinds of systems, generalizing previous developments for higher-order autonomous mechanical systems and first-order non-autonomous mechanical systems. Then, we use this unified formulation to derive the standard Lagrangian and Hamiltonian formalisms, including the Legendre-Ostrogradsky map and the Euler-Lagrange and the Hamilton equations, both for regular and singular systems. As applications of our model, two examples of regular and singular physical systems are studied.Comment: 43 pp. We have corrected and clarified the statement of Propositions 2 and 3. A remark is added after Proposition

Non-local fractional derivatives. Discrete and continuous

We prove maximum and comparison principles for fractional discrete derivatives in the integers. Regularity results when the space is a mesh of length $h$, and approximation theorems to the continuous fractional derivatives are shown. When the functions are good enough, these approximation procedures give a measure of the order of approximation. These results also allows us to prove the coincidence, for good enough functions, of the Marchaud and Gr\"unwald-Letnikov derivatives in every point and the speed of convergence to the Gr\"unwald-Letnikov derivative. The fractional discrete derivative will be also described as a Neumann-Dirichlet operator defined by a semi-discrete extension problem. Some operators related to the Harmonic Analysis associated to the discrete derivative will be also considered, in particular their behavior in the Lebesgue spaces $\ell^p(\mathbb{Z}).

Mechanisms for photon sorting based on slit-groove arrays

Mechanisms for one-dimensional photon sorting are theoretically studied in the framework of a couple mode method. The considered system is a nanopatterned structure composed of two different pixels drilled on the surface of a thin gold layer. Each pixel consists of a slit-groove array designed to squeeze a large fraction of the incident light into the central slit. The Double-Pixel is optimized to resolve two different frequencies in the near infrared. This system shows a high transmission efficiency and a small crosstalk. Its response is found to strongly depend on the effective area shared by overlapping pixels. Three different regimes for the process of photon sorting are identified and the main physical trends underneath in such regimes are unveiled. Optimal efficiencies for the photon sorting are obtained for a moderate number of grooves that overlap with grooves of the neighbor pixel. Results could be applied to optical and infrared detectors.Comment: 12 pages, 4 figure

Nonholonomic constraints in kk-symplectic Classical Field Theories

A $k$-symplectic framework for classical field theories subject to nonholonomic constraints is presented. If the constrained problem is regular one can construct a projection operator such that the solutions of the constrained problem are obtained by projecting the solutions of the free problem. Symmetries for the nonholonomic system are introduced and we show that for every such symmetry, there exist a nonholonomic momentum equation. The proposed formalism permits to introduce in a simple way many tools of nonholonomic mechanics to nonholonomic field theories.Comment: 27 page

Transmittance of a subwavelength aperture flanked by a finite groove array \\ placed near the focus of a conventional lens

One-dimensional light harvesting structures illuminated by a conventional lens are studied in this paper. Our theoretical study shows that high transmission efficiencies are obtained when the structure is placed near the focal plane of the lens. The considered structure is a finite slit-groove array (SGA) with a given number of grooves that are symmetrically distributed with respect to a central slit. The SGA is nano-patterned on an opaque metallic film. It is found that a total transmittance of 80% is achieved even for a single slit when (i) Fabry-Perot like modes are excited inside the slit and (ii) the effective cross section of the aperture becomes of the order of the full width at half maximum of the incident beam. A further enhancement of 8% is produced by the groove array. The optimal geometry for the groove array consists of a moderate number of grooves ($\geq 4$) at either side of the slit, separated by a distance of half the incident wavelength $\lambda$. Grooves should be deeper (with depth $\geq \lambda/4$) than those typically reported for plane wave illumination in order to increase their individual scattering cross section.Comment: 7 pages, 6 figure

Higher-order Mechanics: Variational Principles and other topics

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 derive the Lagrangian and Hamiltonian equations for these kinds of systems. Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding dynamical equations are recovered from this unified framework.Comment: New version of the paper "Variational principles for higher-order dynamical systems", which was presented in the "III Iberoamerican Meeting on Geometry, Mechanics and Control" (Salamanca, 2012). The title is changed. A detailed review is added. Sections containing results about variational principles are enlarged with additional comments, diagrams and summarizing results. Bibliography is update

Lipschitz spaces adapted to Schrödinger operators and regularity properties

Consider the Schrödinger operator L= - Δ + V in Rn, n≥ 3 , where V is a nonnegative potential satisfying a reverse Hölder condition of the type (1|B|∫BV(y)qdy)1/q ≤C|B|∫BV(y)dy, for some q > n/2. We define ΛαL, 0 0‖f(·+z) + f(·-z)-2f(·)‖∞|z|α 0 , we denote by Λ Wα/2 the set of functions f which satisfy ‖ρ(·)-αf(·)‖∞ 0. We prove that for 0 < α ≤ 2 - n/q, ΛαL = Λ Wα/2. As application, we obtain regularity properties of fractional powers (positive and negative) of the operator L, Schrödinger Riesz transforms, Bessel potentials and multipliers of Laplace transforms type. The proofs of these results need in an essential way the language of semigroups. Parallel results are obtained for the classes defined through the Poisson semigroup, Py f = e-y √LfMarta De León-Contreras was partially supported by grant EPSRC Research Grant EP/S029486/1. José L. Torrea was partially supported by Grant PGC2018-099124-B-I00 (MINECO/FEDER

Symmetries in Classical Field Theory

The multisymplectic description of Classical Field Theories is revisited, including its relation with the presymplectic formalism on the space of Cauchy data. Both descriptions allow us to give a complete scheme of classification of infinitesimal symmetries, and to obtain the corresponding conservation laws.Comment: 70S05; 70H33; 55R10; 58A2