The wave equation for stiff strings and piano tuning

We study the wave equation for a string with stiffness. We solve the equation and provide a uniqueness theorem with suitable boundary conditions. For a pinned string we compute the spectrum, which is slightly inharmonic. Therefore, the widespread scale of 12 equal divisions of the just octave is not the best choice to tune instruments like the piano. Basing on the theory of dissonance, we provide a way to tune the piano in order to improve its consonance. A good solution is obtained by tuning a note and its fifth by minimizing their beats.Peer ReviewedPostprint (published version

Canonical Noether symmetries and commutativity properties for gauge systems

For a dynamical system defined by a singular Lagrangian, canonical Noether symmetries are characterized in terms of their commutation relations with the evolution operators of Lagrangian and Hamiltonian formalisms. Separate characterizations are given in phase space, in velocity space, and through an evolution operator that links both spaces.Comment: 22 pages; some references updated, an uncited reference deleted, minor style change

Constraint algorithm for k-presymplectic Hamiltonian systems. Application to singular field theories

The k-symplectic formulation of field theories is especially simple, since only tangent and cotangent bundles are needed in its description. Its defining elements show a close relationship with those in the symplectic formulation of mechanics. It will be shown that this relationship also stands in the presymplectic case. In a natural way, one can mimick the presymplectic constraint algorithm to obtain a constraint algorithm that can be applied to k-presymplectic field theory, and more particularly to the Lagrangian and Hamiltonian formulations of field theories defined by a singular Lagrangian, as well as to the unified Lagrangian-Hamiltonian formalism (Skinner--Rusk formalism) for k-presymplectic field theory. Two examples of application of the algorithm are also analyzed

On Darboux theorems for geometric structures induced by closed forms

This work reviews the classical Darboux theorem for symplectic, presymplectic, and cosymplectic manifolds (which are used to describe regular and singular mechanical systems), and certain cases of multisymplectic manifolds, and extends it in new ways to k-symplectic and k-cosymplectic manifolds (all these structures appear in the geometric formulation of first-order classical field theories). Moreover, we discuss the existence of Darboux theorems for classes of precosymplectic, k-presymplectic, k-precosymplectic, and premultisymplectic manifolds, which are the geometrical structures underlying some kinds of singular field theories. Approaches to Darboux theorems based on flat connections associated with geometric structures are given, while new results on polarisations for (k-)(pre)(co)symplectic structures arise.Comment: improved and extended proofs. 33 p

Calibración de parámetros de infiltración y rugosidad con un modelo numérico para riego con surcos cerrados

Se presenta la calibración de los parámetros que intervienen en las ecuaciones de infiltración de Green y Ampty de rugosidad de Manning, a través del empleo de un modelo hidrodinámico completo en diferencias finitas para riego con surcos cerrados que resuelve las ecuaciones de Saint-Venant. Se muestran los resultados obtenidos en tres ensayos realizados en un surco cerrado de 50 m y, en un apéndice, el ajuste de la curva de avance en un surco cerrado de 150 m. Se concluye que el modelo numérico es aceptable para reproducir las tres fases del riego con surcos cerrados (avance, llenado y receso), por lo que puede ser empleado en el proceso de diseño para riego con surcos cerrados

Erratum: constraint algorithm for singular field theories in the k k -cosymplectic framework

We are indebted to Prof. Dieter Van den Bleeken (Bo˘gazi¸ci University) for having drawn our attention to the error that gave rise to this note. We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovaci´on y Universidades project PGC2018-098265-B-C33, and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017-SGR-932.Peer ReviewedPostprint (author's final draft