Search

L’estudi del caos en sistemes dinàmics, en particular hamiltonians, va motivar la creació del mapa de la separatriu. Aquest es va originar a la segona meitat del segle XX gràcies als físics Zaslavskii i Shilnikov, que el van desenvolupar per separat i considerant sistemes hamiltonians diferents. En aquest projecte aprofundirem i construirem les dues propostes. A més a més, provarem resultats significatius en l’anàlisi del caos en sistemes hamiltonians obtinguts a partir del mapa de la separatriu, com són l’existència d’illes d’estabilitat en lòbuls separatrius i la mida de la capa estocàstica.

En aquest treball de fi de grau, investiguem la teoria de formes simplèctiques i de contacte singulars introduïdes per Guillemin-Miranda-Pires [GMP14] i Nest-Tsygan [NT96], a més de les connexions entre la geometria de contacte i la dinàmica de fluids. L'objectiu és explorar les possibles aplicacions d'aquestes idees a la dinàmica celeste. Comencem examinant nous aspectes del mirall Reeb-Beltrami donat per Etnyre i Ghrist [EG00a], conduint a una versió equivariant d'aquesta correspondència i desvetllant una interpretació de diversos sistemes mecànics Hamiltonians com a fluxos de fluids estacionaris. Destaquem especialment el problema d'N cossos en la mecànica celeste, i en particular el problema de Kepler. Utilitzant les tècniques singulars desenvolupades en aquest treball, també donem fites inferiors per al nombre d'òrbites d'escapament en sistemes dinàmics associats a les formes singulars que apareixen en la mecànica celeste. Aquest treball de fi de grau culmina amb un contraexemple a la conjectura singular de Weinstein sobre l'existència d'òrbites periòdiques singulars formulada per Miranda-Oms a [MO21]., In this undergraduate thesis, we delve into the theory of singular symplectic and contact forms originally introduced by Guillemin-Miranda-Pires [GMP14] and Nest-Tsygan [NT96], as well as the connections between contact geometry and fluid dynamics. Our objective is to explore the potential applications of these ideas in the field of celestial mechanics. Our thesis begins by examining novel aspects of the Reeb-Beltrami mirror given by Etnyre and Ghrist [EG00a], leading to an equivariant version of this correspondence and unveiling an interpretation of various mechanical Hamiltonian systems as stationary fluid flows. Notably, we focus on different incarnations of the n-body problem of celestial mechanics, particularly the Kepler problem. By utilizing the singular techniques developed in this thesis, we also provide lower bounds for escape orbits in dynamical systems associated with the singular forms that appear in celestial mechanics. This undergraduate thesis culminates with a counterexample to the singular Weinstein conjecture regarding the existence of singular periodic orbits formulated by Miranda-Oms in [MO21]., Outgoing

The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers [5, 6, 7, 8] several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence for Euler and Navier-Stokes [28]. Universality of the Euler equations was proved in [7] for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao’s approach relating Turing machines to Navier-Stokes equations, a Turing complete stationary Euler solution on a Riemannian 3-dimensional sphere was constructed in [8]. Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable [30], a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths [8]. In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic, Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the Mar´ıa de Maeztu Programme for Units of Excellence in R& D (MDM-2014-0445) via an FPI grant. Robert Cardona and Eva Miranda are partially supported by the grants PID2019-103849GB-I00 / AEI / 10.13039/501100011033 and the AGAUR grant 2017SGR932. Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Daniel Peralta-Salas is supported by the grants MTM PID2019-106715GBC21 (MICINN) and Europa Excelencia EUR2019-103821 (MCIU). This work was partially supported by the ICMAT–Severo Ochoa grant CEX2019-000904-S., Preprint

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0, This paper presents a new Updated Reference Lagrangian Smooth Particle Hydrodynamics (SPH) algorithm for the analysis of large deformation isothermal elasticity and elasto-plasticity. Taking as point of departure a Total Lagrangian setting and considering as referential configuration an intermediate configuration of the deformation process, the equation of conservation of linear momentum and three geometric conservation laws (for the deformation gradient, its cofactor and its determinant) are re-written leading to a very generic system of first order conservation laws. The key feature of the formulation is a suitable multiplicative decomposition of the conservation variables, leading to a very simple final set of equations with striking similarities to the conventional Total Lagrangian system albeit re-written in terms of alternative Referential Updated conservation variables which are evolved in time. Taking advantage of this new Updated Reference Lagrangian formalism, a second order entropy-stable SPH upwinding stabilisation method will be introduced. With respect to previous publications by the group, a new three-stage Runge–Kutta time integration method is implemented in order to increase the CFL stability restriction. Finally, and to demonstrate the robustness and applicability of the methodology, a wide spectrum of challenging problems will be presented and compared, including some benchmark three-dimensional large deformation elasto-plasticity problems. To facilitate its ease of implementation, the paper explores the use of a series of novel expressions for the evaluation of kernels and the gradients of kernels to provide the SPH user the freedom to choose amongst various options, all leading to equally convincingly robust results., The authors would like to acknowledge the financial support received through the project Marie SkłodowskaCurie ITN-EJD ProTechTion, funded by the European Union Horizon 2020 research and innovation program with Grant Number 764636. The third author also gratefully acknowledges the support provided by the EPSRC Strategic Support Package: Engineering of Active Materials by Multiscale/Multiphysics Computational Mechanics - EP/R008531/1. The fourth author acknowledges the support of the Serra Hunter Programme of the Generalitat de ´ Catalunya, the Spanish Ministry of Economy and Competitiveness (Grant agreement No. PID2020-113463RB-C33) and the Spanish Ministry of Economy and Competitiveness through the Severo Ochoa programme for centres of excellence in RTD (Grant agreement No. CEX2018-000797-S)., Peer Reviewed, Postprint (published version)

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long timescales, they exhibit a strong form of transverse instability in Sobolev spaces Hs(T2) (0

Consider the Restricted Planar Circular 3 Body Problem. If the trajectory of the body of zero mass is defined for all time, it can have the following four types of asymptotic motion when time tends to infinity forward or backward in time: bounded, parabolic (goes to infinity with asymptotic zero velocity), hyperbolic (goes to infinity with asymptotic positive velocity) or oscillatory (the position of the body is unbounded but goes back to a compact region of phase space for a sequence of arbitrarily large times). We consider realistic mass ratio for the Sun-Jupiter pair and Jacobi constant which allows the massless body to cross Jupiter's orbit. This is a non-perturbative regime. We prove the existence of all possible combinations of past and future final motions. In particular, we obtain the existence of oscillatory motions. All the constructed trajectories cross the orbit of Jupiter but avoid close encounters with it. The proof relies on analyzing the stable and unstable invariant manifolds of infinity and their intersections. We construct orbits shadowing these invariant manifolds by the method of correctly aligned windows. The proof is computer assisted., M. C. has been partially supported by the NCN grant 2018/29/B/ST1/00109 2M. G. has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 757802). M. G. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2019. P. M. has been partially supported by the Spanish MINECO-FEDER Grant PGC2018-100928-B-I00 and the Catalan grant 2017SGR1049 T. S. has been also partly supported by the Spanish MINECO-FEDER Grant PGC2018-098676-B100 (AEI/FEDER/UE), the Catalan grant 2017SGR1049 and by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2019. P. Z. has been partially supported by the NCN grant 2019/35/B/ST1/00655, Peer Reviewed, Postprint (author's final draft)

The version of record is available online at: http://dx.doi.org/10.1007/s00605-022-01767-1, Contact geometry allows us to describe some thermodynamic and dissipative systems. In this paper we introduce a new geometric structure in order to describe time-dependent contact systems: cocontact manifolds. Within this setting we develop the Hamiltonian and Lagrangian formalisms, both in the regular and singular cases. In the singular case, we present a constraint algorithm aiming to find a submanifold where solutions exist. As a particular case we study contact systems with holonomic time-dependent constraints. Some regular and singular examples are analyzed, along with numerical simulations., We acknowledge fruitful discussions and comments from our colleague Narciso Román-Roy. MdL acknowledges the financial support of the Ministerio de Ciencia e Innovación (Spain), under grants PID2019-106715GB-C2, “Severo Ochoa Programme for Centres of Excellence in R&D” (CEX2019-000904-S) and EIN2020-112107. JG, XG, MCML and XR acknowledge the financial support of the Ministerio de Ciencia, Innovación y Universidades (Spain), project PGC2018-098265-B-C33., Peer Reviewed, Postprint (author's final draft)

The Restricted 3-Body Problem models the motion of a body of negligible mass under the gravitational influence of two massive bodies, called the primaries. If the primaries perform circular motions and the massless body is coplanar with them, one has the Restricted Planar Circular 3-Body Problem (RPC3BP). In synodic coordinates, it is a two degrees of freedom Hamiltonian system with five critical points, , called the Lagrange points. The Lagrange point is a saddle-center critical point which is collinear with the primaries and is located beyond the largest of the two. In this paper and its sequel [10], we provide an asymptotic formula for the distance between the one dimensional stable and unstable invariant manifolds of when the ratio between the masses of the primaries µ is small. It implies that cannot have one-round homoclinic orbits. If the mass ratio µ is small, the hyperbolic eigenvalues are weaker than the elliptic ones by factor of order . This implies that the distance between the invariant manifolds is exponentially small with respect to µ and, therefore, the classical Poincaré–Melnikov method cannot be applied. In this first paper, we approximate the RPC3BP by an averaged integrable Hamiltonian system which possesses a saddle center with a homoclinic orbit and we analyze the complex singularities of its time parameterization. We also derive and study the inner equation associated to the original perturbed problem. The difference between certain solutions of the inner equation gives the leading term of the distance between the stable and unstable manifolds of . In the sequel [10] we complete the proof of the asymptotic formula for the distance between the invariant manifolds., Peer Reviewed, Postprint (author's final draft)

© 2022 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0, In previous papers, a geometric framework has been developed to describe non-conservative field theories as a kind of modified Lagrangian and Hamiltonian field theories. This approach is that of k-contact Hamiltonian systems, which is based on the k-symplectic formulation of field theories as well as on contact geometry. In this work we present the Skinner–Rusk unified setting for these kinds of theories, which encompasses both the Lagrangian and Hamiltonian formalisms into a single picture. This unified framework is specially useful when dealing with singular systems, since: (i) it incorporates in a natural way the second-order condition for the solutions of field equations, (ii) it allows to implement the Lagrangian and Hamiltonian constraint algorithms in a unique simple way, and (iii) it gives the Legendre transformation, so that the Lagrangian and the Hamiltonian formalisms are obtained straightforwardly. We apply this description to several interesting physical examples: the damped vibrating string, the telegrapher’s equations, and Maxwell’s equations with dissipation terms., We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovación 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 Reviewed, Postprint (published version)

We show that the contact dynamics obtained from the Herglotz variational principle can be described as a constrained nonholonomic or vakonomic ordinary Lagrangian system depending on a dissipative variable with an adequate choice of one constraint. As a consequence, we obtain the dynamics of contact nonholonomic and vakonomic systems as an ordinary variational calculus with constraints on a Lagrangian with a dissipative variable. The variation of the energy and the other dissipative quantities is also obtained, giving the usual results., Peer Reviewed, Postprint (author's final draft)

© 2021 Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0, We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, T kQ × R, whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formalisms for these kinds of systems, including their variational formulation. The variational principle, the contact forms, and the geometric dynamical equations are obtained by using those structures and generalizing the standard formulation of contact Lagrangian and Hamiltonian systems. As an alternative approach, we develop a unified description that encompasses the Lagrangian and Hamiltonian equations as well as their relationship through the Legendre map; all of them are obtained from the contact dynamical equations and the constraint algorithm that is implemented because, in this formalism, the dynamical systems are always singular. Some interesting examples are finally analyzed using these geometric formulations., We acknowledge the financial support from the Spanish Ministerio de Ciencia, Innovación y Universidades project PGC2018-098265-B-C33, the MINECO Grant MTM2016-76-072-P, the ICMAT Severo Ochoa projects SEV-2011-0087 and SEV-2015-0554, and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932. Manuel Laínz wishes to thank MICINN and ICMAT for a FPI-Severo Ochoa predoctoral contract PRE2018-083203., Peer Reviewed, Preprint

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 Reviewed, Postprint (author's final draft)

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 analyzed. Furthermore, a more general framework for the theory is also briefly explained. It is also shown how, from this generic framework, the Lagrangian and Hamiltonian cases of the theory for dynamical systems are recovered, as well as how the model can be extended to other types of physical systems, such as higher-order dynamical systems and (first-order) classical field theories in their multisymplectic formulation., I acknowledge the financial support from project PGC2018-098265-B-C33 of the Spanish Ministerio de Ciencia, Innovación y Universidades and the project 2017–SGR–932 of the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government., Peer Reviewed, Postprint (published version)

A complete geometric classification of symmetries of autonomous Hamiltonian systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied., I acknowledge the financial support from the Spanish Ministerio de Econom´ıa y Competitividad project MTM2014–54855–P, the Ministerio de Ciencia, Innovaci´on y Universidades project PGC2018-098265-BC33, and the Secretary of University and Research of the Ministry of Business and Knowledge of the Catalan Government project 2017–SGR–932. I also greatly appreciate the comments and suggestions of Prof. Mikhail S. Plyushchay. Finally, my thanks to the referees for their extensive and valuable comments that have allowed me to significantly improve the final version of the work., Postprint (author's final draft)

This is the peer reviewed version of the following article: Seoane, M. [et al.]. A combined reduced order-full order methodology for the solution of 3D magneto-mechanical problems with application to magnetic resonance imaging scanners. "International journal for numerical methods in engineering", 1 Gener 2020, vol. 121, núm. 16, p. 3529-3559. , which has been published in final form athttps://onlinelibrary.wiley.com/doi/abs/10.1002/nme.6369. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving., The design of a new magnetic resonance imaging (MRI) scanner requires multiple numerical simulations of the same magneto-mechanical problem for varying model parameters, such as frequency and electric conductivity, in order to ensure that the vibrations, noise, and heat dissipation are minimized. The high computational cost required for these repeated simulations leads to a bottleneck in the design process due to an increased design time and, thus, a higher cost. To alleviate these issues, the application of reduced order modeling techniques, which are able to find a general solution to high-dimensional parametric problems in a very efficient manner, is considered. Building on the established proper orthogonal decomposition technique available in the literature, the main novelty of this work is an efficient implementation for the solution of 3D magneto-mechanical problems in the context of challenging MRI configurations. This methodology provides a general solution for varying parameters of interest. The accuracy and efficiency of the method are proven by applying it to challenging MRI configurations and comparing with the full-order solution., Generalitat de Catalunya, 2017‐SGR‐1278; Marie Sklodowska‐Curie Innovative Training Network AdMoRe, 675919; Spanish Ministry of Economy and Competitiveness, DPI2017‐85139‐C2‐2‐R, Peer Reviewed, Postprint (author's final draft)

© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0, We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Instead of defining these proximity graphs using circles, we use an arbitrary convex shape \(\mathcal {C}\) . Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k- \({DG}_{\mathcal {C}}(S)\) , has vertex set S and edge pq provided that there exists some homothet of \(\mathcal {C}\) with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph k- \({GG}_{\mathcal {C}}(S)\) is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of \(\mathcal {C}\) with p and q on its boundary. We provide upper bounds on the minimum value of k for which k- \({GG}_{\mathcal {C}}(S)\) is Hamiltonian. Since k- \({GG}_{\mathcal {C}}(S)\) \(\subseteq \) k- \({DG}_{\mathcal {C}}(S)\) , all results carry over to k- \({DG}_{\mathcal {C}}(S)\) . In particular, we give upper bounds of 24 for every \(\mathcal {C}\) and 15 for every point-symmetric \(\mathcal {C}\) . We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for \(t \ge 10)\) . These constitute the first general results on Hamiltonicity for convex shape Delaunay and Gabriel graphs., P.B. was partially supported by NSERC. P.C. was supported by CONACyT. M.S. was supported by the Czech Science Foundation, grant number GJ19-06792Y, and by institutional support RVO:67985807. R.S. was supported by MINECO through the Ram´on y Cajal program. P.C. and R.S. were also supported by projects MINECO MTM2015-63791-R and Gen. Cat. 2017SGR1640. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922., Peer Reviewed, Postprint (author's final draft)

The aim of this paper is to develop a constraint algorithm for singular classical field theories in the framework of k-cosymplectic geometry. Since these field theories are singular, we need to introduce the notion of k-precosymplectic structure, which is a generalization of the k-cosymplectic structure. Next k-precosymplectic Hamiltonian systems are introduced in or- der to describe singular field theories, both in Lagrangian and Hamiltonian formalisms. Finally, we develop a constraint algorithm in order to find a sub- manifold where the existence of solutions of the field equations is ensured. The case of affine Lagrangians is studied as a relevant example., Peer Reviewed, Postprint (author's final draft)

We study Hamiltonicity for some of the most general variants of Delaunay and Gabriel graphs. Let S be a point set in the plane. The k-order Delaunay graph of S, denoted k-DGC(S), has vertex set S and edge pq provided that there exists some homothet of C with p and q on its boundary and containing at most k points of S different from p and q. The k-order Gabriel graph k-GGC(S) is defined analogously, except for the fact that the homothets considered are restricted to be smallest homothets of C with p and q on its boundary. We provide upper bounds on the minimum value of k for which k-GGC(S) is Hamiltonian. Since k-GGC(S) ¿ k-DGC(S), all results carry over to k-DGC(S). In particular, we give upper bounds of 24 for every C and 15 for every point-symmetric C. We also improve the bound to 7 for squares, 11 for regular hexagons, 12 for regular octagons, and 11 for even-sided regular t-gons (for t = 10)., Peer Reviewed, Postprint (published version)

We study affine invariant 2D triangulation methods. That is, methods that produce the same triangulation for a point set S for any (unknown) affine transformation of S. Our work is based on a method by Nielson [A characterization of an affine invariant triangulation. Geom. mod, 191-210. Springer, 1993] that uses the inverse of the covariance matrix of S to define an affine invariant norm, denoted AS, and an affine invariant triangulation, denoted DT AS [S]. We revisit the AS-norm from a geometric perspective, and show that DT AS [S] can be seen as a standard Delaunay triangulation of a transformed point set based on S. We prove that it retains all of its well-known properties. In addition, we provide different affine invariant order methods of a point set S and of the vertices of a polygon P that can be combined with well-known algorithms in order to obtain other affine invariant triangulation methods of S and P., ∗P.B. was partially supported by NSERC. P.C. was supported by CONACyT. R.S. was supported by MINECO through the Ram´on y Cajal program. P.C. and R.S. were also supported by projects MINECO MTM2015-63791-R and Gen. Cat. 2017SGR1640. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 734922., Peer Reviewed, Postprint (published version)

This is a work about the collision of 3 celestial bodies which are aligned in a straight line, by following Newton's Gravitational Law. There is an introduction which shows the most important items of the general N-Body Problem in order to introduce the main concepts to work on with the main problem, which is the collinear 3-Body Problem. After this, at the conclusions we will see how the dynamics work for more than 3 bodies.