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The version of record of this article, first published in Journal of fixed point theory and its applications, is available online at Publisher’s website: http://doi.org/10.1007/s11784-024-01105-y, In this article, we study the Hamiltonian dynamics on singular symplectic manifolds and prove the Arnold conjecture for a large class of bm-symplectic manifolds. Novel techniques are introduced to associate smooth symplectic forms to the original singular symplectic structure, under some mild conditions. These techniques yield the validity of the Arnold conjecture for singular symplectic manifolds across multiple scenarios. More precisely, we prove a lower bound on the number of 1-periodic Hamiltonian orbits for b2m-symplectic manifolds depending only on the topology of the manifold. Moreover, for bm-symplectic surfaces, we improve the lower bound depending on the topology of the pair (M, Z). We then venture into the study of Floer homology to this singular realm and we conclude with a list of open questions., Funding OpenAccess funding provided thanks to the CRUE-CSIC agreement with Springer Nature. All the authors are partially supported by the Spanish State Research Agency grant PID2019-103849GB-I00 of MICIU/AEI/10.13039/50110 00110331934 and by the AGAUR project 2021 SGR 00603. J. Brugu´ es is supported by the FWO-FNRS Excellence of Science project G0H4518N “Symplectic techniques in differential geometry” with UA Antigoon Project-ID 36584.E. Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021 and by the Alexander Von Humboldt foundation via a Friedrich Wilhelm Bessel Research Award. E. Miranda is also supported by the Spanish State Research Agency, through the 16 Page 52 of 60 J. Brugués et al. Severo Ochoa and Mar´ ıa de Maeztu Program for Centers and Units of Excellence in R&D (Project CEX2020-001084-M). C. Oms acknowledges financial support from the Juan de la Cierva postdoctoral grant FJC2021-046811-I / MICIU/AEI /10.13039/501100011033 and by the European Union NextGenerationEU/PRTR., Peer Reviewed, Postprint (published version)

A new geometric structure inspired by multisymplectic and contact geometries, which we call multicontact structure, is developed to describe non-conservative classical field theories. Using the differential forms that define this multicontact structure as well as other geometric elements that are derived from them while assuming certain conditions, we can introduce, on the multicontact manifolds, the variational field equations which are stated using sections, multivector fields, and Ehresmann connections on the adequate fiber bundles. Furthermore, it is shown how this multicontact framework can be adapted to the jet bundle description of classical field theories; the field equations are stated in the Lagrangian and the Hamiltonian formalisms both in the regular and the singular cases., Peer Reviewed, Postprint (author's final draft)

This work provides a general overview for the treatment of symmetries in classical field theories and (pre)multisymplectic geometry. The geometric characteristics of the relation between how symmetries are interpreted in theoretical physics and in the geometric formulation of these theories are clarified. Finally, a general discussion is given on the structure of symmetries in the presence of constraints appearing in singular field theories. Symmetries of some typical theories in theoretical physics are analyzed through the construction of the relevant multimomentum maps which are the conserved quantities (by Noether’s theorem) on the (pre)multisymplectic phase spaces., Peer Reviewed, Postprint (published version)

We define the Bohr-Sommerfeld quantization via T -modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [Victor W. Guillemin, Eva Miranda, and Jonathan Weitsman. On geometric quantization of b-symplectic manifolds. Adv. Math., 331:941–951, 2018]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold, The first-named author is supported by the Doctoral INPhINIT-RETAINING grant ID 100010434 LCF/BQ/DR21/11880025 of “la Caixa” Foundation; by the AEI grant PID2019-103849GB-I00 of MCIN/ AEI /10.13039/501100011033; by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP; and by the AGRUPS 2023 grant by UPC. The second-named author is partially supported by the Spanish State Research Agency AEI/10.13039/501100011033, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M); and by the grant PID2019-103849GB-I00. The author is also partially supported by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP; and by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021. The third-named author is supported by a Simons collaboration grant., Peer Reviewed, Preprint

In this article, motivated by the study of symplectic structures on manifolds with boundary and the systematic study of -symplectic manifolds started in Guillemin, Miranda, and Pires Adv. Math. 264 (2014), 864–896, we prove a slice theorem for Lie group actions on -symplectic manifolds., Postprint (published version)

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)

Aquest treball de fi de grau presenta la geometria de contacte, una branca de la geometria diferencial que permet descriure geomètricament certs sistemes dinàmics com ara sistemes amb dissipació. Aquesta formulació permet descriure tant sistemes hamiltonians com lagrangians. S'estudiarà el procés complet des de la formulació geomètrica fins a l'estudi de les equacions de moviment des del punt de vista dels sistemes dinàmics. El primer capítol serà un repàs de la formulació geomètrica clàssica, mentre que el segon presentarà les nocions bàsiques de la geometria de contacte: les varietats de contacte, el formalisme hamiltonià de contacte i el formalisme lagrangià de contacte. En el tercer i últim capítol es realitza l'estudi dinàmic de les equacions de moviment obtingudes i estudiarem l'efecte que hi provoquen la dissipació, l'excitació i una combinació (pseudo)aleatòria d'aquestes.

Motivated by the group of Galilean transformations and the subgroup of Galilean transformations which fix time zero, we introduce the notion of a b-Lie group as a pair where G is a Lie group and H is a codimension-one Lie subgroup. Such a notion allows us to give a theoretical framework for transformations of space-time where the initial time can be seen as a boundary. In this theoretical framework, we develop the basics of the theory and study the associated canonical b-symplectic structure on the b-cotangent bundle ¿ together with its reduction theory. Namely, we extend the minimal coupling procedure to ¿ and prove that the Poisson reduction under the cotangent lifted action of H by left translations can be described in terms of the Lie Poisson structure on ¿ (where is the Lie algebra of H) and the canonical b-symplectic structure on ¿ , where is viewed as a one-dimensional b-manifold having as critical hypersurface (in the sense of b-manifolds) the identity element., Peer Reviewed, Postprint (published version)

We define the Bohr-Sommerfeld quantization via T-modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [GMW18b]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold., We define the Bohr-Sommerfeld quantization via T-modules for a b-symplectic toric manifold and show that it coincides with the formal geometric quantization of [GMW18b]. In particular, we prove that its dimension is given by a signed count of the integral points in the moment polytope of the toric action on the manifold., Finançat pel projecte ICREA ACADEMIA 2016-05 i ICREA ACADEMIA 2021, Preprint

In this article we give a universal model for geometric quantization associated to a real polarization given by an integrable system with non-degenerate singularities. This universal model goes one step further than the cotangent models in [KM17] by both considering singular orbits and adding to the cotangent models a model for the prequantum line bundle. These singularities are generic in the sense that are given by Morse-type functions and include elliptic, hyperbolic and focus-focus singularities. Examples of systems admitting such singularities are toric, semitoric and almost toric manifolds, as well as physical systems such as the coupling of harmonic oscillators, the spherical pendulum or the reduction of the Euler’s equations of the rigid body on T *(SO(3)) to a sphere. Our geometric quantization formulation coincides with the models given in [HM10] and [MPS20] away from the singularities and corrects former models for hyperbolic and focus-focus singularities cancelling out the infinite dimensional contributions obtained by former approaches. The geometric quantization models provided here match the classical physical methods for mechanical systems such as the spherical pendulum as presented in [CS16]. Our cotangent models obey a local-to-global principle and can be glued to determine the geometric quantization of the global systems even if the global symplectic classification of the systems is not known in general., Preprint

The version of record of this article, first published in Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, is available online at Publisher’s website: http://dx.doi.org/10.1007/s13398-021-01136-x, In general, the system of 2nd-order partial differential equations made of the Euler–Lagrange equations of classical field theories are not compatible for singular Lagrangians. This is the so-called second-order problem. The first aim of this work is to develop a fully geometric constraint algorithm which allows us to find a submanifold where the Euler–Lagrange equations have solution, and split the constraints into two kinds depending on their origin. We do so using k-symplectic geometry, which is the simplest intrinsic description of classical field theories. As a second aim, the Einstein–Palatini model of General Relativity is studied using this algorithm., 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)

© 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)

In this article, we consider integrable systems on manifolds endowed with symplectic structures with singularities of order one. These structures are symplectic away from a hypersurface where the symplectic volume goes either to infinity or to zero transversally, yielding either a b-symplectic form or a folded symplectic form. The hypersurface where the form degenerates is called critical set. We give a new impulse to the investigation of the existence of action-angle coordinates for these structures initiated in [36] and [37] by proving an action-angle theorem for folded symplectic integrable systems. Contrary to expectations, the action-angle coordinate theorem for folded symplectic manifolds cannot be presented as a cotangent lift as done for symplectic and bsymplectic forms in [36]. Global constructions of integrable systems are provided and obstructions for the global existence of action-angle coordinates are investigated in both scenarios. The new topological obstructions found emanate from the topology of the critical set Z of the singular symplectic manifold. The existence of these obstructions in turn implies the existence of singularities for the integrable system on Z., Robert Cardona acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in R&D (MDM-2014-0445). Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Both authors are supported by the grants reference number 2017SGR932 (AGAUR) and PID2019-103849GB-I00 / AEI / 10.13039/501100011033., Preprint

© 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

Motivated by Poincare’s orbits going to infinity in the (restricted) three-body problem ´ (see [29] and [7]), we investigate the generic existence of heteroclinic-like orbits in a neighbourhood of the critical set of a b-contact form. This is done by using a singular counterpart [4] of Etnyre– Ghrist’s contact/Beltrami correspondence [11], and genericity results concerning eigenfunctions of the Laplacian established by Uhlenbeck [33]. Specifically, we analyze the b-Beltrami vector fields on b-manifolds of dimension 3 and prove that for a generic asymptotically exact b-metric they exhibit escape orbits. We also show that a generic asymptotically symmetric b-Beltrami vector field on an asymptotically flat b-manifold has a generalized singular periodic orbit and at least 4 escape orbits. Generalized singular periodic orbits are trajectories of the vector field whose a- and ¿-limit sets intersect the critical surface. These results are a first step towards proving the singular Weinstein conjecture., E. M. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2016. Eva Miranda and Cedric Oms are supported by the grants reference number MTM2015-69135-P (MINECO/FEDER) ánd reference number 2017SGR932 (AGAUR) and the project PID2019-103849GB-I00 / AEI / 10.13039/501100011033. C. O. has been supported by an FNR-AFR PhD predoctoral grant (project GLADYSS) until October 2nd, 2020 and by a SECTI-Postdoctoral grant financed by Eva Miranda’s ICREA Academia immediately after. D. P.-S. is supported by the grants MTM PID2019-106715GB-C21 (MICINN) and Europa Excelencia EUR2019- 103821 (MCIU). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant 20205CEX001., Preprint

In this master thesis we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including systems with degenerate singularities which are invariant under a torus action. We also prove the $b$-symplectic analogue of the rigidity results. We illustrate the three basic types of singularities of integrable systems through three models from classical mechanics and we give them as cotangent lifts. Finally we review the focus-focus singularity and the saddle-focus singularity.

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 present a covariant multisymplectic formulation for the Einstein-Palatini (or Metric-Affine) model of General Relativity (without energy-matter sources). As it is described by a first-order affine Lagrangian (in the derivatives of the fields), it is singular and, hence, this is a gauge field theory with constraints. These constraints are obtained after applying a constraint algorithm to the field equations, both in the Lagrangian and the Hamiltonian formalisms. In order to do this, the covariant field equations must be written in a suitable geometrical way, using integrable distributions which are represented by multivector fields of a certain type. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomentum (covariant) Hamiltonian formalism. The gauge symmetries of the model are discussed in both formalisms and, from them, the equivalence with the Einstein-Hilbert model is established., Peer Reviewed, Postprint (author's final draft)

Morse homology studies the topology of smooth manifolds by examining the critical points of a real-valued function defined on the manifold, and connecting them with the negative gradient of the function. Rather surprisingly, the resulting homology is proved to be independent of the choice of the real-valued function and metric defining the negative gradient. This leads to a topological lower bound on the number of critical points. In the 1980s, the construction of Morse homology served as a prototype to define a homology spanned by $1$-periodic Hamiltonian diffeomorphisms on symplectic manifolds. The resulting homology, introduced by Andreas Floer, spectacularly revolutionized the area of symplectic topology and leaded to a proof of the famous Arnold conjecture. Floer theory still is the subject of a lot of active and exciting research and is nowadays an essential technique in symplectic topology.

The problem of reduction of multisymplectic manifolds by the action of Lie groups is stated and discussed, as a previous step to give a fully covariant scheme of reduction for classical field theories with symmetries., Peer Reviewed, Postprint (author's final draft)

En aquest treball es presenten les nocions de geometria simplèctica, de Poisson i b-simplèctica. En cada cas està demostrat un teorema de Arnold-Liouville per a sistemes integrables. Desprès de demostrar una part d'aquests teoremes usant eines diferents de topologia diferencial, ens centre en les varietats fold-simplèctiques. Per aquestes definim la noció de sistema integrable i demostrem un teorema de Arnold-Liouville. Finalment, analitzem l'estructura singular simplèctica de un exemple de Mecànica Celest i l'usem per demostrar un teorema d'existència d'òrbites de col.lisió-ejecció d'una manera més simple.

We present a covariant multisymplectic formulation for the Einstein-Hilbert model of General Relativity. As it is described by a second-order singular Lagrangian, this is a gauge field theory with constraints. The use of the unified Lagrangian-Hamiltonian formalism is particularly interest- ing when it is applied to these kinds of theories, since it simplifies the treatment of them; in particular, the implementation of the constraint algorithm, the retrieval of the Lagrangian description, and the construction of the covariant Hamiltonian formalism. In order to apply this algorithm to the co- variant field equations, they must be written in a suitable geometrical way, which consists of using integrable distributions, represented by multivector fields of a certain type. We apply all these tools to the Einstein-Hilbert model without and with energy-matter sources. We obtain and explain the geometrical and physical meaning of the Lagrangian constraints and we construct the multimomen- tum (covariant) Hamiltonian formalisms in both cases. As a consequence of the gauge freedom and the constraint algorithm, we see how this model is equivalent to a first-order regular theory, without gauge freedom. In the case of presence of energy-matter sources, we show how some relevant geo- metrical and physical characteristics of the theory depend on the type of source. In all the cases, we obtain explicitly multivector fields which are solutions to the gravitational field equations. Finally, a brief study of symmetries and conservation laws is done in this context., Peer Reviewed, Postprint (author's final draft)

En aquesta tesi estudiarem la relació entre les varietats simplèctiques tòriques i els sistemes integrables. Per això, presentem en primer lloc tots els preliminars necessaris de geometria simplèctica. Després presentem el teorema de Delzant i alguns exemples. Finalment, s'estudia i es presenta una demostració alternativa del teorema que relaciona les varietats amb els sistemes integrables: el teorema de Arnold-Liouville.

The consequences of the projectability of Poincar\'e-Cartan forms in a third-order jet bundle $J^3\pi$ to a lower-order jet bundle are analyzed using the constraint algorithm for the Euler-Lagrange equations in $J^3\pi$. The results are applied to the Hilbert Lagrangian for the Einstein equations. Furthermore, the case of higher-order mechanics is also studied as a particular situation., Peer Reviewed, Postprint (author's final draft)

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics., Peer Reviewed, Postprint (author's final draft)