Singular contact geometry and celestial mechanics

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].Outgoin

An equivariant Reeb-Beltrami correspondence and the Kepler-Euler flow

We prove that the correspondence between Reeb and Beltrami vector fields can be made equivariant whenever additional symmetries of the underlying geometric structures are considered. As a corollary of this correspondence, we show that energy levels above the maximum of the potential energy of mechanical Hamiltonian systems can be viewed as stationary fluid flows, though the metric is not prescribed. In particular, we showcase the emblematic example of the $n$-body problem and focus on the Kepler problem. We explicitly construct a compatible Riemannian metric that makes the Kepler problem of celestial mechanics a stationary fluid flow (of Beltrami type) on a suitable manifold, the Kepler-Euler flow.Comment: 16 pages, 3 figures. Overall improvements of the paper. Final section adde

2N or infinitely many escape orbits

In this short note, we prove that singular Reeb vector fields associated with generic Melrose-type $b$-contact forms have either (at least) $2N$ or an infinite number of escape orbits, where $N$ denotes the number of connected components of the critical set. We obtain this result as a corollary of the same statement for the number of escape orbits of singular Beltrami fields using the singular version of Etnyre-Ghrist's contact/Beltrami correspondence.Comment: 12 pages, for Alain Chenciner on his 2N-birthda

Prevalence and characterization of psychological trauma in patients with fibromyalgia: A cross-sectional study

Background: Preliminary evidence suggests that psychological trauma, especially childhood trauma, is a risk factor for the onset of fibromyalgia (FM). Objective: The main objective of this study consisted of evaluating the prevalence and detailed characteristics of psychological trauma in a sample of patients with FM, the chronology of trauma across the lifespan, and its clinical symptoms. We also calculated whether childhood trauma could predict the relationship with different clinical variables. Method: Eighty-eight females underwent an interview to assess sociodemographic data, psychiatric comorbidities, level of pain, FM impact, clinical symptoms of anxiety, depression, insomnia, quality of life, and psychological trauma. Results: The majority of participants (71.5%) met the diagnostic criteria for current post-traumatic stress disorder (PTSD). Participants reported having suffered traumatic events throughout their lifespan, especially in childhood and early adolescence, in the form of emotional abuse, emotional neglect, sexual abuse, and physical abuse. Traumatic events predict both poor quality of life and a level of pain in adulthood. All patients showed clinically relevant levels of anxiety, depression, insomnia, suicidal thoughts, and pain, as well as somatic comorbidities and poor quality of life. Pain levels predicted anxiety, depression, dissociation, and insomnia symptoms. 84% of the sample suffered one or more traumatic events prior to the onset of pain. Conclusions: Our data highlight the clinical complexity of patients with FM and the role of childhood trauma in the onset and maintenance of FM, as well as the high comorbidity between anxiety, depression, somatic symptoms, and FM. Our data also supports FM patients experiencing further retraumatization as they age, with an extremely high prevalence of current PTSD in our sample. These findings underscore the need for multidisciplinary programs for FM patients to address their physical pain and their psychiatric and somatic conditions, pay special attention to the assessment of psychological trauma, and provide trauma-focused interventions. Trial registration: ClinicalTrials.gov NCT04476316. Registered on July 20th, 2020