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Insect flight capabilities provide fascination for humans and fuel aspirations for development and manufacturing of insect-type flapping wing aerial vehicles. However, complex underlying aerodynamic phenomena limit our abilities in understanding insect flight and producing aerial vehicle exploiting same phenomena. To this end, a novel mid-fidelity approach to insect-type flapping vehicles modeling is proposed. Computational model includes Helmholtz-Hodge decompositon of fluid velocity into curl-free and divergence-free parts. Coupled multibody-fluid system equations of motion are derived, including added inertia effects of the environmental fluid and viscous effects arising as an additional aerodynamic load on a multibody system. Curl-free vector field is utilized to accurately model added inertia effects, while expressing coupled system dynamics by using multibody system variables only, after employing symplectic reduction of the coupled multibody-fluid system. On the other hand, unsteady viscous effects included in the divergence-free vector field are modeled by a wake of irrotational point vortices, shed from both leading (important for insect-type flapping flight) and trailing edges of the flapping wing. A proposed computational model is evaluated on two numerical examples involving insect-type flapping flight. The first test case involves standstill hovering of fruit fly in Earth atmospheric environment, propelled by flapping pattern characterized with smooth flapping angle functions. Second test case involves insect-type flapping wing aerial vehicle performing hovering in Mars atmospheric environment, with flapping pattern input in discrete form, resulting from optimization algorithm. It is concluded from results analysis that the proposed computational model exhibited near real time properties with high load prediction accuracy. Let insekata predstavlja inspiraciju za istraživanje u tehničkim i prirodnim znanostima, zbog njihovih izvrsnih performansi letenja, koje se temelje na naglašenom korištenju nestacionarnih ‘fluid-solid’ utjecaja. Mahokrila insektnog tipa omogućuju energetski efikasan let i brze manevre, zadržavajući male dimenzije te predstavljajući letne performanse s kojima se ne mogu usporediti performanse konstruiranih letjelica (tek su nedavno dizajnirane prve letjelice koje u određenoj mjeri oponašaju let insekta). Razlog tomu jest složena nestacionarna aerodinamika karakteristična za let insekata koja se počela razumijevati tek u protekla tri desetljeća. Između ostalog, otkriveno je da insekti koriste vrtlog otpušten s napadnog ruba krila kako bi dodatno povećali uzgon u usporedbi s istim krilom pri istom napadnom kutu u stacionarnom strujanju bez vrtloga. Fokus istraživača u posljednja dva desetljaća usmjeren je na razvoj i izradu mahokrilne letjelice koja bi koristila iste nestacionarne aerodinamičke fenomene karakteristične za let insekata. Složenost nestacionarnih i izrazito nelinearnih aerodinamičkih fenomena otežava zadaću konstruiranja i optimiranja takve letjelice bez odgovarajućih računalnih alata. Zbog visokih frekvencija mahanja i velike amplitude rotacije krila karakterističnih za let insekata računalni modeli spregnutih zadaća temeljeni na metodi konačnih volumena za modeliranje fluida i posebnih (odvojenih) numeričkih rješavača dinamike uronjenog kinematičkog lanca pokazuju neoptimalne performanse i mogućnost pojava računalnih nestabilnosti. Osim toga, računalno vrijeme potrebno za njihovo izvršavanje čini takve pristupe modeliranja neupotrebljivima unutar konstrukcijskih petlji te unutar petlji za optimalno upravljanje. Iz tog razloga javlja se potreba za računalnim modelom mahokrilne letjelice insektnog tipa koji bi omogućio pouzdano modeliranje signifikantnih aerodinamičkih fenomena uz zadržavanje računalne efikasnosti koja omogućuje primjenu unutar konstrukcijskih petlji te unutar petlji optimalnog upravljanja. U tu svrhu spregnuti sustav krilo-fluid promatra se iz perspektive geometrijske mehanike koja omogućava redukcije spregnutog modela na zajedničkoj mognostrukosti, odnosno Lievoj grupi diskretnog mehaničkog sustava i ambijentalnog fluida. Brzina fluida se dekomponira u vektorsko polje bez vrtložnosti te vektorsko polje bez divergencije korištenjem Helmholtz-Hodge dekompozicije. Značajke vektorskog polja bez vrtložnosti se zatim koriste pri simplektičkoj redukciji multi-fizikalnog sustava za modeliranje dodane inercije krila uronjenog u fluid - važan fenomen za mahokrila insektnog tipa zbog velikih vrijednosti ubrzanja i složene kinematike krila insektnog tipa. Vektorsko polje bez divergencije koristi se za modeliranje viskoznih utjecaja vrtložnim tragom.

La dinàmica d'un fluid ideal en un espai 2-dimensional pot ser aproximat mitjançant la evolució de N vortexs singulars a mesura que fem tendir N cap a infinit. Al llarg d'aquest treball estudiem la dinàmica de vortexs singulars, normalment denominats com "point vortices", en una superficie no orientable com és l'espai projectiu real, $\RP^2$. Primer estudiem les equacions de Euler per fluids incompressibles així com la seva generalització a varietats no orientables. L'objectiu es obtenir un nou model de vortexs singulars que sigui compatible amb varietats no orientables. Veurem que la dinamica de N vòretxs en un espai projectiu pot ser visualitzat com la dinàmica de N dipols antipodals en l'esfera $S^2$, la qual és un espai recobridor de dos fulls de l'espai projectiu. En la segona part del treball presentem un estudi dels casos integrables, en el sentit de Liouville, a $\RP^2$ entre els quals trobem tant fenòmens nous com fenòmens similars als que es tenen en el problema estandar a l'esfera. Dins de l'estudi incluim l'estudi de fenòmens de equilibri estàtic, equilibri dinàmic, estabilitat i del fenòmen de col·lapse singular. La dinámica de un fluido ideal en un espacio de dimensión 2 puede ser aproximado mediante la evolución de N vortices singulares a medida que N tiende a infinito. A lo largo de este trabajo estudiamos la dinámica de vortices singulares (point vortices) en una superficie no orientable, el plano proyectivo. Primero estudiamos las equaciones de Euler para fluidos incompressibles asi como su generalización para variedades no orientables. La intencion es obtener un nuevo modelo de vortices singulares que sea compatible con la no-orientabilidad de la variedad. Veremos que la dinámica de N vortices en el espacio proyectivo es equivalente a la dinámica de N dipolos antipodales en la esfera, la cual es un recubrimiento de 2 hojas del espacio proyectivo. En la siguiente parte presentamos una comparación de los casos integrables, en el sentido de Liouville, en RP^2 con dinámicas analogas en S^2. Veremos que la dinámica de N vortices en el espacio proyectivo presenta algunas similitudes con el problema estándar así como algunas diferencias. Dentro del trabajo incluimos el estudio de equilibrios estáticos, relativos, estabilidad y colapso singular. The motion of an ideal two-dimensional fluid can be approximated by the evolution of N point vortices, as N → ∞. Throughout this work, we study the motion of point vortices on a non- orientable surface, the projective plane RP2. First, we study the incompressible Euler equations, as well as their generalization to non-orientable manifolds. The aim is to derive a new model for point vortices that is compatible with non-orientability. We will see that the motion of N vortices on the projective space can be regarded as the motion of N antipodal dipoles on the sphere S2, the orientation cover of RP2. In the next part, we will present a comparison of the integrable cases of the vortex motions on RP2 with analogous motions on S2 describing new phenomena, as well as similarities. Among the cases we study, we include the study of equilibria, relative equilibria, their stability, and collapse. Outgoing

Suppose given a Hamiltonian and holomorphic action of $$G=U(2)$$ G = U ( 2 ) on a compact Kähler manifold M, with nowhere vanishing moment map. Given an integral coadjoint orbit $$\mathcal {O}$$ O for G, under transversality assumptions we shall consider two naturally associated ‘conic’ reductions. One, which will be denoted $$\overline{M}^G_{\mathcal {O}}$$ M ¯ O G , is taken with respect to the action of G and the cone over $$\mathcal {O}$$ O ; another, which will be denoted $$\overline{M}^T_{\varvec{\nu }}$$ M ¯ ν T , is taken with respect to the action of the standard maximal torus $$T\leqslant G$$ T ⩽ G and the ray $$\mathbb {R}_+\,\imath \varvec{\nu }$$ R + ı ν along which the cone over $$\mathcal {O}$$ O intersects the positive Weyl chamber. These two reductions share a common ‘divisor’, which may be viewed heuristically as bridging between their structures. This point of view motivates studying the (rather different) ways in which the two reductions relate to the the latter divisor. In this paper we provide some indications in this direction. Furthermore, we give explicit transversality criteria for a large class of such actions in the projective setting, as well as a description of corresponding reductions as weighted projective varieties, depending on combinatorial data associated to the action and the orbit.

The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

Let $Q$ be a closed manifold admitting a locally-free action of a compact Lie group $G$. In this paper we study the properties of geodesic flows on $Q$ given by Riemannian metrics which are invariant by such an action. In particular, we will be interested in the existence of geodesics which are closed up to the action of some element in the group $G$, since they project to closed magnetic geodesics on the quotient orbifold $Q/G$., The proof of Theorem 1.5 in the previous version of the draft was erroneous. A weaker version of the aforementioned theorem is now stated and proved

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples., Comment: 18 pages, final version

In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincare-Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincare-Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum. (C) 2013 Elsevier B.V. All rights reserved.

In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré–Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincaré–Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.

Let Γ be a finite subgroup of SU2(ℂ). Then the quotient ℂ2/Γ can be embedded in ℂ3 as a surface with an isolated singularity. The quotient ℂ2/Γ is called a Kleinian singularity, after F. Klein who studied them first in 1884. Through their minimal resolutions, these singularities have a deep connection with simply-laced Dynkin diagrams of types Ar, Dr and Er. In the 1970's E. Brieskorn and P. Slodowy took advantage of this connection to describe the resolutions and deformations of these singularities in terms of Lie theory. In 1998 P. Slodowy and H. Cassens constructed the semiuniversal deformations of the Kleinian singularities using quiver theory and work from 1989 by P.B. Kronheimer on symplectic geometry. In Lie theory, the classification of simple Lie algebras allows for a separation in two classes: those simply-laced of types Ar, Dr and Er, and those of types Br, Cr, F4 and G2 called inhomogeneous. With the use of a second finite subgroup Γ’ of SU2(ℂ) such that Γ ⊲ Γ’, P. Slodowy extended in 1978 the definition of a Kleinian singularity to the inhomogeneous types by adding to ℂ2/Γ the group of automorphisms Ω= Γ’/Γ of the Dynkin diagram associated to the singularity. The purpose of this thesis is to generalize H. Cassens' and P. Slodowy's construction to the singularities of types Br, Cr, F4 and G2. It will lead to explicit semiuniversal deformations of inhomogeneous types on the fibers of which the group Ω acts. By quotienting such a map we obtain a deformation of a singularity ℂ2/Γ’, Soit Γ un sous-groupe fini de SU2(ℂ). Alors le quotient ℂ2/Γ peut être plongé dans ℂ3 sous la forme d'une surface munie d'une singularité isolée. Le quotient ℂ2/Γ est appelé singularité de Klein, d'après F. Klein qui fut le premier à les décrire en 1884. A travers leurs résolutions minimales, ces singularités ont un lien étroit avec les diagrammes de Dynkin simplement lacés de types Ar, Dr et Er. Dans les années 1970, E. Brieskorn et P. Slodowy ont tiré profit de cette connection pour décrire les résolutions et les déformations de ces singularités à l'aide de la théorie de Lie. En 1998 P. Slodowy et H. Cassens ont construit les déformations semiuniverselles des ℂ2/Γ à l'aide de la théorie des carquois ainsi que des travaux de P.B. Kronheimer en géométrie symplectique datant de 1989. En théorie de Lie, la classification des algèbres de Lie simples divisent ces dernières en deux classes: les algèbres de Lie de types Ar, Dr et Er qui sont simplement lacées, et celles de types Br, Cr, F4 et G2 appelées non-homogènes. A l'aide d'un second sous-groupe fini Γ' de SU2(ℂ) tel que Γ ⊲ Γ', P. Slodowy a étendu en 1978 la notion de singularité de Klein aux algèbres de Lie non-homogènes en ajoutant à ℂ2/Γ le groupe d'automorphismes Ω= Γ'/Γ du diagramme de Dynkin associé à la singularité. L'objectif de cette thèse est de généraliser la construction de H. Cassens et P. Slodowy à ces singularités de types Br, Cr, F4 et G2. Il en résultera des constructions explicites des déformations semiuniverselles de types inhomogènes sur les fibres desquelles le groupe Ω agit. Le passage au quotient d'une telle application révèle alors une déformation d'une singularité de type ℂ2/Γ'

International audience; The purpose of this article is to generalize a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of types A_r , D_r , E_6 , E_7 and E_8 to the singularities of inhomogeneous types B_r , C_r , F_4 and G_2 defined in 1978 by P. Slodowy. Let Γ be a finite subgroup of SU_2. Then C^2/ Γ is a simple singularity of type ∆(Γ). By studying the representation space of a quiver defined from Γ via the McKay correspondence , and a well chosen finite subgroup Γ ′ of SU_2 containing Γ as normal subgroup, we will use the symmetry group Ω = Γ′/Γ of the Dynkin diagram ∆(Γ) and explicitly compute the semiuniversal deformation of the singularity (C^2/Γ, Ω) of inhomogeneous type. The fibers of this deformation are all equipped with an induced Ω-action. By quotienting we obtain a deformation of a singularity C^2/Γ′ with some unexpected fibers.

International audience; The purpose of this article is to generalize a construction by H. Cassens and P. Slodowy of the semiuniversal deformations of the simple singularities of types A_r , D_r , E_6 , E_7 and E_8 to the singularities of inhomogeneous types B_r , C_r , F_4 and G_2 defined in 1978 by P. Slodowy. Let Γ be a finite subgroup of SU_2. Then C^2/ Γ is a simple singularity of type ∆(Γ). By studying the representation space of a quiver defined from Γ via the McKay correspondence , and a well chosen finite subgroup Γ ′ of SU_2 containing Γ as normal subgroup, we will use the symmetry group Ω = Γ′/Γ of the Dynkin diagram ∆(Γ) and explicitly compute the semiuniversal deformation of the singularity (C^2/Γ, Ω) of inhomogeneous type. The fibers of this deformation are all equipped with an induced Ω-action. By quotienting we obtain a deformation of a singularity C^2/Γ′ with some unexpected fibers.

We consider the motion of a two-dimensional body of arbitrary shape in a planar irrotational, incompressible fluid with a given amount of circulation around the body. We derive the equations of motion for this system by performing symplectic reduction with respect to the group of volume-preserving diffeomorphisms and obtain the relevant Poisson structures after a further Poisson reduction with respect to the group of translations and rotations. In this way, we recover the equations of motion given for this system by Chaplygin and Lamb, and we give a geometric interpretation for the Kutta-Zhukowski force as a curvature-related effect. In addition, we show that the motion of a rigid body with circulation can be understood as a geodesic flow on a central extension of the special Euclidian group SE(2), and we relate the cocycle in the description of this central extension to a certain curvature tensor., Comment: 28 pages, 2 figures; v2: typos corrected

We describe the structure of the Lie groups endowed with a left-invariant symplectic form, called symplectic Lie groups, in terms of semi-direct products of Lie groups, symplectic reduction and principal bundles with affine fiber. This description is particularly nice if the group is Hamiltonian, that is, if the left canonical action of the group on itself is Hamiltonian. The principal tool used for our description is a canonical affine structure associated with the symplectic form. We also characterize the Hamiltonian symplectic Lie groups among the connected symplectic Lie groups. We specialize our principal results to the cases of simply connected Hamiltonian symplectic nilpotent Lie groups or Frobenius symplectic Lie groups. Finally we pursue the study of the classical affine Lie group as a symplectic Lie group. MSC Classes 53D20,70G65 Key words:Symplectic Lie groups,Hamiltonian Lie groups, Symplectic reduction,Symplectic double extension., Comment: 15 pages, added examples, corrected typos, 0 figures

The goal of these notes is to show that the classification problem of algebraically unbiased system of projectors has an interpretation in symplectic geometry. This leads us to a description of the moduli space of algebraically unbiased bases as critical points of a potential functions, which is a Laurent polynomial in suitable coordinates. The Newton polytope of the Laurent polynomial is the classical Birkhoff polytope, the set of double stochastic matrices. Mirror symmetry interprets the polynomial as a Landau-Ginzburg potential for corresponding Fano variety and relates the symplectic geometry of the variety with systems of unbiased projectors., 14 pages

The Goresky-Kottwitz-MacPherson (GKM) graph is a combinatorial analogue of a compact connected symplectic manifold with a Hamiltonian action of a compact torus. This graph has been intensively studied by Guillemin and Zara, who discovered analogues in graph theory of classical results such as: symplectic reduction and "quantization and reduction commute.'' In this paper, we describe the implementation of algorithms illustrating their results.

In a previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity. To prove our main result, we use Lagrangian Floer theory to construct a new class of spectral invariants which satisfy a non-standard triangle inequality., 39 pages, to appear in Annales scientifiques de l'\'Ecole normale sup\'erieure

This paper contains results on geometric Routh reduction and it is a continuation of a previous paper where a new class of transformations is introduced between Lagrangian systems obtained after Routh reduction. In general, these reduced Lagrangian systems have magnetic force terms and are singular in the sense that the Lagrangian does not depend on some velocity components. The main purpose of this paper is to show that the Routh reduction process itself is entirely captured by the application of such a new transformation on the initial Lagrangian system with symmetry., To appear in Journal of Geometric Mechanics

In this paper we study a systematic and natural construction of canonical coordinates for the reduced space of a cotangent bundle with a free Lie group action. The canonical coordinates enable us to compute Poincaré–Birkhoff normal forms of relative equilibria using standard algorithms. The case of simple mechanical systems with symmetries is studied in detail. As examples we compute Poincaré–Birkhoff normal forms for a Lagrangian equilateral triangle configuration of a three-body system with a Morse-type potential and the stretched-out configuration of a double spherical pendulum.

In this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman's results using string topology operations.The main result can be applied to the de Rham complex of a smooth manifold as well as the Dolbeault resolution of the endomorphisms of a holomorphic bundle on a Calabi-Yau manifold., 14 Pages