Benvinguts a la pàgina web del Seminari Teen de Geometria

Inverse problem and control

8 de jul. 2019, 11:01 publicada per Xavier Gràcia   [ actualitzat el 8 de jul. 2019, 23:50 ]

Speaker: Marta Farré Puiggalí (University of Michigan) 
Date and place: Friday 12 July 2019, 13:00; FME (UPC), room 102
Title: Inverse problem and control

Abstract
The inverse problem of the calculus of variations consists in determining whether or not a given system of second order differential equations is equivalent to some regular Lagrangian system. I will explain how to apply some results of the inverse problem to the stabilization of controlled Lagrangian systems.

References
[1] M. Farré Puiggalí, A.M. Bloch, "An extension to the theory of controlled Lagrangians using the Helmholtz conditions", J. Nonlinear Sci. 29 (2019) 345-376.
[2] M. Farré Puiggalí, T. Mestdag, "The inverse problem of the calculus of variations and the stabilization of controlled Lagrangian systems", SIAM J. Control Optim. 54 (2016) 3297-3318.

To special relativity from newtonian gravitation? A plausible uchronia

26 de febr. 2019, 6:26 publicada per Xavier Gràcia   [ actualitzat el 26 de febr. 2019, 6:28 ]

Speaker: Mariano Santander (Universidad de Valladolid)
Date and place: Thursday 7 March 2019, 12:00h; FME (UPC), room 004
Title: To special relativity from newtonian gravitation? A plausible uchronia

Abstract
Could the historical temporal sequence of discovery: first 'special relativity'; then after eight year's struggle 'general relativity' have developed otherwise? Is it physically sound, from a conceptual viewpoint, to imagine that general relativity could have appeared whitout special relativity, starting directly from newtonian gravitation?
The talk describes the uchronia (or Gedankenexperiment in history) which unfolds from asking ourselves: What if the concept of connection had been available by the 1900s, before special relativity, and not in 1917 (as it was actually first introduced), after general relativity was completed?
In the newtonian theory, both gravitational and inertial forces admit a description from a principle of stationary action similar to the one now standard but first written (for electromagnetism) by Schwarzschild in 1904. If the fictional character of this uchronia, call him Schwarzstein, had pursued ---before Einstein 1905--- the same ideas for the newtonian gravitation, he would have been led to recognize that the 'potentials' of this theory cannot be a simple scalar nor a four vector (as it is in electromagnetism) but necessarily a second order, covariant symmetrical tensor in space time. When there are no gravitational nor inertial effects (say, in absence of gravitation and in an `inertial frame'), the theory thus obtained, which stems from Newtonian gravitation, reduces precisely and directly not to the old Galileo-Newton space-time but instead to special relativity.

Routh reduction for field theory

12 de des. 2018, 16:48 publicada per Xavier Gràcia

Speaker: Santiago Capriotti (Universidad Nacional del Sur, Bahía Blanca) 
Date and place: Monday 17 December 2018, 15:30; FME (UPC), room 103
Title: Routh reduction for field theory

Abstract
Routh reduction is a reduction theory which takes into account the conservation of momenta. Although this kind of reduction has been extensively studied in the case of classical mechanics, for the field theory case it has remained almost unexplored.
The purpose of this talk is to present some aspects of the joint work with Eduardo García-Toraño Andrés regarding the formulation of Routh reduction for first order field theories.

Global properties of manifolds with integral curvature conditions

7 de des. 2018, 9:34 publicada per Xavier Gràcia   [ actualitzat el 14 de des. 2018, 1:43 ]

Speaker: Xavier Ramos (University of California, Riverside)  web page
Date and place: Monday 17 December 2018, 12:30; FME (UPC), room 101
Title: Global properties of manifolds with integral curvature conditions

Abstract
Since the 1950's, a trend in Riemannian geometry has been to study how curvature affects global properties of a manifold. Assuming that a manifold satisfies certain curvature bounds has consequences on global metric notions like its diameter, on topological invariants like its fundamental group, and on analytic properties like the spectrum of the Laplacian or the heat kernel.
Many of these results rely on a pointwise lower bound of the Ricci curvature. In recent years, there has been an increasing interest in weakening this assumption to an Lp curvature assumption, which is more natural to estimate topological invariants, and more suitable in the study of geometric flows like the Ricci flow. We will discuss some examples of results that hold under this weaker assumption, including a recent Zhong-Yang type estimate for the first eigenvalue of the Laplacian under integral curvature conditions.
This is joint work with Shoo Seto, Guofang Wei and Qi S. Zhang.

References

[1] P. Li, S.T. Yau; "On the parabolic kernel of the Schrödinger operator", Acta Math. 156 (1986) 153–201.
[2] X. Ramos Olivé, S. Seto, G. Wei, Q.S. Zhang; "Zhong-Yang type eigenvalue estimate with integral curvature condition"; arXiv:1812.00579 (2018).
[3] X. Ramos Olivé; "Neumann Li--Yau gradient estimate under integral Ricci curvature bounds"; Proc. Amer. Math. Soc. 147 (2019) 411--426.
[4] J.Q. Zhong, H.C. Yang; "On the estimate of the first eigenvalue of a compact Riemannian manifold"; Sci. Sinica Ser. A 27 (1984) 1265–1273.

Two equivalent non-linear stabilization methods

7 de des. 2018, 9:12 publicada per Xavier Gràcia   [ actualitzat el 7 de des. 2018, 9:36 ]

Speaker: Leandro Salomone (Universidad Nacional de La Plata)
Date and place: Tuesday 11 December 2018, 12:30; FME (UPC), room 101
Title: Two equivalent non-linear stabilization methods

Abstract
The energy shaping method (ES) consists of a family of non-linear (asymptotic) stabilization methods for stabilizing unstable equilibrium points of control mechanical systems. It is possible to show that any two members of this family are equivalent to each other. The idea behind all of these methods is that of feedback equivalence. It can be used to find a set of PDEs known as matching conditions, which is the key ingredient of the ES method.

On the other hand, the Lyapunov constraint-based method (LCB) is another non-linear (asymptotic) stabilization method but, in this case, the idea originating the control strategy is that of stabilization by means of the imposition of kinematic constraints. In this framework, the control law is obtained as the constraint force associated to some (apropriately chosen) set of kinematic constraints. As it occurs with the ES, the key ingredient of the LCB is another set of PDEs.

In this talk we will show that every instance of the ES is "contained" in the LCB. Moreover, in the case of simple Hamiltonian systems (i.e., those with kinetic plus potential energy Hamiltonian), we will see that there is a distinguished member of the ES family that is equivalent to the LCB, in the sense that every control law of one method can also be produced by the other method, and viceversa. In particular, we will show that the PDEs characterizing both methods are exactly the same.

References
[1] S.D. Grillo, L.M. Salomone, M. Zuccalli, "On the relationship between the energy shaping and the Lyapunov constraint based methods", J. Geom. Mech. 9 (2017), 459–486.
[2] D.E. Chang, "The method of controlled Lagrangians: energy plus force shaping", SIAM J. Control Optimization, 48 (2010), 4821–4845.
[3] D. Chang, A.M. Bloch, N.E. Leonard, J.E. Marsden, C. Woolsey, "The equivalence of controlled Lagrangian and controlled Hamiltonian systems", ESAIM: Control, Optimisation and Calculus of Variations, (2001).

Geometry and dynamics

2 d’oct. 2018, 4:16 publicada per Xavier Gràcia   [ actualitzat el 7 de des. 2018, 9:35 ]

Speaker: Manuel de León (CSIC and RACEFyN)
Date and place: Thursday 4 October 2018, 12:30; FME (UPC), room 101
Title: Geometry and dynamics

Abstract
In this talk, we will discuss how different background geometries produce completely different dynamics.
More precisely, we show how symplectic and cosymplectic geometries produce the Hamilton equations for a given Hamiltonian function, but dissipative Hamiltonian equations need to use contact geometry.
In the second part of the talk, we will discuss a coisotropic reduction theorem for contact geometry and its applications to mechanics.

Matched pair of Hamiltonian and Lagrangian dynamics

20 de jul. 2017, 13:25 publicada per Xavier Gràcia   [ actualitzat el 21 de jul. 2017, 3:32 ]

Speaker: Oğul Esen (Gebze Technical University)
Date and place: Wednesday 26 July 2017, 12:45; FME (UPC), room S01
Title: Matched pair of Hamiltonian and Lagrangian dynamics

Abstract
In this talk, equations governing the motion of two mutually interacting systems (whose configuration spaces are Lie groups) will be presented both in the Lagrangian formalism and the Hamiltonian formalism. This theory is called the dynamics of matched pairs. Under the presence of the symmetry, the matched Euler--Poincaré equations and the matched Lie--Poisson equations will be derived. Two concrete examples will be provided. The first one is in finite dimension, namely the matched pair dynamics on SL(2,C). The second one is in infinite dimension, coupling of hydrodynamics with electromagnetic field.

References
[1] O. Esen and S. Sütlü, "Lagrangian Dynamics on Matched Pairs", J. Geom. Phys. 111, 142--157, 2017.
[2] O. Esen and S. Sütlü, "Hamiltonian Dynamics on Matched Pairs", Int. J. Geom. Methods Mod. Phys. 13, 1650128, 2016.
[3] O. Esen, M. Pavelka and M. Grmela, "Hamiltonian Coupling of Electromagnetic Field and Matter", Int. J. Adv. Eng. Sci. Appl. Math., DOI 10.1007/s12572-017-0179-4, 2017.
[4] O. Esen, "Dinamik Sistemlerin Eslenmesi", SAU Fen Bilimleri Ens. Dergisi 21, 469--480, 2017.

Using momentum polytopes to examine the SU(3) action on (CP^2)^2 and (CP^2)^3

5 de jul. 2017, 2:48 publicada per Xavier Gràcia

Speaker: Amna Shaddad (University of Manchester)
Date and place: Thursday 6 July 2017, 12:30; FME (UPC), room S05
Title: Using momentum polytopes to examine the SU(3) action on (CP2)2 and (CP2)3

Abstract:
The momentum polytope provides geometric information on the image of the momentum map. I will discuss recent developments to momentum polytopes that came about from the investigation of a system of vortices.
The results we have derived are powerful in that they provide a new means for evaluating N-body systems: we present the classification of the systems according to their respective weightings and their dynamics according to their momentum polytopes.

Geometric structures and Lie systems: some examples

3 de des. 2015, 1:27 publicada per Xavier Gràcia

Speaker: Silvia Vilariño (Centro Universitario de la Defensa, Zaragoza)
Date and place: Thursday 10 December 2015, 12:30; FME (UPC), room 102
Title: Geometric structures and Lie systems: some examples

Abstract:
A Lie system is a system of first-order differential equations whose general solution can be expressed as a function, the superposition rule, of a generic finite collection of particular solutions and a set of constants. Thus an important questions is: how can you get a superposition rule for a given system?
In searching for the answer to this question, an important role is played by Lie systems admitting a Vessiot-Guldberg Lie algebra of Hamiltonian vector fields (with respect to some geometric structure). This particular type of Lie systems gives new methods to derive superposition rules.
In this seminar we comment on some of these methods using several geometric structures: Poisson, Dirac, k-symplectic... We present some examples and in particular, in one of them, we compare the method of deriving superposition rules using k-symplectic structures and other geometric structures.

References

  1. Cariñena, J. F.; de Lucas, J.;
    Lie systems: theory, generalisations, and applications.
    Dissertationes Math. (Rozprawy Mat.) 479 (2011), 162 pp.
  2. Cariñena, J. F.; Grabowski, J.; de Lucas, J.; Sardón, C.;
    "Dirac-Lie systems and Schwarzian equations".
    J. Differential Equations 257 (2014) 2303–2340.
  3. de Lucas, J.; Sardón, C.
    "On Lie systems and Kummer-Schwarz equations".
    J. Math. Phys. 54 (2013) 033505, 21 pp.
  4. de Lucas, Javier; Tobolski, Mariusz; Vilariño, Silvia
    "A new application of k-symplectic Lie systems".
    Int. J. Geom. Methods Mod. Phys. 12 (2015) 1550071, 6 pp.
  5. de Lucas, J.; Vilariño, S.
    "k-symplectic Lie systems: theory and applications".
    J. Differential Equations 258 (2015) 2221–2255.

Lie–Hamilton systems: properties, classification, and applications

25 de nov. 2015, 2:40 publicada per Xavier Gràcia   [ actualitzat el 30 de nov. 2015, 0:45 ]

Speaker: Cristina Sardón (Dept. of Fundamental Physics, Universidad de Salamanca)
Date and place: Thursday 3 December 2015, 12:30-14:00h; FME (UPC), room 102
Title: Lie–Hamilton systems: properties, classification, and applications

Abstract:
This seminar is concerned with the definition and analysis of a new class of Lie systems on Poisson manifolds that enjoy rich geometric properties: the so called Lie–Hamilton systems.
We start by their definition, statement of their geometric fundamentals and focus on their wide applicability by exposing a number of examples found throughout the Mathematics, Physics, Biology literature and other scientific disciplines. We devise a new method to derive their superposition rules: the so called coalgebra method, and illustrate its usefulness with concrete examples. In particular, we will study Lie–Hamilton systems on the plane and will provide their complete classification on the plane. This classification will give rise to a number of diffeomorphisms between well known systems in the Physics literature that will serve us as a canonical classification of certain phenomena occurring in nature. Solutions for such phenomena will be obtained in form of superposition rules.
To finish, we will introduce the notion of structure-Lie systems, being the latter those Lie systems admitting different geometric backgrounds whose theory, applications and properties can be retrieved from the point of view of another geometric structure.

References

  1. A. Ballesteros, A. Blasco, J.F. Cariñena, F.J. Herranz, J. de Lucas, C. Sardón, "Lie-Hamilton systems on the plane: properties, classification and applications", J. Differential Equations 258, 2873–2907 (2015).
  2. A. Ballesteros, J.F. Cariñena, F.J. Herranz, J. de Lucas, C. Sardón, "From constants of motion to superposition rules of Lie–Hamilton systems", J. Phys. A: Math. Theor. 46, 285203 (2013).
  3. A. Blasco, F.J. Herranz, J. de Lucas, C. Sardón, "Lie–Hamilton systems on the plane: applications and superposition rules", J. Phys. A: Math. Theor. 48, 345202 (2015).
  4. J.F. Cariñena, J. de Lucas, C. Sardón, "Lie–Hamilton systems: theory and applications", Int. J. Geom. Methods Mod. Phys. 10, 0912982 (2013).

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