DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_60

Video: http://www.youtube.com/watch?v=leK1MggION0

Slides: Justh_Enlargement geodesics collectives.pdf

Presentation: https://www.see.asso.fr/node/14343

Creative Commons Attribution-ShareAlike 4.0 International

Abstract:

We investigate optimal control of systems of particles on matrix Lie groups coupled through graphs of interaction, and characterize the limit of strong coupling. Following Brockett, we use an enlargement approach to obtain a convenient form of the optimal controls. In the setting of drift-free particle dynamics, the coupling terms in the cost functionals lead to a novel class of problems in subriemannian geometry of product Lie groups.

DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_59

Video: http://www.youtube.com/watch?v=Dm1ht6tNJbo

Slides: Machado_Rolling Symmetric Spaces.pdf

Presentation: https://www.see.asso.fr/node/14342

Creative Commons Attribution-ShareAlike 4.0 International

Abstract:

Riemannian symmetric spaces play an important role in many areas that are interrelated to information geometry. For instance, in image processing one of the most elementary tasks is image interpolation. Since a set of images may be represented by a point in the Graßmann manifold, image interpolation can be formulated as an interpolation problem on that symmetric space. It turns out that rolling motions, subject to nonholonomic constraints of no-slip and no-twist, provide efficient algorithms to generate interpolating curves on certain Riemannian manifolds, in particular on symmetric spaces. The main goal of this paper is to study rolling motions on symmetric spaces. It is shown that the natural decomposition of the Lie algebra associated to a symmetric space provides the structure of the kinematic equations that describe the rolling motion of that space upon its affine tangent space at a point. This generalizes what can be observed in all the particular cases that are known to the authors. Some of these cases illustrate the general results.

DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_58

Video: http://www.youtube.com/watch?v=uiOKLhoh3IA

Slides: Dubois Pontryagin calculus Riemannian.pdf

Presentation: https://www.see.asso.fr/node/14340

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Abstract:

In this contribution, we study systems with a finite number of degrees of freedom as in robotics. A key idea is to consider the mass tensor associated to the kinetic energy as a metric in a Riemannian configuration space. We apply Pontryagin’s framework to derive an optimal evolution of the control forces and torques applied to the mechanical system. This equation under covariant form uses explicitly the Riemann curvature tensor.

DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_57

Video: http://www.youtube.com/watch?v=wTxIdmALzjo

Slides: Barbaresco_symplectic structure.pdf

Presentation: https://www.see.asso.fr/node/14339

Creative Commons Attribution-ShareAlike 4.0 International

Abstract:

We introduce the Symplectic Structure of Information Geometry based on Souriau’s Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. In the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. Finally, we conclude on Balian Gauge theory of Thermodynamics compatible with Souriau’s Model.

DOI URL: http://dx.doi.org/10.1007/978-3-319-25040-3_56

Video: http://www.youtube.com/watch?v=7czAJrfFfgo

Slides: DeSaxce_Entropy structures thermodynamical.pdf

Presentation: https://www.see.asso.fr/node/14338

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Abstract:

With respect to the concept of affine tensor, we analyse in this work the underlying geometric structure of the theories of Lie group statistical mechanics and relativistic thermodynamics of continua, formulated by Souriau independently one of each other. We reveal the link between these ones in the classical Galilean context. These geometric structures of the thermodynamics are rich and we think they might be source of inspiration for the geometric theory of information based on the concept of entropy.

DIO URL: http://dx.doi.org/10.1007/978-3-319-25040-3_55

Video: http://www.youtube.com/watch?v=RXF12CcGOwU

Slides: Boyer_ Poincaré equations Cosserat shells.pdf

Presentation: https://www.see.asso.fr/node/14356

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Abstract:

In 1901 Henri Poincaré proposed a new set of equations for mechanics. These equations are a generalization of Lagrange equations to a system whose configuration space is a Lie group, which is not necessarily commutative. Since then, this result has been extensively refined by the Lagrangian reduction theory. In this article, we show the relations between these equations and continuous Cosserat media, i.e. media for which the conventional model of point particle is replaced by a rigid body of small volume named microstructure. In particular, we will see that the usual shell balance equations of nonlinear structural dynamics can be easily derived from the Poincaré’s result. This framework is illustrated through the simulation of a simplified model of cephalopod swimming.