14th International Young Researchers Workshop on Geometry, Mechanics and Control

V. I. Arnold found out in the 1960s that the motion of inertia rigid objects in Classical Mechanics and the incompressible flow of some ideal fluid can be described by the same mathematical approach: both can be recast as a geodesic flow on a suitable Lie group. While the configuration space for a rigid three-dimensional body is the Lie group SO(3), fluid flow can be modeled on suitable diffeomorphism groups. In this talk, we explain how the geometric reformulation of a variant of the Degasparis-Procesi equation occurring in shallow water theory can be used to obtain results on well-posedness, conservation laws and blow up.

The existence of an exact discrete lagrangian function for nonholonomic systems is still an open problem in the field of geometric integration. In the last few decades, an effort has been made to introduce geometric numerical methods, such as variational integrators, which preserve geometric structure. In the case of variational integrators, we discretize the principal action of the system and then we apply a discrete variational principle to obtain the discrete-time equations of motion, whose solutions are sequences of points which approximate the solution for the continuous-time problem. In this talk we will restrict to nonholonomic mechanics. Although there are plenty of discrete descriptions at our disposable in the existing literature, the problem of finding an exact discrete lagrangian function for onholonomic mechanics or, more generally, the problem of integrating exactly the continuous-time nonholonomic problem was not yet solved. We will show how to produce a family of nonholonomic integrators with the special property that we can distinguish one of them that exactly integrates the continuous-time non-holonomic problem. This discovery could make an advance to the study of error analysis of numerical methods and would have many applications on subjects such as optimal control.

Let H be a C¹ function defined in the 2-sphere of the Euclidean space R³. An oriented surface M is said to have prescribed mean curvature equal to H if its mean curvature function at every point is given by the value of the function H at the image of the Gauss map at p. In this talk we give a structure result concerning the existence and uniqueness of properly embedded surfaces with prescribed mean curvature. This is a joint work with José Antonio Gálvez and Pablo Mira.

A sub-Finslerian manifold is, roughly speaking, a manifold endowed with a Finsler type metric which is defined on a k-dimensional smooth distribution only, not on the whole tangent manifold. Our purpose is to construct a generalized non-linear connection for a sub-Finslerian manifold, called L-connection by the Legendre transformation which characterizes normal extremals of a sub-Finsler structure as geodesics of this connection. We also wish to investigate some of its properties like normal, adapted, partial and metrical.

In this talk, we will discuss about 'representations up to homotopy' which are considered to be the 'correct' notion of representations of higher Lie algebroids. We will construct an important class of representations called 'adjoint representation' and see an interesting connection between this class and higher Poisson and symplectic structures.

Methods such as deep learning and reinforced learning have been successful in modeling and controlling many dynamical systems. For such methods to achieve the desired performance which appears in the real world, but we don't know which dynamics follow, we often require multiple systems runs over a large sum of data that sometimes may not be available in several scenarios. In this poster, we present a method based on quadratic programming to approximate the Lagrangian associated with the distance-based formation problem from limited data from an observation of the trajectory. We further show how to obtain bounds for the approximation errors. This is joint work together with Leonardo J. Colombo

The aim of this work is to combine multiobjective optimization with the feedback optimal control problem via solving the Hmailton Jacobi Bellman (HJB) PDE. A method used to solve the feedback optimal control problem has been introduced by Park, Guibout and Scheeres, 2006. This method is called the generating function technique which allows to solve the associated Hamilton-Jacobi-Bellman equation directly. Our aim is to intertwine this technique with numerical methods for feedback optimal control problem. In order to overcome the nonlinearity of the most of the physical problems we transform the nonlinear feedback optimal control problem to iterative linear problems by a sequence approximating series method. The solutions derived by the proposed method have the property to be robust to perturbations and errors in the initial conditions: in this way we do not solve a single optimal control problem but rather we find the whole family of optimal control laws. We can apply this technique with both soft and hard constraints according to the physical model of the problem in hand.

We study the dynamics and geometry of contact Hamiltonian systems with nonholonomic constraints. The equations of motion can be obtained from Herglotz's variational principle when we constrain the variations to lie on a given distribution. We prove that this dynamics can also be understood as a projection of the non-constrained dynamics. Finally, we construct a bracket that provides the dynamics on the constrained system that generalizes the Jacobi bracket of the contact manifold on the unconstrained case. This new bracket does not fulfill the Jacobi identity.

The symmetry of a Lagrangian mechanical system, given by the action of a Lie group G, may sometimes come from different sources. In this context, it is natural to reduce by stages, that is, reducing first by a normal subgroup N of G and afterwards by the quotient. In order to perform successive steps of reduction it was introduced [2] the category of Lagrange- Poincaré bundles. The aim of this poster is to explore if there exists a category of bundles playing an analogous role in the Field Theory setting developed in [1,3].
[1] M. Castrillón López, P. L. García Pérez, T.S. Ratiu: Euler-Poincaré Reduction on Principal Bundles. Lett. Math. Phys 58, 167-180, 2001.
[2] H. Cendra, J.E. Marsden, T.S. Ratiu: Lagrangian reduction by stages. Mem. Amer. Math. Soc. 152, no. 722 (2001).
[3] F.Gay-Balmaz, D. Holm, and T. S. Ratiu: Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions J. Braz. Math. Soc. 42(4), (2011), 579–606.

In this poster, we study how the equations of motions for a quadrotor arise naturally from variational principles on Lie groups for systems with external forces. We extend the analysis to the motion of four quadrotors in formation, where a constraint is introduced to keep the formation between the quadrotors. We formulate the problem of inexact interpolation for the centroid dynamics of the team, that is, a trajectory planning problem, for the quadrotors moving in the space while they keep the formation and the planned (smooth) trajectory passes close enough to the centroid of the formation. This problem can bee seen as an optimization problem with constraints and moreover, as a constrained higher-order variational problem. We derive necessary conditions for optimal solutions. This is joint work together with Leonardo Colombo.

A hypersurface M immersed in the Euclidean space has prescribed linear mean curvature if its mean curvature function is given as a linear function defined in the n-sphere, depending on its Gauss map. These hypersurfaces are related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. They also can be characterized as critical points of a variational problem involving the weighted area and volume functionals. In this work, we give a classification of such hypersurfaces that are rotational. This is a joint work with Irene Ortiz.

In this work we study hypersurfaces in R^{n+1} with linear prescribed mean curvature which have constant curvature. By classical theorems of Liebmann, Hilbert and Hartman-Nirenberg, any such hypersurface must be flat, hence invariant by an (n − 1)-group of translations and described as the riemannian product α × R^{n+1}, where α is a plane curve called the base curve. We classify such hypersurfaces by giving explicit parametrizations of the base curve. This is a joint work with Antonio Bueno.

In the classical theory of thermodynamics, thermal signals propagate with infinite speed, local actions and cumulative behaviour are neglected; the history, even the very recent history is not taken into account. A way to introduce a memory is first to introduce a so called memory function. By this memory function β the history will be taken into account by taking a averaging the past with β and leads to a so-called Volterra equation. The theory of rough path introduced by Terry Lyons in his seminal work as an extension of the classical theory of controlled differential equations. In this work we show existence and uniqueness of mild solution for an infinitesimal semilinear Volterra equations driven by a rough path perturbation. The first step we give some maximal regularity results of the Ornstein Uhlenbeck process with memory term driven by a rough path using the Nagy Dilation Theorem.

My talk will focus on inverting the covering map from SL(2,R) to So+(2,1) and simultaneously provide the polar decomposition for matrices in So+(2,1). Extensions to higher dimensions will also be discussed.

In Riemannian geometry, the rolling system is a well-known framework for comparing certain geometric information of two Riemannian manifolds. Typically, one is well understood and something needs to be said about the other one. In a series of papers, Chitour, Kokkonen, Godoy (2015, 2016, 2017) and others, found a deep link between the holonomy of an appropriate vector bundle connection and the controllability of the rolling system. In this poster, I will present some results that extend this, but for rolling pseudo-Riemannian manifolds, as introduced by Markina and Silva-Leite (2016). These results are part of my Ph.D. thesis in Mathematics at Universidad de La Frontera (Temuco, Chile).

A classic point of view to study sub-Riemannian manifolds is to find a Riemannian metric taming the sub-Riemannian one. Controlling this taming is necessary if one hopes to be able to prove something. One possibility is to consider sub-Riemannian metrics obtained via Riemannian submersions. The aim of this talk is to present necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvature. Additionally, it is possible to describe a canonical extension of the sub-Riemannian metric and study geometric properties of the obtained Riemannian manifold. This is joint work with Erlend Grong (Orsay) and Irina Markina (Bergen).

Abstract. In this poster, we describe a variational derivation of the forced Euler-Poincaré equations [1] using a duplication of variables technique originated in [2, 3]. We show that the underlying geometry is related with the notion of a Poisson groupoid. This is applied to the variational construction of geometric integrators for forced systems and allows us to apply variational error analysis, extending the results of [4] to the reduced case. This is a joint work with David Martín de Diego.
[1] D. Martín de Diego and R. T. Sato Martín de Almagro. Variational order for forced Lagrangian systems II: Euler-poincaré equations with forcing. (preprint, arXiv:1906.09819), 2019.
[2] C. R. Galley. Classical mechanics of nonconservative systems. Phys. Rev. Lett., 110:174301, 4 2013.
[3] C. R. Galley, D. Tsang, and L. C. Stein. The principle of stationary nonconservative action for classical mechanics and field theories. (preprint, arXiv:14 12.3082), 12 2014.
[4] D. Martín de Diego and R. T. Sato Martín de Almagro. Variational order for forced Lagrangian systems. Non-linearity, 31(8):3814-3846, 2018.