**Plenary Speakers**

**Angel Ballesteros (University of Burgos, Spain)**

**Integrable anisotropic oscillator and Hénon-Heiles systems on curved spaces**

A method aimed to generalize Euclidean Hamiltonian systems to the two-dimensional sphere and the hyperbolic plane by preserving their (super)integrability is presented. The constant Gaussian curvature of the underlying spaces is introduced as an explicit deformation parameter, thus allowing the construction of new integrable Hamiltonians in a unified geometric setting, in which the Euclidean systems are obtained in the vanishing curvature limit. In particular, the constant curvature analogue of the anisotropic oscillator Hamiltonian is presented, and its superintegrability for commensurate frequencies is shown. As a second example, an integrable version of the Hénon-Heiles system on the sphere and the hyperbolic plane is introduced. Further applications of this approach are sketched.

A. Ballesteros, F.J. Herranz, F. Musso

The anisotropic oscillator on the two-dimensional sphere and the hyperbolic plane

Nonlinearity, vol. 26, (2013) 971

A. Ballesteros, A. Blasco, F.J. Herranz, F. Musso

An integrable Hénon-Heiles system on the sphere and the hyperbolic plane

Nonlinearity, vol 28, (2015) 3789

**Santiago Capriotti (Universidad Nacional del Sur, Argentina)**

**Tulczyjew’s triples and a formulation of the inverse problem in classical field theory**

Tulczyjew’s triples are geometrical structures useful in describing either mechanical systems and classical field theory. In this approach, equations of motion are characterized by Lagrangian submanifolds in a premultisymplectic manifold; these submanifolds, in turn, are defined through the dynamical data of the theory (namely, the Lagrangian or Hamiltonian densities). In the present talk we will show how to construct the Lagrangian submanifolds directly from the equations of motion of field theory. This remarkable link between equations and submanifolds will allow us to discuss a formulation of the inverse problem both for mechanics and classical first order field theory, in the context of Tulczyjew’s triple.

**Renato Calleja (UNAM, Mexico)**

**Symmetries and choreographies in families that bifurcate **

**from the polygonal relative equilibrium of the n-body problem**

In my talk I will describe numerical continuation and bifurcation techniques in a boundary value setting used to follow Lyapunov families of periodic orbits. These arise from the polygonal system of n bodies in a rotating frame of reference. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship, the orbit is also periodic in the inertial frame. We prove that a dense set of Lyapunov orbits, with frequencies satisfying a Diophantine equation, correspond to choreographies. We present a sample of the many choreographies that we have determined numerically along the Lyapunov families and bifurcating families, namely for the cases n=4,6,7,8 and, 9. We also present numerical results for the case where there is a central body that affects the choreography, but that does not participate in it. This is joint work with Eusebius Doedel and Carlos García Azpeitia.

**Elena Celledoni (NTNU, Norway)**

**Shape analysis on Lie groups and homogeneous manifolds.**

We propose a way to generalize the definition of Square Root Velocity Transform (SRVT), (by Srivastava et al. 2011), from vector spaces and Lie groups to homogeneous manifolds.

This approach is alternative to what earlier proposed in (Su et al 2104) and the main idea is, following (Celledoni et al. 2015}, to take advantage of the Lie group acting transitively on the homogeneous manifold (Elena Celledoni, Markus Eslitzbichler and Alexander Schmeding).

Shape Analysis of Elastic Curves in Euclidean Spaces.

IEEE Transactions on Pattern Analysis and Machine Intelligence, 33(7):1415 –1428, July 2011.

**Gonzalo Contreras (CIMAT, Mexico)**

**The C2 Mañé’s Conjecture on Surfaces**

Given a closed surface M and a convex and superlinear lagrangian L on TM, we prove that there is an open and dense set of function f in C2(M,R) such that the lagrangian L+f has a unique minimizing measure supported on a hyperbolic periodic orbit.

**Francesco Fassó (Università di Padova, Italy)**

**Integrable and nearly-integrable Hamiltonian systems in almost-symplectic manifolds**

How much do systems which are Hamiltonian with respect to an almost-symplectic structure (a nondegenerate but non-closed two form) differ from standard Hamiltonian systems? This question is motivated, e.g., by nonholonomic mechanics. If the two-form is non-closed, then Hamiltonian vector fields are not automatically symmetries of the almost-symplectic structure, and do not form a Lie algebra. This has deep consequences on the relation symmetry-conservation laws, on integrability etc. And of course, on the dynamics; for instance, these systems need not preserve a volume form.

The very special subclass of Hamiltonian vector fields which do preserve the almost-symplectic two-form are called “strongly Hamiltonian” and form a Lie algebra. These vector fields have been studied in [1,2]. A theory of integrability for them has been developed under the very strong hypotheses that all dynamical symmetries are strong Hamiltonian; the resulting picture is very close to the standard one.

The study of small perturbations of strongly Hamiltonian integrable almost-symplectic systems has been initiated in [3]. There are two very different cases. If the perturbation is strongly Hamiltonian, then the system reduces, under an almost-symplectic version of the Meyer-Marsden-Weinstein symplectic reduction, to a standard Hamiltonian system on a reduced symplectic manifold. In particular, the renown KAM and Nekhoroshev theorems of Hamiltonian perturbation theory on the (approximate or exact) persistence of invariant tori still apply, with only some subtle difference. The situation is very different, and not yet completely understood, if the perturbation is only Hamiltonian.

[1] F. Fassò and N. Sansonetto, Integrable almost-symplectic Hamiltonian systems. J.Math. Phys. 48 (2007), 092902, 13 pp.

[2] I. Vaisman, Hamiltonian vector fields on almost symplectic manifolds. J. Math. Phys. 54 (2013), 092902, 11 pp.

[3] F. Fassò and N. Sansonetto, Nearly-integrable almost-symplectic Hamiltonian systems. Preprint (2016)

**David Martín de Diego (ICMAT, Madrid)**

**Why are groupoids important in (my) real life?**

“. . . groupoids should perhaps be renamed ‘groups’, and those special groupoids with just one base pointgiven a new name to reflect their singular nature.” (F. W Lawvere,)

we will propose a rigorous construction of the exact discrete Lagrangian formulation associated to a continuous Lagrangian problem. This construction of an exact discrete Lagrangian is of considerable interest for the analysis of the error between an exact trajectory and the discrete trajectory derived by a variational integrator.

**James Montaldi (University of Manchester, UK)**

*A 3-form in some non-holonomic systems with symmetry and the resulting Casimirs*

Non-holonomic mechanical systems are not Hamiltonian, but there is nonetheless an almost Poisson structure (due to van der Schaft and Maschke) which gives an almost Hamiltonian structure to the equations of motion. Here ‘almost’ means the brackets do not satisfy the Jacobi identity. If the system has symmetries that lead to so-called gauge momenta, then I describe how to define a 3-form on the phase space which, through a standard procedure, allows a modification of the almost Poisson structure for which the equations are still almost Hamiltonian. For this new structure, the gauge momenta become Casimirs. This gives a general procedure for producing the modified almost Poisson structure, which previously had been found on an ad hoc basis for individual examples. This is joint work with Luis Garcia-Naranjo.

**Miguel Rodríguez Olmos (Universidad Politécnica de Cataluña)**

**Classification and stability of relative equilibria for the two-body problem in the hyperbolic space of dimension 2**

Using methods from geometric mechanics, we classify and analyze the stability of all relative equilibria for the two-body problem in the hyperbolic space of dimension 2.

Bibliography: Journal of Differential Equations, Vol. 260, No. 7, pp 6375-6404, 2016

**Nicola Sansonetto (Università di Padova, Italy)**

*Nonholonomic Systems with Affine Constraints, (Moving) Energies and Integrability*

Nonholonomic systems with constraints that are affine functions of the velocities do not typically conserve energy. Nevertheless there might exists modifications of the energy, called moving energies, that under certain conditions are first integrals of the systems.

In this talk we will discuss the conservation of the energy for nonholonomic systems with affine constraints and investigate the existence of moving energies if the energy is not preserved, furthermore we will relate the existence of moving energies to the presence of symmetries. Then we will apply the obtained results to a class of nonholonomic systems with affine constraints. Eventually, after the extension to nonholonomic systems with affine constraints of the gauge method to look for first integrals, we will investigate the integrability of the system of a heavy homogeneous sphere that rolls without sliding inside a convex surface of revolution in uniform rotation around its vertical figure axis.

**Short Lectures**

**Marta Batoreo (UFES – Universidade Federal do Espírito Santo, Brazil)**

*On periodic orbits of symplectomorphisms *

We will talk about some results on the existence of periodic points of symplectomorphisms on compact manifolds. One of the main results we plan to present focuses on the existence of such points on compact surfaces.

**Leonardo Colombo (University of Michigan, USA)**

*Optimal Control Problems with Symmetry Breaking Cost Functions *

Symmetry reduction of optimal control problems (OCPs) for left-invariant control systems on Lie groups has been studied extensively over the past couple of decades. Such symmetries are usually described as an invariance under an action of a Lie group and the system can be reduced to a lower-dimensional one or decoupled into subsystems by exploiting the symmetry. Symmetry reduction of OCPs is desirable from a computational point of view as well. Given that solving OCPs usually involves iterative/numerical methods such as the shooting method (as opposed to solving a single initial value problem), reducing the system to a lower-dimensional one results in a considerable reduction of the computational cost as well.

Symmetry breaking is common in several physical contexts, from classical mechanics to particle physics. The simplest example is the heavy top dynamics, where due to the presence of gravity, we get a Lagrangian that is SO(2)-invariant but not SO(3)-invariant, contrary to what happens for the free rigid body.

The goal of this talk is to explain new results in symmetry reduction of OCPs for left-invariant control systems on Lie groups, with partially broken symmetries, more specifically, cost functions that break some of the symmetries but not all. In the context of motion planning, the symmetry breaking appears naturally in the form of a barrier function.

**Viviana Díaz (Universidad Nacional del Sur, Bahía Blanca, Argentina)**

*Coordinate equations from orbit reduction*

It is known that a process of orbit reduction in a Hamiltonian system with symmetry can be performed. This reduction can be developed in one stage considering the whole symmetry group, or in two stages in the case in which the group has a normal subgroup. Following this theory, we shall write the reduced symplectic two-form in coordinates and the consequent equations of motion in the case of one and two stages.

**Luís Miguel Faustino Machado, (UTAD; ISR, Coimbra, Portugal)**

**Lie Theory Approach in the Geometry of Rolling Motions**

Interpolation is one of the most elementary tasks that is often employed in pattern recognition and image processing. When the data is represented on some manifolds, one approach that is quite effective is based on the notion of rolling motions, subject to nonholonomic constraints of no-slip and no-twist, of a manifold over another one. The main idea behind these algorithms is to use the rolling motions to project the data from the manifold into a simpler space, where classical methods can be applied, and then roll back the solution in order to solve the initial interpolation problem on the manifold. In this talk, our attention goes to the rolling motion of Riemannian symmetric spaces over the affine tangent space at a point. It is shown how the natural decomposition of the Lie algebra associated to the symmetric space provides the structure of the kinematic equations that describe the rolling motion. This clarifies why many particular examples scattered through the existing literature always show a common pattern.

[1] Krakowski, K. A., Machado, L., and Silva Leite, F. Rolling Symmetric Spaces. Springer International Publishing Cham, Geometric Science of Information, Second International Conference, GSI 2015, F. Nielsen and F. Barbaresco (Eds.), October 28-30 2015, pp. 550–557.

[2] Krakowski, K. A., Machado, L., and Silva Leite, F., Lie Theory Approach in the Study of Rolling Motions (submitted).

**Eduardo García-Toraño (Universidad Nacional del Sur, Bahia Blanca, Argentina)**

*Unreduction for SODEs*

Reduction has been one of the most influential techniques in geometric mechanics. In the case of a Lagrangian system, reduction makes use of the symmetries of the Lagrangian to reduce it to a Lagrangian-type dynamical system on a quotient manifold. Recently, there has been some interest in the following inverse problem: given a Lagrangian system on a manifold and a principal bundle over that manifold, find a Lagrangian system on the principal bundle whose solutions project. We will take a look at this problem in the context of second-order differential equations (SODEs), show how the Lagrangian case fits in the picture and give examples out of the realm of Lagrangian systems.

**Ramiro Lafuente (Münster University, German and National University of Córdoba, Argentina)**

*Moment maps and homogeneous Riemannian geometry*

The aim of this talk will be to explain briefly how the notion of a moment map from symplectic geometry can be extended to the setting of non-compact real Lie groups acting linearly on vector spaces. We will then focus on one particular example for such actions, and show a remarkable link between said moment map and the curvature of a homogeneous Riemannian manifold.

**Juan Margalef (UC3M-CSIC, Madrid)**

**When continuous meets discrete: quantization of field systems with point-masses**

In this talk I will discuss a simple system consisting of a 1D field coupled to point-masses. Despite its simplicity, it is necessary to introduce appropriate functional spaces defined with non standard measures and use appropriate geometric methods to deal with it. Some interesting questions arise at both classical and quantum levels. We will show, in particular, that the quantum Fock space cannot be written in a natural way as a tensor product of Hilbert spaces associated with the field and the point masses respectively. This is somehow unexpected and poses some questions regarding the physical interpretation of the system.

**Alessia Mandini (PUC-Rio, Brazil)**

*Bending systems for polygons spaces*

**Leandro Salomone (Universidad Nacional de la Plata, Argentina)**

**The Lyapunov constraint-based method for the asymptotic stabilization of mechanical systems**

The Lyapunov constraint-based method (or LCB method for short) is a recent global non-linear method for the asymptotic stabilization of unstable equilibria of underactuated mechanical systems. It is based on the imposition of a kinematic constraint on the unactuated system, where the control law to stabilize the system is obtained as the related constraint force. The method has the additional advantage of producing a Lyapunov function for the closed-loop system, proving the (asymptotic) stability. When compared with other stabilization procedures that also exhibit Lyapunov functions, the LCB method turns out to be a maximal method, in the sense that all the control laws produced by other methods of this kind can be also constructed using the LCB method. The family of stabilization methods enclosed under the name of energy shaping is one of these and, in particular, the version developed by Chang is equivalent to the LCB method. In this talk we will present the LCB method for simple mechanical systems. We will see that the applicability of the method reduces to solving a set of PDEs (the kinetic and potential equations) and we will describe the mentioned relation with the energy shaping method. To illustrate the convenience of the LCB as an alternative to other methods, we show that it can be used to prove necessary and sufficient conditions for asymptotic stabilization of systems with two degrees of freedom.

**Cristina Sardón (ICMAT, Madrid, Spain)**

**Cosymplectic and contact structures to resolve time-dependent and dissipative hamiltonian systems**

In this talk, we apply the geometric Hamilton–Jacobi theory to obtain solutions of classical hamiltonian systems that are either compatible with a cosymplectic or a contact structure. As it is well known, the first structure plays a central role in the theory of time-dependent hamiltonians, whilst the second is here used to treat classical hamiltonians including dissipation terms. The interest of a geometric Hamilton–Jacobi equation is the primordial observation that if a hamiltonian vector field XH can be projected into a configuration manifold by means of a 1-form dW, then the integral curves of the projected vector field XdW H can be transformed into integral curves of XH provided that W is a solution of the Hamilton– Jacobi equation. In this way, we use the geometric Hamilton–Jacobi theory to derive solutions of physical systems with a time-dependent hamiltonian formulation or including dissipative terms. Explicit, new expressions for a geometric Hamilton–Jacobi equation are obtained on a cosymplectic and a contact manifold. These equations are later used to solve physical examples containing explicit time dependence, as it is the case of a unidimensional trigonometric system, and two dimensional nonlinear oscillators as Winternitz–Smorodinsky oscillators. For explicit dissipative behavior, we solve the example of a unidimensional damped oscillator.

[1] D.E. Blair, Contact manifolds in Riemannian Geometry, Lecture Notes in Mathematics, Springer, 1976.

[2] W.H. Boothby and H.C. Wang, On contact manifolds, Annals of Mathematics, 68, 721–734 (1958).

[3] A. Bravetti, H. Cruz and D. Tapias, Contact hamiltonian Mechanics, https://arxiv.org/abs/1604.08266. (2016).

[4] J.B. Etnyre, Introductory lectures on contact manifolds, Unspecified book Proceeding series, people.math.gatech.edu/ etnyre/preprints/papers/contlect.pdf.

**Miguel Vaquero (ICMAT, Madrid, Spain)**

**Hamilton-Jacobi, symmetries and Poisson manifolds**

In this talk we are going to review some results about reduction of the Hamilton-Jacobi theory for systems with symmetry and Hamiltonian systems on Poisson manifolds. Some applications to sympectic and Poisson integrators will we shown along the talk. In the end, some present lines of research will be shown.

**Silvia Vilariño (Centro Universitario de la Defensa, Zaragoza)**

*Multisymplectic Lie systems*

A Lie system is a nonautonomous system of first-order differential equations describing the integral curves of non-autonomous vector field taking values in a finite dimensional Lie algebra of vector fields, a Vessiot-Gulberg Lie algebra.

The study of Lie systems with associated geometric structures is very important. For instance Lie-Hamilton systems, Dirac-Lie systems, k-symplectic- Lie systems, etc. Using these geometric structures one can obtain time-independent constants of motions and superpositions rules.

In this talk we present a particular type of Lie systems, those admitting a Vessiot-Gulberg Lie algebra of Hamiltonian vector field relative to a multisymplectic structure. We define these systems and their principal properties. Moreover we present some examples of these systems.

### Posters

Eber | Chuño | UFPE | Brazil | On the bi-Hamiltonian structure of Coxeter-Toda lattices | |

William | Clark | University of Michigan | United States of America | Time minimal control for a quantum Hamiltonian system under Lindblad dissipation | |

Daniel | De la Fuente | Universidad de Granada | España | Relativistic Kinematics of certain accelerated observers in spacetimes with symmetries | |

Marine | Fontaine | The University of Manchester | United Kingdom | Explicit symmetry breaking for equivariant Hamiltonian systems | |

Xavier | Gràcia | Universitat Politècnica de Catalunya | Spain | k-precosymplectic systems and Darboux theorem | |

Iván | Gutiérrez | Sagredo | Spain | On Hamiltonians with position-dependent mass from Kaluza-Klein compactifications | |

Antonio | Hernández-Garduño | Universidad Autónoma Metropolitana – Iztapalapa | México | Reconstruction phases in the three- and four-vortex problem | |

Maximiliano | Palacios Amaya | CONICET – Centro atómico bariloche | Argentina | A variational formalism for finite dimensional thermo-mechanical systems | |

Juan Jesús | Salamanca | Universidad de Granada | España | Rational formulation of the pendulum system | |

José Antonio | Sánchez Pelegrín | Universidad de Granada | Spain | Maximal hypersurfaces in pp-wave spacetimes |