# SCIENTIFIC PROGRAMS AND ACTIVITIES

June 27, 2017

THE FIELDS INSTITUTE
FOR RESEARCH IN MATHEMATICAL SCIENCES

20th ANNIVERSARY YEAR

 July 3-29, 2012 FOCUS PROGRAM ON GEOMETRY, MECHANICS AND DYNAMICS the Legacy of Jerry Marsden at The Fields Institute, Toronto Speaker Abstracts Call for abstracts Back to main program index

Felipe Arrate, Basque Center for Applied Mathematics
Cardiac Electrophysiology Model on the moving heart

As part of a continuing effort by the scientific community to develop reliable models of the electrical waves that contracts the heart muscle, a combined approach is presented that includes an approximated deformation model for the moving heart using medical images of the heart muscle, a particle method for parabolic PDE's, and variational integrators for calculation of the deformation of the images. The electrical response of the myocardium will be approximated using a Monodomain model, and the motion of the heart will be initially interpolated following a diffeomorphic spline approach, solved using a gradient descent on the initial momentum. The meshless particle method developed involves the motion of the nodes (particles) by the time dependent vector field defined by the image registration, adding an extra difficulty to known electrophysiology meshless models.

Twisted brackets in nonholonomic mechanics

As it is known, nonholonomic systems are characterized by the failure of the Jacobi identity of the bracket describing the dynamics. In this talk I will present different (geometric) technics to deal with the failure of the Jacobi identity and we will see how twisted Poisson structures might appear once we reduce the system by a group of symmetries.

Anthony Bloch, Univ. of Michigan
Continuous and Discrete Embedded Optimal Control Problems

In this talk I shall a discuss a general class of optimal control problems which we call embedded optimal control problems and which allow for a parametrized family of well defined associated optimal control problems. Embedded and associated optimal control problems are related by a projection and the embedded problem is often easier to solve. This class of problems include many control problems of interest including the Clebsch problem and various geodesic flows modeled by Lie-Poisson or symmetric type equations. The extension to the discrete case gives useful variational integrators. This includes joint work with Peter Crouch and Nikolaj Nordkvist.

The geometry of integrable and gradient flows and dissipation

In this talk I will discuss the dynamics and geometry of various systems that exhibit asymptotic stability and dissipative behavior. This includes integrable systems, gradient flows, and dissipative perturbations of integrable systems. Examples include the finite Toda lattice, the dispersionless Toda equations, gradient flows on loop groups and certain nonholonomic systems. I will describe the geometric structures, including metric and complex structures, that give rise to some of these flows and determine their behavior. The talk includes recent work with P. Morrison and T. Ratiu.

Ale Cabrera, Universidade Federal do Rio de Janeiro
Geometric phases in partially controlled mechanical systems

We study mechanical systems where part of the degrees of freedom are being controlled in a known way and determine the motion of the rest of the variables due to the presence of constraints/conservation laws. More concretely, we consider the configuration space to be a G-bundle Q \to Q/G in which the base Q/G variables are being controlled. The overall system's motion is considered to be induced from the base one due to the presence of general non-holonomic constraints or conservation laws. We show that the overall solution can be factorized into dynamical and geometrical contributions (geometric phase), yielding a so called reconstruction phase formula. Finally, we apply this results to the study of concrete mechanical systems like a self-deforming satellite in space.

Higher order covariant Euler-Poincaré

Given a Lagrangian $L$ defined in the r-jet $J^r P$ of a principal G-bundle $P\to M$, the reduction of the variational principle when $L$ is $G$ invariant is studied. In particular, this generalizes the Euler-Poincaré reduction shceme given in the literature when $r=1$ or $M=\mathbb{R}$. A particular interest is put on the constraints of the reduced problem.

Dong Eui Chang, University of Waterloo
Damping-Induced Self Recovery Phenomenon in Mechanical Systems with an Unactuated Cyclic Variable

The conservation of momentum is often used in controlling underactuated mechanical systems with symmetry. If a symmetry-breaking force is applied to the system, then the momentum is notconserved any longer in general. However, there exist forces linear in velocity such as the damping force that breakthe symmetry but induce a new conserved quantity in place of the original momentum map. We formalize the new conserved quantity which can be constructed by combining the time integral of a general damping force and the original momentum map associated with the symmetry. From the perspective of stability theories, the new conserved quantity implies the corresponding variable possesses the self recovery phenomenon, i.e. it will be globally attractive to the initial condition of the variable. We discover that what is fundamental in the damping-induced self recovery is not the positivity of the damping coefficient but certain properties of the time integral of the damping force. The self recovery effect and theoretical endings are demonstrated by simulation results using the two-link planar manipulator and the torque-controlled inverted pendulum on a passive cart. (This is an outcome of the collaboration with Soo Jeon at the University of Waterloo)

Graciela Chichilnisky, Columbia University
Statistic Dynamics with Catastrophic Events

New axioms for statistic dynamics extend classical dynamics by requiring sensitivity to rare & catastrophic events. The minicourse will focus on the Geometry and Topology of this extension emphasizing (i) how classic dynamics is insensitive to catastrophic events (ii) how the new axioms extend classic theory and require sensitivity to rare events (iii) how the new axioms relate to classic results of Von Neumann and Morgenstern, Arrow, Milnor and Godel (iv) characterize the new distributions that satisfy the new axioms, which contain both countably and purely finitely additive terms (v) Characterize statistical processes based on those distributions - fump diffusion processes (vi) Implications for Bayesian analysis updating samples with new information (vii) new foundations of probability and statistics and the dynamic process they imply (viii) presentation of existing experimental and empirical results.

Leonardo Colombo, Instituto de Ciencias Matemáticas
Higher-order Lagrange-Poincaré reduction for optimal control of underactuated mechanical systems

In this talk we will describe a geometric setting for the reduction of higher-order lagrangian systems with symmetries.  We will deduce a suitable framework to study higher-order systems with higher order constraints (see [2] for the original case without constaints) based in the classical lagrangian reduction theory devoloped by Cendra, Marsden and Ratiu in [1]. Interesting applications as, for instance, a derivation of the higher-order Lagrange-Poincaré equations for systems with higher-order constraints, optimal control of underactuated control systems with symmetries, etc, will be considered.

Geodesic chaos around black-holes with magnetic fields

Some exact solutions to the Einstein equations representing a stationaryblack hole surrounded by a magnetic field is considered. Time-likegeodesic are numerically integrated and dynamically analyzed by means ofPoincaré sections. We find chaotic motion induced my the magnetic field.The onset of chaos is studied and the influence of the magnetic field onthe system integrability is estimated.

Holger Dullin, University of Sydney
The Lie-Poisson structure (and integrator) of the reduced N-body problem

We reduce the classical $n$-body problem in $d$-dimensional space by its full Galilean symmetry group using the method of invariants. As a result we obtain a reduced system with a Lie-Poisson structure which is isomorphic of $\sp(2n-2)$, independently of $d$. The reduction preserves the natural form of the Hamiltonian as a sum of kinetic energy that depends on velocities only and a potential that depends on positions only. Hence we proceed to construct a Poisson integrator for the reduced $n$-body problem using a splitting method. The method is illustrated by computing special periodic solutions (choreographies) of the 3-body problem for $d=2$ and $d=3$.

Geometric Phase in Aerial Motion

Gymnasts and divers in aerial motion use their shape to control their orientation. Utilising shape change it is possible to turn even with vanishing angular momentum, as the falling cat testifies. We will show that in certain cases the optimal shape change which maximises the overall rotation can be found using a variational principle. These ideas will be illustrated in a number of settings including the shape-changing equilateral pentagon, planar motion in trampolining, and three dimensional motion of divers performing a twisting somersault.

Andrea Dziubek and Edmond Rusjan, SUNY IT
A Model for the Retina Including Blood Flow and Deformation

We model retinal blood flow by Darcy flow equations using discrete exterior calculus. The model is important in Ophthalmology. Without a mathematical model for the oxygen transport it is not possible to use oximetry images in clinical diagnoses, about many conditions and diseases in the body, not just diseases of the eye itself. Discrete exterior calculus aims at preserving the structures present in the underlying continuous model by reformulating the problem in the language of exterior calculus and then discretizing the operators present in the equations. We outline extensions of this model, which include coupling the blood flow model to elastic deformations of the retina, based on a Kirchhoff-Love shell model.

David Ebin, SUNY Stoneybrook
Reflections on the paper, "Groups of diffeomorphisms and the motion of an incompressible fluid"

The above paper by Ebin and Marsden was an important milestone for both authors. In fact according to Google Scholar it is the most cited of Ebin's papers and the 2nd most cited of Marsden's -- after his paper "Reduction of symplectic manifolds with symmetry" which he wrote together with Alan Weinstein. The paper was partly a sequel to a small work of Abraham and Marsden which was a restating of Arnol'd's paper on perfect fluids using tangent spaces rather than their duals. It's creation was inspired by a course of Smale on various topics in mechanics.

The paper shows how one can construct solutions of the Euler equations by using a Picard iteration -- or basic ODE. At the time PDE people were either incredulous or thought that somehow the magic of differential geometry was brought to bear. We shall explain how we came upon the technique and mention how it has subsequently been used in a number of other equations.

Lyudmyla Grygor'yeva, Juan-Pablo Ortega and Stanislav Zub
Problems of non-contact confinement of rigid bodies

I Spatial magnetic potential well (MPW) and magnetic levitation in the system of magnetic dipole - superconductive sphere. Part I.
(i) History of the problem.
(ii) Earnshaw's theorem and unreasonable conclusions about the principal instability of magnetic systems.
(iii) "Combined" cases (levitation) and "pure" cases of the static equilibrium.
(iv) Magnetic potential well (MPW).
(v) Spatial MPW.
(vi) Constructive proof of the MPW existence.
II Spatial magnetic potential well (MPW) and magnetic levitation in the system of magnetic dipole - superconductive sphere. Part II.
(i) Mathematical explanation of the Kapitsa-Arkadyev experiment.
III Lagrangian formalism for description of non-contact magnetic interaction of rigid bodies in the systems with permanent magnets and superconductive elements (quasi-stationary approximation).
(i) The principle of Hertz.
(ii) Magnetic potential energy of a system of the above type.
IV Examples of constructive proof of the MPW existence.
V Magnetic levitation based on the MPW as the perspective variant of non-contact suspension for the Levitated Dipole Experiment (LDX).
VI Orbitron. Stable orbital motion of a magnetic dipole in the field of permanent magnets.
VII Stable orbital motion in the problem of two magnetic "dumbbells" (in addition - with the Monte-Carlo simulations).
VIII Levitated dipole experiment (LDX). Magnetic levitation of the superconductive elements allows to solve the problems of extremely low energy conversion efficiency of the existing LDX variant and of thermal pollution of the environment.
IX New statement for the problem of the Levitron. New questions about existence of stable orbital motion.

Elisa Guzmán, Universidad de La Laguna
Reduction of Lagrangian submanifolds and Tulczyjew's triple

In 1976, W. Tulczyjew introduced different canonical isomorphisms between the spaces T^*TQ, TT^*Q and T^*T^*Q of a smooth manifold Q. These mappings are of furthermost importance since they allow to formulate the dynamics of a mechanical system as Lagrangian submanifold of the symplectic manifold TT^*Q. This includes in particular the case
if the Lagrangian fuction L is singular. In this talk we consider the
case that a Lie group is acting freely and properly on Q. I will
present someideas about how the reduced dynamics can be formulated again as Lagrangian submanifold.

Antonio Hernández-Garduño, UAM-I, Mexico
Algebra and reduction of the three vortex problem

We will describe a Lie-algebra interpretation of the three vortex problem. This interpretation looks at the dynamics in an enlargement of the space of square-distances, which admits a Hamiltonian structure. This allows to identify the problem as a coadjoint-orbit reduction in a Poisson manifold. In this manner, an analogy with the rigid-body-reduction is established. Interpretations and further developments of this point of view will be discussed.

Darryl D Holm, Imperial College London
Fermat's Principle and the Geometric Mechanics of Ray Optics

According to Fermat's principle (1662): The path between two points taken by a ray of light leaves the optical length stationary under variations in a family of nearby paths. These summer school lectures illustrate how the modern ideas of reduction by symmetry, Lie-Poisson brackets and dual pairs of momentum maps help characterize the properties of geometric ray optics.

Momentum Maps, Image Analysis & Solitons

This survey talk discusses some opportunities for applied mathematics and, in particular, for geometric mechanics in the problem of registration of images, e.g., comparison of planar closed curves. It turns out that many aspects of geometric mechanics apply in this problem, including soliton theory and momentum maps. Much of this talk is based on work done with Jerry Marsden (1942 - 2010). Some trade secrets will be revealed.

Henry Jacobs, California Institute of Technology
Is swimming a limit cycle

It has been surmised repeatedly that animal locomotion incorporates a few passive mechanisms. In the case of swimming, this suggests that swimming could be interpreted as a stable limit cycle in some space. The question we ask is, "what space?" Upon further inspection, the idea that swimming is a limit cycle is preposterous. After each period, the animal would occupy a new position in space and therefore would not close an orbit in the phase space. However, one can perform a reduction by the symmetry of R^3. Upon performing this reduction, the idea that swimming is a limit cycle appears reasonable once again.

The creation and analysis of particle methods for ideal fluid

Since the 60s we have known that one can describe an ideal fluid as a geodesic equation on a diffeomorphism group with respect to a right invariant Riemannian metric. Using this insight we may perform Lagrange-Poincare reduction with respect to the isotropy group of a finite set of points. The resulting equations of motion are known as the Lagrange-Poincare equations and are decomposed into two parts, a horizontal part and a vertical part. The horizontal equation evolves on the configuration manifold for the N-body problem. By ignoring the vertical equation one arrives at a particle method. In this lecture we will explore the possibility of constructing error bounds for these particle methods.

Jair Koiller, Getulio Vargas Foundation
A gentle introduction to Microswimming: geometry, physics, analysis

Low Reynolds number swimming theory started in 1951 with a paper by G.I. Taylor in which a cartoon spermatozoon was modeled as a swimming sheet. Sixty years after the subject is thriving in analyhtic and computational sophisitcation, offering interesting avenues for mathematicians to interact with biologists and engineers. The lectures will attempt to provide a gentle introduction to the area. I start recollecting Jerry's stimulus for our first steps on the theme (together with Richard Montgomery and Kurt Ehlers). In the third presentation some of the work by leading research groups with be propandized and some open problems presented. If there is further interest, an extra lecture could be added on another tack could - acoustic streaming, a physical effect involving the compressibility of the fluid.

Wang Sang Koon, California Institute of Technology
Control of a Model of DNA Division Via Parametric Resonance

We study the internal resonance, the energy transfer, the actuation mechanism, and the control of a model of DNA division via parametric resonance. Our results not only may advance the understanding on the control of real DNA division by electric-magnetic fields, they may also reveal the role that enzymes play in the DNA open states dynamics. The model is a chain of pendula in a Morse potential, with torsional springs between pendula, that mimic real DNA. It exhibits an intriguing phenomenon of structural actuations observed in many
bio-molecules: while the system is robust to noise, it is sensitive to certain specific fine scale modes that can trigger the division.

By using Fourier modal coordinates in our study, the DNA model can be seen as a small perturbation of n harmonic oscillators. The reactive mode, i.e., the 0th mode, forms a nearly 0:1 resonance with any other mode, each of which has an O(1) frequency. This fact leads to small denominators or coupling terms in the corresponding averaged equations or normal forms. By developing the method of partial averaging, we are able to obtain the average equations for a reduced model of this chain of Morse oscillators up to nonlinear terms of very high degree. These equations not only reveal clearly the coupling between the energy of the excited mode and the dynamics of the reactive mode, they also shed lights on the phase space structure of the actuation mechanism.
Moreover, they enable us to estimate analytically the minimum actuation energy, the time to DNA division, and the reaction rate for each excited mode. The results not only match well with those obtained from numerical simulations of the full DNA model, but also uncovers an interesting relationship between frequencies of the excited modes and their corresponding minimum actuation energies for DNA division.

Furthermore, by building on our understanding of the internal resonant dynamics of our model and the techniques of parametric resonance, we are able to control and induce the division of this NDA model, via parametric excitation, that is in resonance with its internal trigger modes. Hopefully, our results may provide insights and tools to understand and control the dynamics and the rates of real DNA division by low intensity electric-magnetic fields. They may also reveal the action of enzymes that may use the external-internal resonance to pump energy into the trigger modes and cause the DNA division via the internal nearly 0:1 resonance.

Jeffery Lawson, Western Carolina University
A heuristic approach to geometric phase

Geometric phase measures the holonomy of a mechanical connection on an $SO(2)$ fiber bundle. Elroy's Beanie (see [Marsden, Montgomery, and Ratiu, 1990]) is a simple mechanical system in which the computation of geometric phase is distilled down to obtaining a one-form from a single conservation law. This emphasizes that geometric phase is primarily kinematic with only a minimum of dynamic information required. Through elementary student-friendly examples we illustrate that in many systems geometric phase can be computed through kinematics alone, using a single constraint,. We conclude by showing how geometric phase can be used to introduce concepts in differential equations, geometry, and topology to an undergraduate audience.

Melvin Leok, University of California, San Diego
General Techniques for Constructing Variational Integrators

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi's solution of the Hamilton--Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.

Discrete Hamiltonian Variational Integrators and Discrete Hamilton--Jacobi Theory

We derive a variational characterization of the exact discrete Hamiltonian, which is a Type II generating function for the exact flow of a Hamiltonian system, by considering a Legendre transformation of Jacobi's solution of the Hamilton--Jacobi equation. This provides an exact correspondence between continuous and discrete Hamiltonian mechanics, which arise from the continuous and discrete-time Hamilton's variational principle on phase space, respectively. The variational characterization of the exact discrete Hamiltonian naturally leads to a class of generalized Galerkin Hamiltonian variational integrators, which include the symplectic partitioned Runge--Kutta methods. This extends the framework of variational integrators to Hamiltonian systems with degenerate Hamiltonians, for which the standard theory of Lagrangian variational integrators cannot be applied. We also characterize the group invariance properties of discrete Hamiltonians which lead to a discrete Noether's theorem.

Christian Lessig, California Institute of Technology
A Primer On Geometric Mechanics for Scientists and Engineers

Geometric mechanics is a reformulation of mechanics that employs the tools of modern differential geometry, such as tensor analysis on manifolds and Lie groups, to gain insight into physical systems. Additionally, the theory is also of great importance for numerical computations. However, in many applied fields, geometric mechanics has so far not been appreciated, arguably because it is traditionally considered as part of mathematical physics and applied mathematics and formulated in a language that is difficult to appreciate by practitioners. We develop the mathematics and physics of geometric mechanics at a level accessible to scientists and engineers. We introduce Lagrangian and Hamiltonian mechanics and show how this naturally leads to a geometric formulation of mechanics. The central role of symmetries and conserved quantities is discussed, and how this can lead to simplified descriptions. Throughout, the discussion employs concrete physical systems to motivate and clarify abstract ideas. We also discuss the importance of geometric mechanics for numerical computations and why good numerical techniques have to respect the geometric structure of a continuous theory.

Andrew Lewis, Queen's University
Problems in geometric control theory

The problem of controllability has a long history in geometric control theory and, along with optimal control theory, has played a central role in the development of geometric control. Another important research area in control theory, particularly where applications are concerned, is the theory of stabilisation. This area is dominated by Lyapunov theory, and has not really been a subject of great interest to the geometric control community. In this talk, connections between controllability theory and stabilisation theory are discussed, and some open research directions are indicated.

An overview of control theory for mechanical systems

Differential geometry has been successfully applied to nonlinear control theory, resulting in geometric control theory which was born in the mid 1960's. Around that same time, differential geometric methods were systematically applied to the formulations of classical mechanics. In the mid 1990's these two areas of research were fused with the result that significant advances were made in the control theory for mechanical systems. This continues to be an active area of research today.

An introduction to geometric control theory

These lectures will provide, at a level suitable for graduate students, the basic background of geometric control theory. The emphasis will be on the study of control theoretic problems where the intrinsic methods of differential geometry have proven valuable. Topics will include: (1) geometric formulations of control systems; (2) distributions and the Orbit Theorem; (3) the Sussmann/Jurdjevic theory of accessibility; (4) an introduction to the theory of controllability.

Debra Lewis, University of California Santa Cruz
Relative critical points

Relative equilibria of Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic, Poisson, variational - generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings.

Treating the (dual) algebra elements as parameters yields functions invariant only with respect to the isotropy subgroup of the given parameter; if the algebra elements are regarded as variables transformed by the (co)adjoint action, the relevant functions are invariant with respect to the full symmetry group. A generating set of invariant functions can be used to reverse the usual perspective: rather than seeking the critical points of a specific function, one can determine famililies of functions that are critical on specified orbits. This approach can be used in the design of conservative models when the underlying dynamics must be inferred from limited quantitative and/or qualitative information.

Optimal control with moderation incentives

Optimal solutions of generalized time minimization problems, with purely state-dependent cost functions, take control values on the boundary of the admissible control region. Augmenting the cost function with a control-dependent term rewarding sub-maximal control utilization moderates the response. A moderation incentive is a cost term of this type that is identically zero on the boundary of the admissible control region.

Two families of moderation incentives on spheres are considered here: the first, constructed by shifting a quadratic control cost, allows piecewise smooth solutions with controls moving on and off the boundary of the admissible region; the second yields solutions with controls remaining in the interior of the admissible region. Two simple multi-parameter control problems, a controlled velocity interception problem and a controlled acceleration evasion problem, illustrate the approach.

Jaume Llibre, Universitat Autonoma de Barcelona,
On the central con gurations of the N-body problem and its geometry

Since Euler found the rst central con guration in the 3{body problem in 1767 our knowledge on them has grown over the years, but as we shall see it remains many open questions. Our talk will be on the following items.
-Introduction to the central con gurations.
-Central con gurations of the coorbital satellite problem.
-Central con gurations of the p nested n-gons.
-Central con gurations of the p nested regular polyhedra.
-Piramidal central con gurations.

Robert Lowry, SUNY Suffolk
A Bundle Approach to the Hamiltonian Structure of Compressible Free Boundary Fluid Flows

Building on Richard Montgomery's “bundle picture” (using a non-canonical Hamiltonian framework on principal bundles) and the work of Ratiu and Mazer, I will explore the example of a perfect compressible fluid with a free boundary. Here, I illustrate how the Euler equations for a compressible fluid with a free boundary (such as that observed for instance in weather systems and oceanographic problems) can be derived from Lie-Poisson reduction scheme using a general formula for brackets on reduced principle bundles.

Eder Mateus, Universidade Federal de Sergipe, Brazil
Spatial isosceles three body problem with rotating axis of symmetry

Joint work with Andrea Venturelli and Claudio Vidal. We consider the spatial isosceles newtonian three-body problem, with one particle on a fixed plane, and the other two particles (with equal masses) are symmetric with respect to this plane. Using variational methods, we find a one parameter family of collision solutions of this systems.

Klas Modin, Chalmers University of Technology
Higher dimensional generalisation of the $\mu$--Hunter--Saxton equation

A higher dimensional generalisation of the $\mu$--Hunter--Saxton equation is presented. This equation is the Euler-Arnold equation corresponding to geodesics in $Diff(M)$ with respect to a right invariant metric. It is the first example of a right invariant non-degenerate metric on Diff(M) that descends properly to the space of densities $Dens(M) = Diff(M)/Diffvol(M)$. Some properties and results related to this equation are discussed.

Richard Montgomery, University of California Santa Cruz
Classical Few-body Progress

We review progress in the classical N-body problem N= 3, 4, 5 over the last three to four decades. Then we will focus on my contributions over the last 13 years to the zero-angular momentum three-body problem: the eight solution, the existence of infinitely many syzygies (= collinearities), and how the Jacobi-Maupertuis metric can give a hyperbolic structure.

Juan Carlos Marrero, University of La Laguna
Hamilton-Jacobi equation and  integrability

It is well-known that Hamilton-Jacobi theory is closely related with the theory of completely integrable systems. In fact, from a complete solution of the Hamilton-Jacobi equation for a Hamiltonian system with n degrees of freedom one may obtain a set of n independent rst integrals which are in involution. In this talk, I will discuss some recent advances on the extension of the previous theory to the reduction of Hamiltonian systemswhich are invariant under the action of a symmetry Lie group. In the last part of the talk, I will present some ideas on the extension of this theory to nonholonomic mechanics.

Marcel Oliver, Jacobs University
Backward error analysis for symplectic Runge–Kutta Methods on Hilbert spaces

In this talk, I will review classical backward error analysis for symplectic Runge--Kutta methods for Hamiltonian ODEs and explain the difficulties when applying similar ideas in the context of PDEs. I will then explain two stragegies for making backward error analysis work on infinite dimensional Hilbert spaces as well. The first approach is based on exploiting the regularity of the original PDE system and yields, under sufficiently strong assumptions, results which are almost as strong as those available for ODEs. The second approach involves a new construction of modified equations within the framework of variational integrators. This approach is still work-in-progress, but initial numerical tests support the validity and point to possible analytic advantages of this approach. (Joint work with C. Wulff and S. Vasylkevych.)

Edith Padron, University of La Laguna
Hamilton-Jacobi equation and nonholonomic dynamics

In this talk,  I will present recent advances about a new formalism which allows describe Hamilton-Jacobi equation for a great variety of mechanical systems (nonholonomic systems subjected to
linear or affine constraints, dissipative systems subjected to external forces, time-dependent mechanical systems...). Several examples will illustrate this theory.

Geometry and the analysis of geometric numerical algorithms

Variational integrators are numerical algorithms formulated geometrically on a manifold of dynamical states. Just as the construction of such integrators benefits from geometry, their analysis also benefits.

Geodesic equations on contactomorphism groups

Contact structures are the odd-dimensional analogues of symplectic structures. We study Riemannian geometries on the diffeomorphism groups that preserve either a contact form (quantomorphisms) or a contact structure (contactomorphisms). In the former case, the geodesic equation ends up being a generalization of the quasigeostrophic equation in the $f$-plane approximation, while in the latter case, the geodesic equation generalizes the Camassa-Holm equation. We discuss the structures of these groups as infinite-dimensional Sobolev manifolds and use this structure to obtain local existence results (following Ebin-Marsden). We also obtain global existence for the quantomorphism equation and some conservation laws for the contactomorphism equation. This is joint work with David Ebin.

On violins with rubber strings, or contact chaos caused by the perfect friction contact of elastic rods

One of the most important and challenging problems of elastic rod-based models of polymers is to accurately take into account the self-intersections. Normally, such dynamics is treated with an introduction of a suitable short-range repulsive potential to the elastic string. Inevitably, such models lead to a sliding contact, because of the very nature of the potential interaction between two parts of the string. Such models, however, fail to take into account situations where the small scale structure of the polymer's "surface" is very rough, as is the case with e.g dendronized polymers. Such polymers are more likely to incur the rolling contact dynamics, or at the very least some combination of rolling and sliding contact. It is generally believed to be impossible to model rolling contact, even in the simplest cases, by introducing a contact potential.

We derive a consistent motion of two elastic strings in perfect rolling contact, a situation that can be easily visualized by putting two rubber strings in contact. We show that even the contact dynamics is essentially nonlinear, and even if the string's motion away from contact is assumed linear, the contact dynamics leads to strongly nonlinear motion, which we call "contact chaos". We also derive exact motion of contact when the string consists of discrete particles. We finish by presenting some exact solutions of the problem, as well as numerical simulations.

Tudor S. Ratiu, Ecole Polytechnique Fédérale de Lausanne (abstract)

Vortices on the triaxial ellipsoid: a movie show

Joint work with Cesar Castilho. We present a demo implementing the motion of vortex pairs on a triaxial ellipsoid. In the limit of a vortex dipole the motion approaches Jacobi's geodesic system. It seems that the vortex pair is a KAM perturbation, interpreting M x M near the diagonal ~ T*M with a proper time rescaling.

Miguel Rodriguez-Olmos, Technical University of Catalonia
Hamiltonian bifurcations from stable branches of relative equilibria

It was shown by Arnold that if a non-degenerate relative equilibrium of a symmetric Hamiltonian system is regular (i.e. it doesn't have phase space isotropy) it persists without bifurcation continuously to nearby momentum values. We show that, under some conditions, if the relative equilibrium is in particular formally stable and exhibits continuous isotropy then Hamiltonian bifurcations must occur for every point in the persisting branch. This effect is therefore purely produced by the existence of isotropy, or singularities of the Lie symmetry action on phase space. Joint work with J. Montaldi

Shane Ross, Virginia Tech
Lagrangian coherent structures

The concept and study of Lagrangian coherent structures (LCS) have evolved from a need to formally define intrinsic structures within fluid flows that govern flow transport. Roughly speaking, LCS are distinguished material lines or surfaces that delineate regions of fluid for which the long-term evolution of a tracer particle is qualitatively very different. Jerry Marsden helped lead the development of efficient mathematical tools for identifying the presence and form of these structures in complex numerical and experimental data sets, which are becoming commonplace in fluid dynamics research. This ability significantly advances our capability to both understand and exploit fluid flows in engineering and natural systems.

Tube dynamics and applications

Hamiltonian systems with rank-one saddles can exhibit a mechanism of phase space transport known as 'tube dynamics', which goes back to work of Conley and McGehee, and further explored by Marsden and co-workers. This mechanism, based on stable and unstable manifolds of normally hyperbolic invariant manifolds, has seen application to celestial mechanics as well as chemistry. Recently, it has been applied to the motion of ships near capsize, geometrically interpreted as surfaces separating states leading to capsize from those which are not, with some practical implications.

Tanya Schmah, University of Toronto
Reduction of systems with configuration space isotropy

We consider Lagrangian and Hamiltonian systems with lifted symmetries, near points with configuration space isotropy. Using twisted parametrisations of phase space, we deduce reduced equations of motion. On the Lagrangian side, these are a hybrid of the Euler-Poincar\'e and Euler-Lagrange equations, and correspond to a constrained variational principle. We specialise the equations of motion to simple mechanical systems, for which, on the Hamiltonian side, we state a relative equilibrium criterion in terms of an \textit{augmented-amended potential}.

Brian Seguin, McGill University
A transport theorem for irregular evolving domains

The Reynolds transport theorem is fundamental in continuum physics. In this theorem, the evolving domain of integration is given by a time-dependent family of diffeomorphisms. There are, however, applications in which an evolving domains evolution cannot be described such a family. Examples of such domains include those that, among other things, develop holes, split into pieces, or whose fractal dimension changes in time. I will present a transport theorem that holds for evolving domains that can have these kinds of irregularities. Possible applications include phase transitions, fracture mechanics, diffusion, or heat conduction.

From the Geometry of Extreme Value Distributions to 'Laws of Mechanics' in Financial Markets

A new geometric invariant explains and unifies the landmark results of Extreme Value Theory. This invariant also provides an intrinsic measure of the rate of convergence of tails of probability distributions to their Extreme Value limits. Tail models that converge rapidly over quantile ranges that are practical in statistical applications are highly efficient. They reveal previously unobservable regularities and anomalies in financial market data. This allows the formulation of some 'laws of mechanics' in markets. Among other things, these give early warning signals of asset price bubbles as well as a measure of their severity.

Banavara Shashikanth, New Mexico State University
Vortex dynamics of classical fluids in higher dimensions

The talk will focus on vortex dynamics of classical fluids in $\mathbb{R}^4$. In particular, I will discuss the geometry and dynamics of singular vortex models---termed vortex membranes---which are the analogs of point vortices and vortex filaments. Some basic facts about the vorticity two-form and the curvature induced dynamics of vortex filaments will be recalled. Following this, the main result will be presented--namely, that the self-induced velocity field of a membrane, using the local induction approximation, is proportional to the skew mean curvature vector field of the membrane. Time permitting, the dynamics of the four-form $\omega \wedge \omega$ and an application to Ertel's vorticity theorem in $\mathbb{R}^3$ will be briefly discussed.

Jedrzej Sniatycki, University of Calgary
Differential Geometry of Singular Spaces and Reduction of Symmetries (abstract)

Ari Stern, University of California, San Diego
Symplectic groupoids and discrete constrained Lagrangian mechanics

The subject of discrete Lagrangian mechanics concerns the study of certain discrete dynamical systems on manifolds, whose geometric features are analogous to those in classical Lagrangian mechanics. While these systems are quite mathematically interesting, in their own right, they also have important applications to structure-preserving numerical simulation of dynamical systems in geometric mechanics and optimal control theory. In fact, these structure-preserving properties are intimately related to the geometry of symplectic groupoids, Lagrangian submanifolds, and generating functions. In this talk, we describe how a more general notion of generating function can be used to construct Lagrangian submanifolds, and thus discrete dynamics, even for systems with constraints. Within this framework, Lagrange multipliers and their dynamics are shown to arise in a natural way.

Cesare Tronci, University of Surrey
Collisionless kinetic theory of rolling molecules

A collisionless kinetic theory is presented for an ensemble of molecules undergoing nonholonomic rolling dynamics. Nonholonomic constraints lead to problems in generalizing the standard methods of statistical physics. For example, no invariant measure is available. Nevertheless, a consistent kinetic theory is formulated by using Hamilton's variational principle inLagrangian variables. Also, a cold fluid closure is presented.

Tomasz Tyranowski, California Institute of Technology
Space-adaptive geometric integrators for field theories

Moving mesh methods (also called r-adaptive methods) are space-adaptive strategies used for the numerical simulation of time-dependent partial differential equations. The spatial mesh consists of a constant number of nodes with fixed connectivity, but nodes can be redistributed to follow the areas where a higher mesh point density is required. There are a very limited number of methods designed for solving field-theoretic partial differential equations, and the numerical analysis of the resulting schemes is challenging. In this talk we present two ways to construct r-adaptive variational integrators for (1+1)-dimensional Lagrangian field theories. Some numerical results for the Sine-Gordon equation are also presented. (Joint work with Mathieu Desbrun)

Joris Vankerschaver, University of California, San Diego
Fluid-Structure Interactions: Geometric and Numerical Aspects

In this talk, I will show how symplectic reduction can be used to study various aspects of the dynamics between rigid bodies and ideal flows. After discussing the general framework, I will focus on a simple example: that of a planar rigid body with circulation. I will show that the equations of motion arise by either considering a central extension of SE(2), or by reducing with respect to exact diffeomorphisms, and that the resulting system can be viewed as a fluid-dynamical analogue of the Kaluza-Klein equations. Finally, we will see how this framework can be used to construct in a systematical way variational integrators for fluid systems.

Miguel Vaquero, Instituto de Ciencias Matemáticas
Hamilton-Jacobi for Generalized Hamiltonian Systems

It is well-known the important role played by the classical Hamilton-Jacobi theory in the integration of the equations of motion of mechanical systems. In this talk we will introduce a Hamilton-Jacobi theory in the context of hamiltonian systems defined on almost-Poisson manifolds with a bundle structure. This is a very general framework that allows us to recover, in a very geometric fashion, the classical Hamilton-Jacobi equation and even the nonholonomic Hamilton-Jacobi theory developed by Iglesias et al. and later studied by Ohsawa and Bloch. Future directions will be also given.

Olivier Verdier, NTNU
Geometric Generalisations of the Shake and Rattle methods

Constrained mechanical systems (robots, rod models) have to be simulated with care. In particular, it is important to design numerical integrators which preserve the “mechanical structure” of the system. Those integrators are known, for instance, to approximately preserve energy and other invariants. I will give a geometric description of the existing structure preserving integrators for constrained mechanical systems (called “Shake” and “Rattle”). Finally I will explain how to extend those methods to handle cases that were out of reach for the current solvers. (Joint work with K. Modin, R.I. McLachlan and M. Wilkins).

Francois-Xavier Vialard
Geometric Mechanics for Computational Anatomy: From geodesics to cubic splines and related problems

In this talk, I will present applications of geometric mechanics to Computational Anatomy, whose goals are among others, developping geometrical and statistical tools to quantify the biological shape variablility. In this direction, we will present a model that uses right-invariant metrics on the group of diffeomorphisms of the ambient space. We will present in particular two different models to account for time dependent shape evolutions: geodesic regression and cubic splines. We will conclude the talk with experimental results.

Hiroaki Yoshimura, Waseda University
Dirac Structures, Variational Principles and Reduction in Mechanics --- Toward Understanding Interconnection Structures in Physical Systems

In this talk, we survey the fundamentals of Dirac structures and their applications to mechanics, including the case of degenerate Lagrangians in the context of implicit Lagrangian systems, together with some examples of nonholonomic mechanics, electric circuits and field theories. We also show some recent advances in interconnecting distinct Dirac structures and associated dynamical systems, in which we emphasize the idea of "interconnections" in physical systems can be fit into the setting of Dirac geometry and plays an essential role in understanding the system as a network.

Dmitry Zenkov, North Carolina State University
Hamel's Formalism and Variational Integrators

Variational integrators are obtained by discretizing a variational principle of continuous-time mechanics. It has been observed recently that such discretizations may lead to a lack of preservation of system's relative equilibria and their stability. This behavior is not desirable for long-term numerical integration. The talk will discuss that measuring system's velocity components relative to a suitable frame leads to the integrators that keep system's relative equilibria and their stability intact.