Felipe Arrate, Basque Center for Applied
Mathematics
C
ardiac 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.
Paula Balseiro, Universidade Federal
Fluminense
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.
Marco Castrillon Lopez, Universidad
Complutense de Madrid
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.
Gabriela Depetri, Universidade Estadual
de Campinas
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.
The Geometry of Radiative Transfer
Founded on Lambert's radiometry from the 18th century, radiative
transfer theory describes the propagation of visible light energy
in macroscopic environments. While already in 1939 the theory
was characterized as "a case of `arrested development' [that]
has remained basically unchanged since 1760", no re-formulation
has been undertaken since then. Following recent literature, we
develop the geometric structure of radiative transfer from Maxwell's
equations by studying the short wavelength limit of a lifted representation
of electromagnetic theory on the cotangent bundle. This shows
that radiative transfer is a Hamiltonian system with the transport
of the light energy density, the phase space representation of
electromagnetic energy, described by the canonical Poisson bracket.
The Hamiltonian function of radiative transfer is homogeneous
of degree one, enabling to reduce the system from the cotangent
bundle to the cosphere bundle, while a non-canonical Legendre
transform relates radiative transfer theory to Fermat's principle
and geometric optics. By considering measurements, as did Lambert
in his experiments, and using the tools of modern tensor analysis,
we are also able to obtain classical concepts from radiometry
from the phase space light energy density. In idealized environments
where the Hamiltonian vector field is defined globally, we show
that radiative transfer is a Lie-Poisson system for the group
Diff_{can}(T^*Q) of canonical transformations. The Poisson bracket
then describes the infinitesimal coadjoint action in the Eulerian
representation while the momentum map in the convective representation
recovers the classical law that "radiance is constant along
a ray" with the convective light energy density as Noetherian
quantity. The group structure also unveils a tantalizing similarity
between ideal radiative transfer and the ideal Euler fluid, warranting
to consider the systems as configuration and phase space analogues
of each other. A functional analytic description of the time evolution
of ideal light transport is obtained using Stone's theorem, yielding
a unitary flow on the space of phase space light energy densities
instead of the nonlinear time evolution on the cotangent bundle.
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.
George Patrick, University of Saskatchewan
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.
Stephen Preston, University of Colorado
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.
Vakhtang Putkaradze, University
of Alberta
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)
Reduced Variatonal Principles for Free-Boundary Continua
Adriano Regis, Universidade Federal
Rural de Pernambuco, Brazil
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.
William Shadwick, Omega Analysis
Limited
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.
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