# Geometric Mechanics: Continuous and discrete, finite and infinite dimensional (07w5068)

## Organizers

Jerrold Marsden (California Institute of Technology)

Juan-Pablo Ortega (CNRS, Universite de Franche-Comte)

George Patrick (University of Saskatchewan)

Mark Roberts (University of Surrey)

Jedrzej Sniatycki (University of Calgary)

Cristina Stoica (Wilfrid Laurier University)

## Objectives

STATEMENT OF OBJECTIVES

The academic scope of Geometric Mechanics is wide and interdisciplinary. Dispersed groups in Canada, the United States, and Europe, have ongoing independent collaborations. The objective of the workshop is to bring these collaborative groups together, to assist and enable their specific projects, and to encourage inter-group communication and appreciation. Another objective is to generate new projects and collaborations.

The participants for the workshop will be engaged in an ongoing collaborative work on specific, well defined projects. These projects will be interlinked, and at a rather well developed stage, when the workshop is in progress. Because the 2007 timeline is far, the projects list and participant list will require ongoing work, should this proposal be successful. The talks in this workshop will be linked to these ongoing projects when the schedule is completed, some 6 months prior to the workshop date.

Our conception of the projects is outlined below. This list represents the current opportunities as we see them, and it will be reduced considerably for the workshop. Following the list, we detail a few the projects which are at an advanced stage of development.

1. Bifurcation and stability of Hamiltonian systems with symmetry.

Bifurcations of relative equilibria of generic Hamiltonian systems.

Buono, Chossat, Field, Hernandez, Lamb, Marsden, Montaldi, Ortega,

Patrick, Planas-Bielsa, Roberts, Schmah, Stoica, Wulff

Bifurcations of relative equilibria. Newtonian mechanical systems.

Buono, Field, Hernandez, Lamb, Marsden, Montaldi, Patrick, Ratiu,

Schmah, Stoica, Wulff

Bifurcations of relative periodic orbits.

Doedel, Lamb, Roberts, Way, Wulff

Dynamics near relative equilibria. Normal form theory.

Roberts, Schmah, Stoica

Stability of symmetric relative equilibria. Newtonian mechanical systems.

Chossat, LewisD, Patrick, Ortega, Planas-Bielsa, Roberts,

Rodriguez-Olmos, Schmah, Stoica, Wulff

Numerics and bifurcation.

Schebesch, Wulff, Way

Topological stability theory of symmetric relative equilibria for

non-compact group actions

Patrick, Planas-Bielsa, Roberts, Wulff

2. Control theory

Geometry and control theory

Bullo, Junge, LewisA, Marsden, Ober-Bloebaum

3. Example systems

Dynamics and control of rigid body systems. Coordinated groups of vehicles.

Leonard, Marsden, Mezic, Montgomery, Patrick, Roberts, Wulff

Geometric continuum mechanics.

Epstein, Kanso, Marsden, Sniatycki, West, Yavari

Lagrangian coherent structures in fluids.

Lekien, Marsden, Padberg, Shadden

Transport properties (as in celestial and chemical reaction rates).

Dellnitz, Koon, Gabern, Grubbits, Marsden

4. Numerics and discretization

Discrete fluid mechanics.

Desbrun, Kanso, Marsden, Shkoller, Tong, West

Holonomic and nonholonomic discrete systems.

Cuell, Marsden, Patrick

5. Theoretical aspects; symplectic and Poisson geometry

Blowing-Up Poisson structures.

Patrick, Roberts, Stoica, Wulff

Convexity of standard and non-standard momentum maps.

Birtea, Ortega, Ratiu

Differential spaces and reduction of symmetries. Reduction of

symmetries for improper actions.

Sniatycki

Groupoid theoretical approach to Poisson geometry with symmetries.

Crainic, Loja-Fernandes, Lu, Ortega, Ratiu, Weinstein

Non-holonomic constraints.

Bloch, Cushman, Marsden, Patrick, Sniatycki

Reduction theory. Dirac structures.

Marsden, Ratiu, Yoshimura

Symplectic reduction without standard momentum maps.

Ortega, Ratiu

Topology of momentum maps and energy-momentum maps.

Giacobbe, Montaldi, Plummer, Roberts

Variational principles for symmetric mechanical systems.

Marsden, McCord, Montaldi, Montgomery, Sbano, Southall

DETAILED PROJECTS

1. Bifurcations of Relative Equilibria

Although there is a very well established equivariant bifurcation theory for relative equilibria of general dynamical systems, there is nothing comparable for Hamiltonian systems, except in the case of free abelian group actions. Symmetries come with associated conserved momenta that act as 'internal parameters' with respect to which relative equilibria will bifurcate. In the case of free abelian group actions, the bifurcations reduce to those of equilibria of functions which depend generically on these parameters, and hence to 'elementary catastrophe theory'. However for general non-abelian groups the analogous reduction leads to a 'catastrophe theory' problem coupled to generalized rigid body dynamics. The bifurcations of such systems are very poorly understood.

1.1 Bifurcations of Relative Equilibria - Generic Hamiltonian Systems

An approach that has been pioneered by Patrick and Roberts [2000] for free compact group actions, and Wulff [2003] for free non-compact group actions, is to show that the set of relative equilibria of a Hamiltonian system can be obtained as the inverse image of the singular algebraic variety of `commuting velocity-momentum pairs' by a map constructed from the Hamiltonian and momentum map. Transversality theory then enables us to describe the structure of the set of relative equilibria, and in particular its singularities (bifurcation points), for generic Hamiltonians. Our current aim is to extend this to non-free group actions - i.e. relative equilibria with symmetries - by using the equivariant transversality theory developed by Bierstone [1977] and Field [1977]. Time-reversing symmetries will also be considered, leading to a Hamiltonian analogue of a bifurcation theory for 'time-reversal equivariant systems' currently being developed by Buono, Lamb and Roberts [2005].

1.2 Bifurcations of Relative Equilibria - Newtonian Mechanical Systems

The ubiquitous systems that are derived from Newtons second law have a special structure: they are defined on the tangent bundle of the configuration space of the system and have Lagrangians that incorporate kinetic energy terms coming from metrics on the configuration space, potential energy functions for conservative forces, and terms pairing velocity variables with one forms for magnetic forces. Roberts, Schmah and Stoica [2005] have recently described the general structure of bifurcation equations for relative equilibria of Newtonian mechanical systems on cotangent bundles invariant under general proper group actions. Our current aim is to use these to develop a theory to describe the generic structure of the set of relative equilibria of Newtonian systems. It is likely that (equivariant) transversality theory will also play a central role in this. The theory will be illustrated by application to molecules and other N-body problems. It will also be compared with an alternative approach due to Hernandez-Garduno and Marsden [2006], and Birtea et.al. [2003].

2. Convexity of standard and non-standard momentum maps

The convexity properties of the image of the momentum map have been a very active field of research. Recent results on this subject indicate that the essential attributes underlying the convexity theorems that one finds in the literature have to do with deep topological properties that are much more primitive than the analytical and Morse theoretical arguments that have been used so far. This suggests the extension of the convexity results to some of the generalizations of the notion of momentum map available in the literature.

3. Discrete Fluid Mechanics

Since Arnold's work on fluid mechanics, a discrete version of the diffeomorphism group has been long sought that can be used for geometric integration of fluid mechanics. An exciting candidate has recently emerged, namely the group of invertible volume preserving transfer operators on a given mesh. After choosing a candidate discrete Lagrangian, one can make use of discrete Euler-Poincare theory. If successful, this will open some new avenues for a true geometric integration approach to computational fluid mechanics. It is expected that this area will be still in the emergent state during the meeting.

4. Numerics and Bifurcation.

The bifurcation theory and numerics of equilibria and periodic orbits of general dynamical systems is well developed, and in recent years there has been rapid progress in the development of a bifurcation theory for relative equilibria and relative periodic orbits in dynamical systems with structure, like symmetry or symplecticity, see e.g. Wulff, Lamb, and Melbourne [2001], and Patrick and Roberts [2000]. But there are few results on the numerical computation of those bifurcations yet. Gatermann and Hohmann [1991] developed mixed symbolic and numerical methods for the computation of symmetry-breaking bifurcations of equilibria and implemented these methods in the package SYMCON. In Wulff and Schebesch [2005a] methods are developed for the numerical computation of bifurcations of periodic orbits which preserve the isotropy, and in Wulff and Schebesch [2005b] non-degenerate Hamiltonian relative periodic orbits with regular drift-momentum pair are continued in energy and momentum. These methods have been implemented in the program package SYMPERCON which is still under development. The aim of this project is the parallel development of the theory and numerical analysis of isotropy-breaking bifurcations in symmetric Hamiltonian systems.

5. Groupoid theoretical approach to Poisson geometry with symmetries.

The treatment of the canonical symmetries of a Poisson manifold cannot be directly obtained from the existing well developed theory for symplectic manifolds. Key notions in the symplectic context as, for example, the momentum map, are not well adapted to this category. Recent important developments in groupoid theory allow to associate to each Poisson manifold a unique symplectic manifold (called its symplectic groupoid) that 'integrates it' in a very specific sense. Its use could shed some light in the study of Poisson symmetries.

6. Stability of relative equilibria

There is a well established stability theory for relative equilibria of Hamiltonian systems which are invariant under the action of a compact symmetry group. This is based on a principle of energy-momentum confinement, due originally to Arnold [1969] and Marsden and Weinstein [1974], that states that if the relative equilibrium is an isolated point in its energy-momentum level set, then it is Liapounov stable with respect to all perturbations, including those that do not preserve the momentum. A topological formulation of this result has been given by Montaldi [1997] and a sufficient condition on the restriction of the Hessian of the Hamiltonian in a co-moving frame to the 'sympletic normal space' by Lerman and Singer [1998] and Ortega and Ratiu [1999].

6.1 Topological Stability Theory for Non-Compact Group Actions

It is a venerable result that a rigid body will rotate stably about its long and short axes of inertia, and unstably about its middle axis. Patrick, Roberts, and Wulff [2004] demonstrated that it is impossible to use energy-momentum Lyapunov functions to establish the same statement if the body is immersed in a fluid. They further demonstrated that, even if stable, the orientation of these motions would not be stable i.e. that, although the rigid body would forever spin about its long or short principle axis, that axis would tipped relative to its original orientation. Patrick [2003], using desingularizations of Poisson structures, normal form, and KAM confinement, demonstrated that such motions are in fact stable, and numerically verified the orientation instability.

These methods are topologically rooted and are applicable to relative equilibria occur with nontrivial isotropy at which the reduced phase spaces are singular. Our aim will be to obtain a necessary and sufficient topological confinement criterion for relative equilibria with symmetries. Preliminary work suggests that the presence of isotropy eases obstructions to energy-momentum confinement: our plans include making this precise. We will also derive a sufficient condition on the Hessian of the Hamiltonian in a co-moving frame for

this confinement to hold.

6.2 Topological Stability Theory and Newtonian Mechanical Systems

In the case of compact group actions the restriction of the Hessian of the Hamiltonian in a co-moving frame to the symplectic normal space has been decomposed into a block-diagonal form by Marsden et.al. [1989] and Lewis [1992]. The aim of this project is to describe the structure of the full Hessian for all proper non-compact group actions. This will then be used to 'decompose' the Hessian criteria of the general stability theory in the special case of Newtonian

mechanical systems.

REFERENCES

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P Birtea, M Puta, T.S. Ratiu, R Tudoran [2003] Symmetry breaking for

toral actions in simple mechanical systems. P. Birtea, M. Puta,

T.S. Ratiu, R Tudoran. Arxiv preprint math.DS/0311451.

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F. Bullo and A. Lewis [2005]. Geometric control of mechanical

systems. Modeling, analysis, and design for simple mechanical control

systems. Texts in Applied Mathematics, 49. Springer-Verlag, New York,

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P-L. Buono, J.S.W. Lamb, and R.M. Roberts [2005]. Bifurcation and

branching of equilibria in reversible equivariant vector fields.

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M.J. Field [1977]. Transversality in G-manifolds. Trans. AMS 231:429-450.

K. Gatermann and A. Hohmann [1991]. Symbolic exploitation of symmetry

in numerical pathfollowing. IMPACT Comput. Sci. Engrg. 3:330--365.

A. Hernández-Garduño and J.E. Marsden [2005]. Regularization of the amended

potential and the bifurcation of relative equilibria. J. Nonlinear

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O. Junge, J.E. Marsden, and S. Ober-Blobaum [2005]. Discrete

mechanics and optimal control, Proc. IFAC Conf, Prague, June, 2005. To

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equilibria at singular values of the moment map. Nonlinearity

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J.-P. Ortega and T.S. Ratiu [1999]. Stability of Hamiltonian relative

equilibria. Nonlinearity 12:693--720.

R.M. Roberts, T. Schmah and C. Stoica [2005]. Relative equilibria in systems

with configuration space isotropy. J. Geom. Phys. Available online.

G.W. Patrick [2003]. Stability by KAM confinement of certain wild,

nongeneric relative equilibria of underwater vehicles with coincident

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G.W. Patrick and R.M. Roberts [2000]. The Transversal Relative

Equilibria of a Hamiltonian System with Symmetry. Nonlinearity

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G.W. Patrick, R. M. Roberts and C. Wulff [2004]. Stability of Poisson

Equilibria and Hamiltonian Relative Equilibria by Energy

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C. Wulff [2003]. Persistence of relative equilibria in Hamiltonian

systems with non-compact symmetry. Nonlinearity 16:67--91.

C. Wulff, J. Lamb, and I. Melbourne [2001]. Bifurcation from relative

periodic solutions. Ergodic Th. Dyn. Syst. 21:605--635.

C. Wulff and A. Schebesch [2005a]. Numerical continuation of symmetric

periodic orbits. Preprint. 45 pages.

C. Wulff and A. Schebesch [2005b]. Numerical continuation of Hamiltonian

relative periodic orbits. Preprint. 41 pages.