# Applied Dynamics and Geometric Mechanics

### Jerrold E. Marsden

California Institute of Technology, Pasadena, United States### Jürgen Scheurle

TU München, Germany

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## Abstract

\noindent The meeting was inspired by the fact that the number of interesting new applicable basic tools and techniques in applied dynamical systems and geometric mechanics continues to grow and be developed in interesting ways, both from the point of view of fundamental theory as well as from their applicability. Some examples of specific areas are basic advances in geometric mechanics, techniques for dealing with complex interconnected systems, new optimization methods, new geometric structures such as Dirac structures, discrete mechanics and its related geometric integrators, particle methods, variational principles, etc. Applications included fluid mechanics, stochastic mechanics, nonholonomic systems, plasticity, atmospheric dynamics, impact mechanics, nonlinear geometric control, and classical field theory. The specific themes of the workshop were as follows: \medskip \noindent {\bf Core Dynamical Systems.} The basic theory of dynamical systems continues to develop with numerous new ideas that have importance in applications. For example, the concept of Finite Time Lyapunov Exponents and their derivative, Lagrangian Coherent Structures are tools that extend invariant manifold theory in a nice way to the time dependent case. These ideas have found application in, for example, the detection of recirculation zones in the heart and atmospheric barriers to transport. Another example is the extension of classical concepts from bifurcation and stability theory to time dependent systems with applications to phase transitions and materials with memory. \medskip \noindent {\bf Complex Interconnected Systems.} The computational limitations of dealing with complex interconnected systems, such as fuel cells, aircraft, etc have already been reached. New methods are being developed that move away from the monolithic approach (that is, thinking of systems of thousands of coupled equations, be they ode's or pde's) to the idea of many computations running in parallel with message passing and information exchange. This is an area in which dynamical systems theory is playing a key role. Set oriented methods and multiobjective optimization is an example of a classical area that has seen significant applications of dynamical systems. \medskip \noindent {\bf Structured Model Reduction.} Related to the previous topic is that of forming dimensionally reduced models for computational feasibility. This is important in, for example, complex fluid flows, where resolvability of the Navier-Stokes equations is simply not possible and where reduced order models have been developed. POD (proper orthogonal decomposition) methods are also undergoing continued development and their limitations being better understood. In carrying out such reductions, it is of special interest to do so in a way that preserves structure (such as symmetry structures, mechanical structures, etc). \medskip \noindent {\bf Uncertainty and Stochastic Methods.} Another area in which dynamical systems can play a key role is how to deal with systems whose very models are uncertain; for example, think of modeling an asteroid moving mainly in the field of the sun and Jupiter. How much error is introduced by neglecting the effects of Saturn? Is this more important than errors in initial conditions? How do error balls propagate under the dynamics? Of course in many systems, such as laboratory based mechanical systems, there are sources of uncertainty due to noise as well and one needs techniques deal with this and to distinguish this from the other sources of uncertainty and from numerical uncertainties. \medskip \noindent {\bf Core Geometric Mechanics.} Despite its maturity, especially over the last few decades, the basic theory of geometric mechanics continues to thrive and develop. For example, the reduction of mechanical systems with symmetry continues to grow and find applications in, for instance, computation and control. Areas that are currently undergoing particularly interesting growth that have links with applications are reduction by stages (for example, applied to fluid--solid interactions), singular cotangent bundle reduction (relevant for instance, to the dynamics of multiple pendula), and the development of Dirac structures. Also, the theory of integrable systems continues to be a valuable link between geometric mechanics and pure mathematics. Another area that is of great interest is the application of ideas from geometric mechanics to classical field theories such as electromagnetism. \medskip \noindent {\bf Geometric Integrators.} The area of structured integrators for mechanical systems continues to undergo strong growth for problems in which preserving structure is important, such as symplectic structures for backward error analysis, in fluid systems in which it is important to conserve circulation, and in the development of asynchronous integrators. New insight into the development of methods that are robust to uncertainty are also quite promising. Both variational integrators and particle methods were represented at the meeting. \medskip \noindent {\bf Optimization and Control of Mechanical Systems.} New methods for optimization for mechanical systems such as DMOC (discrete mechanics and optimal control) are showing promise for the optimization of, for example, complex systems of vehicles---for instance a swarm of micro-air vehicles that is sent to investigate a biohazard. In these methods, the use of techniques that are successful already in internet congestion control as well as the use of parallel computation are quite attractive. Other types of control, such as stabilization, continue to benefit from basic advances in the theory and to make strong links with, for example, geometric integrators and discrete mechanics. \medskip \noindent {\bf Structure of the Meeting.} The meeting had a balance of senior researchers, postdoctoral fellows and graduate students. Consistent with the general approach advocated by Oberwolfach, there were only about 20 main lectures at the meeting. These senior people suggested students, postdoctoral fellows and junior faculty all of whom participated through two poster sessions. \medskip \noindent {\bf Posters.} The poster session was one of the most interesting aspects of the meeting. The organizers decided to award prizes for the best poster and an ad hoc committee (consisting of Reich, Ratiu, Marsden and Scheurle) was formed to choose the four best posters. This number matched the number of gifts that were available. The winners were (in alphabetical order): \medskip \begin{itemize} \item Ueli Aeberhard (ETH Z\"urich), {\it Perfect multi-contact collisions\/} \\ \item Philip Du Toit (Caltech), {\it Hurricanes, Horseshoes, and Homoclinic Tangles\/} \\ \item Andreas Johann (TU M\"unchen), {\it Spiral Waves and Spiral Solitons in Lattice Differential Equations\/} \\ \item Sigrid Leyendecker and Sina Ober-Bl\"obaum (Caltech), {\it Dynamic Optimization of a Three-Dimensional Walker\/}. \\ \end{itemize} \bigskip \noindent {\bf Impressions, Connections, Insights.} The mixture of participants, from those who were very applied and those who came from more of a dynamical systems or geometric mechanics background, but all with a strong mathematical dedication, was very fruitful. Dynamical systems and geometric mechanics methods are now used in a spectacular array of very diverse application areas from complex fluids and plasticity to impact mechanics and spacecraft control. There is a wide range of research opportunities, including the possibility for new theoretical and methodological progress. Amongst the specific new connections and insights that were gained, the power of dynamical systems ideas, such as invariant manifolds or Lyapunov exponents, seemed to be quite impressive as well as useful in the context of atmospheric science as well as oceanography. The use of dynamical systems ideas in control of mechanical systems was also quite interesting for a number of the participants.