Symplectic techniques in conservative dynamics

The roots of symplectic geometry lie in the study of conservative dynamical systems: the space of positions and velocities of the solutions of a system of Hamiltonian differential equations admits a natural symplectic structure. This underlying geometric structure, together with the topology of the energy levels, determines the dynamics. In the last thirty years, symplectic geometry (together with its ‘sister’, contact geometry) has developed into a prominent field of research in its own right, and the efforts of an increasing and very active community have brought about many new and exciting results. Questions in conservative dynamics have again been among the major driving forces behind these developments. In particular, variational principles have been used to define new symplectic invariants, which have shed light on the nature of the relationship between dynamical questions and questions in symplectic geometry.

It is our belief, though, that the potential of the new symplectic techniques for the solution of dynamical problems has been far from fully exploited. On the one hand these techniques are undeniably very sophisticated and not easily accessible, on the other hand the main focus in the construction of invariants like contact homology, symplectic homology and symplectic field theory has been on their application in classification problems in symplectic and contact topology. We think that there are still plenty of very interesting problems to be explored in dynamics with the help of these new tools. Examples of such problems include:

• existence of quasi-periodic solutions and solutions of arbitrary rotation numbers for higher dimensional symplectic twist maps and/or Lagrangian systems;

• existence of planetary orbits of given braid type in celestial mechanics;

• existence of periodic orbits in Hamiltonian systems with non-compact energy levels and related multiplicity issues;

• variational methods for (viscosity) solutions for the Hamilton-Jacobi equation;

• the problem of finding geodesics in Lorentzian manifolds;

• forcing results for periodic orbits and chaotic systems;

• possible implications of the theory of J-holomorphic curves for harmonic maps.

With this workshop we hope to bring together researchers working in symplectic geometry and in different areas of conservative dynamics: we strongly believe that this can lead to a fruitful interaction, from which their fields of expertise can gain in strength and insight. Therefore we do not intend to focus on a specialized topic and its depth, but the aim of the workshop will rather be to foster discussions that can broaden the participants’ horizons. What we ultimately intend to achieve is a more comprehensive view of the interactions between symplectic geometry and dynamics and of the challenges and possibilities found at this interface.