The main objective of this project is to advance significantly the understanding of the symplectic geometry of cotangent bundles endowed with lifted actions, from the point of view of reduction, mechanics and applications.
The usual methods of symplectic geometry and symplectic reduction applied to cotangent bundles with Hamiltonian lifted symmetries are not fine enough to treat this important case since the extra bundle structure that makes a cotangent bundle different from other symplectic manifolds are not taken into account.
Recently, adapted methods for the study of singular reduction of cotangent bundles at zero momentum have been introduced by the proposer, opening the field to a big amount of future original research. In this project this program is continuated, obtaining generalizations not only to arbitrary momentum values but also to other related geometries, like Poisson and contact geometry.
This study will be done both globally and locally. Globally, the various classic results on regular cotangent bundle reduction will be generalized to the singular case within the framework of stratified spaces. Locally, invariant tubular neighbourhoods and normal forms for the momentum map adapted to the extra bundle structure will be found.
The geometric results obtained will produce a core of applications, which are an important part of this project. These include new insights in the stability and bifurcations of relative equilibria in classical mechanical systems, the geometry of symmetric non-holonomic systems with singularities, and the interplay between the schemes of singular Poisson reduction of cotangent bundles and the classical theory of the geometry and representations of compact Lie groups, among others.
Finally, this fellowship will provide the pro poser advanced training mainly in the aspects of Hamiltonian actions and reduction in symplectic, Poisson and contact cases, infinite-dimensional geometry, and qualitative dynamics of Hamiltonian systems.
Call for proposal
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