## Final Report Summary - SILGA (Singularities of Lie Group Actions in Geometry and Dynamics)

Introduction.

The objectives of the EU funded project SILGA were organized along two main lines in the general framework of Hamiltonian actions of Lie groups in symplectic geometry.

The study of symmetries of differential equations and dynamical systems has a long story going back to Newton, Lagrange, Hamilton, Poincare and other masters of the last centuries. In the modern formulations of physics and mechanics it is shown that some geometric structures, such as symplectic and Poisson structures, play a fundamental role in these fields, and that the correct geometric picture of the situation corresponds, mathematically, to a Lie group acting on different smooth manifolds preserving one of these geometric objects. The smooth manifold realizes the phase space or “physical arena” where the dynamics or the physical theory lives. The geometric structure is the distinguishing feature of the theory, and the action of the Lie group preserving the geometric structure realizes the symmetries of the theory. This far reaching interpretation has significantly evolved since then, and the geometry of these Lie group actions has emerged as the fundamental object of study in this field. From a global point of view, the main construction is know as reduction theory, and it consists in the process of elimination of these symmetries, constructing from the original models new simpler, and lower dimensional manifolds, which still encode all the mechanical and physical properties of the systems under study. From the local point of view, the main interest lies in how using the symmetries of the problem, we can give useful information of a given mechanical or physical system in a small neighbourhood of a solution, obtaining then important qualitative information, such as long-term stability and bifurcations, which is extremely important in applications, like control theory or engineering. This local study is a particularly active research topic these days. As for the reduction theory, the current trends are focused in applying the techniques previously used to deal with symplectic and Poisson structures, to other related geometric objects, such as cotangent bundles, Dirac structures, generalized complex geometry, or groupoids, which happen to be extremely relevant to new developments in mathematical physics, like supersymmetry and string theory. The global and local aspects of the research are articulated into two main different objectives.

A. Reduction Theory, and

B.Hamiltonian Relative Equilibria.

Description of the Work

----------------------------------

The strategy employed towards the objectives of this project has been a systematic use of semilocal methods based on the Marle-Guillemin-Sternberg normal form (or MGS for short). The MGS form provides a tubular neighborhood of an orbit of a Lie group acting on a symplectic manifold by symplectomorphisms. In this model both the geometry and the action have very simple expressions, which facilitate their study and interactions. Since all the geometric and dynamic problems involved in this problem are local, this approach has been very well suited to this research.

The implementation is actually very simple. On the one hand, we have worked towards developing a MGS form applicable to situations where the symplectic manifold is a cotangent bundle, and used this tool to attack objective A. Reduction Theory, therefore studying the reduced spaces of these fibered space and the interaction between the fixed points of the group action and the singularities of the reduced spaces. On the other hand, we have rewritten Hamilton's equations for a generic symmetric Hamiltonian system in the coordinates offered by the MGS form in order to attack objective B. Hamiltonian Relative Equilibria, therefore obtaining a universal local model for any Hamiltonian system with symmetry which would allows us to study the natural questions of qualitative local dynamics of these systems.

Main Results

----------------

Objective A. We have obtained a MGS form specific for cotangent bundles and have successfully used it towards obtaining a stratified description of their symplectic quotients at arbitrary momenta. Additionally we have developed a theory of gauging Dirac structures that would be applicable to the study of the Poisson quotients of cotangent bundles. Also we have obtained a reduction theory for symplectic and prolonged Lie algebroids.

Objective B. Using the MGS local approach we have provided a common framework that has been used to prove, in a unified way, virtually every previous result about relative equilibria, and also advance the theory with new results. As an example of this, we have given precise conditions on which a formally stable branch of relative equilibria can exhibit bifurcations for every parameter value, and linked this with the classic example of the sleeping Lagrange top, giving the first explanation for the fast-superfast transition observed in 1992.

Ongoing Work and Future Results

---------------------------------------------------------

As ongoing work and future results we can expect a complete description of the singular reduced symplectic and Poisson spaces for actions on cotangent bundles together with a geometric description of their strata. This is already very advanced and it is likely that the developed theory of gauged Dirac structures will be a key ingredient. Also, we expect to apply to concrete case studies the theory of Hamiltonian relative equilibria developed during this project. Two potential candidates are the Levitron, and the problem of Riemann Ellipsoids governing planetary evolution. In both cases the complete description of the stability and bifurcations of their relative equilibria has been classically studied but is not completely well understood and therefore they constitute perfect examples for testing and applying our results.

The objectives of the EU funded project SILGA were organized along two main lines in the general framework of Hamiltonian actions of Lie groups in symplectic geometry.

The study of symmetries of differential equations and dynamical systems has a long story going back to Newton, Lagrange, Hamilton, Poincare and other masters of the last centuries. In the modern formulations of physics and mechanics it is shown that some geometric structures, such as symplectic and Poisson structures, play a fundamental role in these fields, and that the correct geometric picture of the situation corresponds, mathematically, to a Lie group acting on different smooth manifolds preserving one of these geometric objects. The smooth manifold realizes the phase space or “physical arena” where the dynamics or the physical theory lives. The geometric structure is the distinguishing feature of the theory, and the action of the Lie group preserving the geometric structure realizes the symmetries of the theory. This far reaching interpretation has significantly evolved since then, and the geometry of these Lie group actions has emerged as the fundamental object of study in this field. From a global point of view, the main construction is know as reduction theory, and it consists in the process of elimination of these symmetries, constructing from the original models new simpler, and lower dimensional manifolds, which still encode all the mechanical and physical properties of the systems under study. From the local point of view, the main interest lies in how using the symmetries of the problem, we can give useful information of a given mechanical or physical system in a small neighbourhood of a solution, obtaining then important qualitative information, such as long-term stability and bifurcations, which is extremely important in applications, like control theory or engineering. This local study is a particularly active research topic these days. As for the reduction theory, the current trends are focused in applying the techniques previously used to deal with symplectic and Poisson structures, to other related geometric objects, such as cotangent bundles, Dirac structures, generalized complex geometry, or groupoids, which happen to be extremely relevant to new developments in mathematical physics, like supersymmetry and string theory. The global and local aspects of the research are articulated into two main different objectives.

A. Reduction Theory, and

B.Hamiltonian Relative Equilibria.

Description of the Work

----------------------------------

The strategy employed towards the objectives of this project has been a systematic use of semilocal methods based on the Marle-Guillemin-Sternberg normal form (or MGS for short). The MGS form provides a tubular neighborhood of an orbit of a Lie group acting on a symplectic manifold by symplectomorphisms. In this model both the geometry and the action have very simple expressions, which facilitate their study and interactions. Since all the geometric and dynamic problems involved in this problem are local, this approach has been very well suited to this research.

The implementation is actually very simple. On the one hand, we have worked towards developing a MGS form applicable to situations where the symplectic manifold is a cotangent bundle, and used this tool to attack objective A. Reduction Theory, therefore studying the reduced spaces of these fibered space and the interaction between the fixed points of the group action and the singularities of the reduced spaces. On the other hand, we have rewritten Hamilton's equations for a generic symmetric Hamiltonian system in the coordinates offered by the MGS form in order to attack objective B. Hamiltonian Relative Equilibria, therefore obtaining a universal local model for any Hamiltonian system with symmetry which would allows us to study the natural questions of qualitative local dynamics of these systems.

Main Results

----------------

Objective A. We have obtained a MGS form specific for cotangent bundles and have successfully used it towards obtaining a stratified description of their symplectic quotients at arbitrary momenta. Additionally we have developed a theory of gauging Dirac structures that would be applicable to the study of the Poisson quotients of cotangent bundles. Also we have obtained a reduction theory for symplectic and prolonged Lie algebroids.

Objective B. Using the MGS local approach we have provided a common framework that has been used to prove, in a unified way, virtually every previous result about relative equilibria, and also advance the theory with new results. As an example of this, we have given precise conditions on which a formally stable branch of relative equilibria can exhibit bifurcations for every parameter value, and linked this with the classic example of the sleeping Lagrange top, giving the first explanation for the fast-superfast transition observed in 1992.

Ongoing Work and Future Results

---------------------------------------------------------

As ongoing work and future results we can expect a complete description of the singular reduced symplectic and Poisson spaces for actions on cotangent bundles together with a geometric description of their strata. This is already very advanced and it is likely that the developed theory of gauged Dirac structures will be a key ingredient. Also, we expect to apply to concrete case studies the theory of Hamiltonian relative equilibria developed during this project. Two potential candidates are the Levitron, and the problem of Riemann Ellipsoids governing planetary evolution. In both cases the complete description of the stability and bifurcations of their relative equilibria has been classically studied but is not completely well understood and therefore they constitute perfect examples for testing and applying our results.