We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings fr... more We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case Emb(R, R) which turns out to be Burgers' equation. Then we derive the geodesic equation, the curvature, and the Jacobi equation of a right invariant Riemannian metric on an infinite dimensional Lie group, which we apply to Diff(R), Diff(S 1 ), and the Virasoro-Bott group. Many of these results are well known, the emphasis is on conciseness and clarity. Table of contents 1. Introduction .
The purpose of this paper is finding the essential attributes underlying the convexity theorems f... more The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of a topological nature; more specifically, we show that convexity follows if the map is open onto its image and has the so-called local convexity data property. These conditions are satisfied in all the classical convexity theorems and hence they can, in principle, be obtained as corollaries of a more general theorem that has only these two hypotheses. We also prove a generalization of the socalled Local-to-Global Principle that only requires the map to be closed and to have a normal topological space as domain, instead of using a properness condition. This allows us to generalize the Flaschka-Ratiu convexity theorem to noncompact manifolds.
HAL (Le Centre pour la Communication Scientifique Directe), 2002
We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endow... more We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle and Guillemin and Sternberg for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model, the so-called Chu map, can be used instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will ¢nd situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.
The symplectic induction procedure is extended to the case of weak symplectic Banach manifolds. U... more The symplectic induction procedure is extended to the case of weak symplectic Banach manifolds. Using this procedure, one constructs hierarchies of integrable Hamiltonian systems related to the Banach Lie-Poisson spaces of k-diagonal trace class operators.
In this note we clarify the relationship between the local and global definitions of dual pairs i... more In this note we clarify the relationship between the local and global definitions of dual pairs in Poisson geometry. It turns out that these are not equivalent. For the passage from local to global one needs a connected fiber hypothesis (this is well known), while the converse requires a dimension condition (which appears not to be known). We also provide examples illustrating the necessity of the extra conditions.
We generalize the sufficient condition for the stability of relative periodic orbits in symmetric... more We generalize the sufficient condition for the stability of relative periodic orbits in symmetric Hamiltonian systems presented in [J.-P. Ortega, T.S. Ratiu, J. Geom. Phys. 32 (1999) 131-1591 to the case in which these orbits have non-trivial symmetry. We also describe a block diagonalization, similar in philosophy to the one presented in [J.-P. Ortega, T.S. Ratiu, Nonlinearity 12 (1999) 693-7201 for relative equilibria, that facilitates the use of this result in particular examples and shows the relation between the stability of the relative periodic orbit and the orbital stability of the associated singular reduced periodic orbit.
We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endow... more We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle [Mar84, and Guillemin and Sternberg [GS84] for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the construction of the tubular model. The present work shows that in the construction of the tubular model it can be used the so called Chu map [Chu75] instead, which exists for any canonical action, unlike the momentum map. Hamilton's equations for any invariant Hamiltonian function take on a particularly simple form in these tubular variables. As an application we will find situations, that we will call tubewise Hamiltonian, in which the existence of a standard momentum map in invariant neighborhoods is guaranteed.
A Class of Integrable Geodsic Flows on the Symplectic Group
arXiv (Cornell University), Dec 30, 2005
In this paper we show that the left-invariant geodesic flow on the symplectic group with metric g... more In this paper we show that the left-invariant geodesic flow on the symplectic group with metric given by the Frobenius norm is an integrable system that is not contained in the Mishchenko-Fomenko class. We show that this system may be expressed as a flow on symmetric matrices and that the system is bi-Hamiltonian.
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Papers by Tudor Ratiu