Tudor Ratiu | Ecole Polytechnique Federale de Lausanne (original) (raw)

Papers by Tudor Ratiu

Research paper thumbnail of Bifurcation of relative equilibria in mechanical systems with symmetry

Advances in Applied Mathematics, Jul 1, 2003

Research paper thumbnail of Extensions of Banach Lie–Poisson spaces

Journal of Functional Analysis, Dec 1, 2004

Research paper thumbnail of Hamiltonian Hopf Bifurcation with Symmetry

Archive for Rational Mechanics and Analysis, May 1, 2002

Research paper thumbnail of The momentum map in Poisson geometry

American Journal of Mathematics, 2009

Research paper thumbnail of Asymptotic and Lyapunov stability of Poisson equilibria

arXiv (Cornell University), Apr 26, 2004

Research paper thumbnail of The Geometric Nature of the Flaschka Transformation

Communications in Mathematical Physics, Mar 20, 2017

Research paper thumbnail of A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems

Journal of Geometry and Physics, Dec 1, 1999

Research paper thumbnail of On the geometry of saddle point algorithms

There has been great deal of innovative work in recent years relating discrete algorithms to cont... more There has been great deal of innovative work in recent years relating discrete algorithms to continuous flows. Of particular interest are flows which are gradient flows or Hamiltonian flows. Hamiltonian flows do not have asymptotically stable equilibria, but a restriction of the system to a certain set of variables may have such an equilibrium. In nonlinear optimization and game theory there is an interest in systems with saddle point equilibria. The authors show that certain flows with such equilibria can be both Hamiltonian and gradient and discuss the relationship of such flows with the gradient method for finding saddle points in nonlinear optimization problems. These results are compared with gradient flows associated with the Toda lattice.<<ETX>>

Research paper thumbnail of Pseudogroups and Groupoids

The action \(\Phi :{\kern 1pt} \,\;G \times M \to M\) of Lie group G on a manifold M can be seen ... more The action \(\Phi :{\kern 1pt} \,\;G \times M \to M\) of Lie group G on a manifold M can be seen as the choice of a subgroup \({A_G}: = \{ {\Phi _g}|g \in G\}\) of Diff(M), that is, the globally defined diffeomorphisms of M. There are mathematical structures, such as distributions and foliations, where the transformations of the manifold M that naturally appear in the problem are only locally defined. It is in the study of those structures that the objects constituting the subject of this chapter become relevant.

Research paper thumbnail of Singular Reduction and the Stratification Theorem

This chapter studies the structure of the symplectic reduced spaces introduced in Chapter 6 when ... more This chapter studies the structure of the symplectic reduced spaces introduced in Chapter 6 when the hypothesis on the freeness of the canonical group action is dropped. In this new scenario, standard momentum maps are not submersions anymore and consequently, the reduced spaces are not necessarily smooth manifolds, but just quotient topological spaces. The main result proved here shows that these quotients are symplectic Whitney stratified spaces in the sense that the strata are symplectic manifolds in a very natural way; moreover, the local properties of this Whitney stratification make it into a cone space in the sense of Definition 1.7.3. This statement is referred to as the Symplectic Stratification Theorem. This symplectic stratification is well adapted to the study of G-invariant dynamics since the flows of Hamiltonian vector fields associated to G-invariant Hamiltonian functions naturally reduce to Hamiltonian systems on these strata.

Research paper thumbnail of The reduced spaces of a symplectic Lie group action

Annals of Global Analysis and Geometry, Aug 22, 2006

Research paper thumbnail of The Banach Poisson geometry of multi-diagonal Toda-like lattices

arXiv (Cornell University), Oct 20, 2003

Research paper thumbnail of Curvature of the Virasoro-Bott group

arXiv (Cornell University), Jan 26, 1998

Research paper thumbnail of The Banach Poisson geometry of the infinite Toda lattice

arXiv (Cornell University), Oct 20, 2003

The rigorous functional analytic description of the infinite Toda lattice is presented in the fra... more The rigorous functional analytic description of the infinite Toda lattice is presented in the framework of the Banach Lie-Poisson structure of trace class operators. The generic coadjoint orbits of the Banach Lie group of bidiagonal bounded operators are studied. It is shown that the infinite dimensional generalization of the Flaschka map is a momentum map.

Research paper thumbnail of A Class of Integrable Geodesic Flows on the Symplectic Group and the Symmetric Matrices

arXiv (Cornell University), Dec 30, 2005

Research paper thumbnail of Asymptotic and Lyapunov stability of constrained and Poisson equilibria

Journal of Differential Equations, Jul 1, 2005

Research paper thumbnail of A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map

Bulletin of the American Mathematical Society, 1990

Research paper thumbnail of Banach Lie-Poisson Spaces and Reduction

Communications in Mathematical Physics, Nov 1, 2003

Research paper thumbnail of Banach Lie-Poisson Spaces

WORLD SCIENTIFIC eBooks, 2005

Research paper thumbnail of Integration on Manifolds

Applied mathematical sciences, 1988

The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ... more The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ℝn using a partition of unity subordinate to an atlas. The change-of-variables theorem guarantees that the integral is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral calculus, the change-of-variables theorem and Stokes’ theorem, are discussed in detail along with some applications.

Research paper thumbnail of Bifurcation of relative equilibria in mechanical systems with symmetry

Advances in Applied Mathematics, Jul 1, 2003

Research paper thumbnail of Extensions of Banach Lie–Poisson spaces

Journal of Functional Analysis, Dec 1, 2004

Research paper thumbnail of Hamiltonian Hopf Bifurcation with Symmetry

Archive for Rational Mechanics and Analysis, May 1, 2002

Research paper thumbnail of The momentum map in Poisson geometry

American Journal of Mathematics, 2009

Research paper thumbnail of Asymptotic and Lyapunov stability of Poisson equilibria

arXiv (Cornell University), Apr 26, 2004

Research paper thumbnail of The Geometric Nature of the Flaschka Transformation

Communications in Mathematical Physics, Mar 20, 2017

Research paper thumbnail of A Dirichlet criterion for the stability of periodic and relative periodic orbits in Hamiltonian systems

Journal of Geometry and Physics, Dec 1, 1999

Research paper thumbnail of On the geometry of saddle point algorithms

There has been great deal of innovative work in recent years relating discrete algorithms to cont... more There has been great deal of innovative work in recent years relating discrete algorithms to continuous flows. Of particular interest are flows which are gradient flows or Hamiltonian flows. Hamiltonian flows do not have asymptotically stable equilibria, but a restriction of the system to a certain set of variables may have such an equilibrium. In nonlinear optimization and game theory there is an interest in systems with saddle point equilibria. The authors show that certain flows with such equilibria can be both Hamiltonian and gradient and discuss the relationship of such flows with the gradient method for finding saddle points in nonlinear optimization problems. These results are compared with gradient flows associated with the Toda lattice.<<ETX>>

Research paper thumbnail of Pseudogroups and Groupoids

The action \(\Phi :{\kern 1pt} \,\;G \times M \to M\) of Lie group G on a manifold M can be seen ... more The action \(\Phi :{\kern 1pt} \,\;G \times M \to M\) of Lie group G on a manifold M can be seen as the choice of a subgroup \({A_G}: = \{ {\Phi _g}|g \in G\}\) of Diff(M), that is, the globally defined diffeomorphisms of M. There are mathematical structures, such as distributions and foliations, where the transformations of the manifold M that naturally appear in the problem are only locally defined. It is in the study of those structures that the objects constituting the subject of this chapter become relevant.

Research paper thumbnail of Singular Reduction and the Stratification Theorem

This chapter studies the structure of the symplectic reduced spaces introduced in Chapter 6 when ... more This chapter studies the structure of the symplectic reduced spaces introduced in Chapter 6 when the hypothesis on the freeness of the canonical group action is dropped. In this new scenario, standard momentum maps are not submersions anymore and consequently, the reduced spaces are not necessarily smooth manifolds, but just quotient topological spaces. The main result proved here shows that these quotients are symplectic Whitney stratified spaces in the sense that the strata are symplectic manifolds in a very natural way; moreover, the local properties of this Whitney stratification make it into a cone space in the sense of Definition 1.7.3. This statement is referred to as the Symplectic Stratification Theorem. This symplectic stratification is well adapted to the study of G-invariant dynamics since the flows of Hamiltonian vector fields associated to G-invariant Hamiltonian functions naturally reduce to Hamiltonian systems on these strata.

Research paper thumbnail of The reduced spaces of a symplectic Lie group action

Annals of Global Analysis and Geometry, Aug 22, 2006

Research paper thumbnail of The Banach Poisson geometry of multi-diagonal Toda-like lattices

arXiv (Cornell University), Oct 20, 2003

Research paper thumbnail of Curvature of the Virasoro-Bott group

arXiv (Cornell University), Jan 26, 1998

Research paper thumbnail of The Banach Poisson geometry of the infinite Toda lattice

arXiv (Cornell University), Oct 20, 2003

The rigorous functional analytic description of the infinite Toda lattice is presented in the fra... more The rigorous functional analytic description of the infinite Toda lattice is presented in the framework of the Banach Lie-Poisson structure of trace class operators. The generic coadjoint orbits of the Banach Lie group of bidiagonal bounded operators are studied. It is shown that the infinite dimensional generalization of the Flaschka map is a momentum map.

Research paper thumbnail of A Class of Integrable Geodesic Flows on the Symplectic Group and the Symmetric Matrices

arXiv (Cornell University), Dec 30, 2005

Research paper thumbnail of Asymptotic and Lyapunov stability of constrained and Poisson equilibria

Journal of Differential Equations, Jul 1, 2005

Research paper thumbnail of A new formulation of the generalized Toda lattice equations and their fixed point analysis via the momentum map

Bulletin of the American Mathematical Society, 1990

Research paper thumbnail of Banach Lie-Poisson Spaces and Reduction

Communications in Mathematical Physics, Nov 1, 2003

Research paper thumbnail of Banach Lie-Poisson Spaces

WORLD SCIENTIFIC eBooks, 2005

Research paper thumbnail of Integration on Manifolds

Applied mathematical sciences, 1988

The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ... more The integral of an n-form on an n-manifold is defined by piecing together integrals over sets in ℝn using a partition of unity subordinate to an atlas. The change-of-variables theorem guarantees that the integral is well defined, independent of the choice of atlas and partition of unity. Two basic theorems of integral calculus, the change-of-variables theorem and Stokes’ theorem, are discussed in detail along with some applications.