Tudor Ratiu - Profile on Academia.edu (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

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of th... more The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [Nonlinearity 11 (1998) 1637-1649] is generalized to the framework of Abelian proper actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a study of bifurcations with maximal isotropy are presented. An elementary example illustrates the use of this approach.

Research paper thumbnail of Extensions of Banach Lie–Poisson spaces

Journal of Functional Analysis, Dec 1, 2004

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special cl... more The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W Ã-algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.

Research paper thumbnail of Hamiltonian Hopf Bifurcation with Symmetry

Archive for Rational Mechanics and Analysis, May 1, 2002

In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf... more In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.

Research paper thumbnail of The momentum map in Poisson geometry

American Journal of Mathematics, 2009

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on it... more Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.

Research paper thumbnail of Asymptotic and Lyapunov stability of Poisson equilibria

arXiv (Cornell University), Apr 26, 2004

This paper includes results centered around three topics, all of them related with the nonlinear ... more This paper includes results centered around three topics, all of them related with the nonlinear stability of equilibria in Poisson dynamical systems. Firstly, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account certain asymptotically stable behavior that may occur in the Poisson category. This method is adapted to Poisson systems obtained via a reduction procedure and we show in examples that the kind of stability that we propose is appropriate when dealing with the stability of the equilibria of some constrained systems. Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick et al. [Paal02].

Research paper thumbnail of The Geometric Nature of the Flaschka Transformation

Communications in Mathematical Physics, Mar 20, 2017

We show that the Flaschka map, originally introduced to analyze the dynamics of the integrable To... more We show that the Flaschka map, originally introduced to analyze the dynamics of the integrable Toda lattice system, is the inverse of a momentum map. We discuss the geometrical setting of the map and apply it to the generalized Toda lattice systems on semisimple Lie algebras, the rigid body system on Toda orbits, and to coadjoint orbits of semidirect products groups. In addition, we develop an infinite-dimensional generalization for the group of area preserving diffeomorphisms of the annulus and apply it to the analysis of the dispersionless Toda lattice PDE and the solvable rigid body PDE.

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

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.

Research paper thumbnail of On the geometry of saddle point algorithms

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

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

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

There exist three main approaches to reduction associated with canonical Lie group actions on a s... more There exist three main approaches to reduction associated with canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case, it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.

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

arXiv (Cornell University), Oct 20, 2003

The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integral... more The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integrals in involution is studied. It is shown that these systems can be considered as generalizing the semi-infinite Toda lattice which is an example of a bidiagonal system, a case to which special attention is given. 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. Action-angle variables for the Toda system are constructed.

Research paper thumbnail of Curvature of the Virasoro-Bott group

arXiv (Cornell University), Jan 26, 1998

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.

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

This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Fr... more This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Frobenius metric is an integrable system that is not contained in the Mishchenko-Fomenko class of rigid body metrics. This system may be expressed as a flow on symmetric matrices and is bi-Hamiltonian. This analysis is extended to cover flows on symmetric matrices when an isomorphism * Research partially supported by the NSF. † Research partially supported by the California Institute of Technology and NSF. ‡ Research partially supported by the Swiss NSF. 1 Introduction 2 with the symplectic Lie algebra does not hold. The two Poisson structures associated with this system, including an analysis of its Casimirs, are completely analyzed. Since the system integrals are not generated by its Casimirs it is shown that the nature of integrability is fundamentally different from that exhibited in the Mischenko-Fomenko setting.

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

Journal of Differential Equations, Jul 1, 2005

This paper includes results centered around three topics, all of them related with the nonlinear ... more This paper includes results centered around three topics, all of them related with the nonlinear stability of equilibria in constrained dynamical systems. First, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account the asymptotically stable behavior that may occur in certain constrained systems, such as Poisson and Leibniz dynamical systems. Second, this method is specifically adapted to Poisson systems obtained via a reduction procedure and we show in examples that the kind of stability that we propose is appropriate when dealing with the stability of the equilibria of some constrained mechanical systems. Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick et al. (Arch. Rational Mech. Anal., 2004, preprint arXiv:math.DS/0201239v1, to appear).

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

The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the categor... more The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W *-algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are given. The relationship between classical and quantum reduction is discussed.

commutes. By a theorem of Wigner, the automorphism » is of the form =(p) = UpU*, where U is a unitary or anti-unitary operator on M. Due to the hypothesis that im K is linearly dense in 6, if such an automorphism © exists, it is necessarily unique. It is natural to interpret © as the quantization of o. Denote by Aut(h!(M)) the linear Poisson isomorphisms of h'(M). The set of all Poisson diffeomorphisms o for which a  » as above exists, form a subgroup Diffx(P, {-,-}) of the Poisson diffeomorphism group Diff(P, {-,-}) of P. The map

Research paper thumbnail of Banach Lie-Poisson Spaces

Banach Lie-Poisson Spaces

WORLD SCIENTIFIC eBooks, 2005

Research paper thumbnail of Integration on Manifolds

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

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of th... more The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a neighborhood of a given relative equilibrium. The structure of the reduced equations is studied in a few relevant situations. In particular, a persistence result of Lerman and Singer [Nonlinearity 11 (1998) 1637-1649] is generalized to the framework of Abelian proper actions. Also, a Hamiltonian version of the Equivariant Branching Lemma and a study of bifurcations with maximal isotropy are presented. An elementary example illustrates the use of this approach.

Research paper thumbnail of Extensions of Banach Lie–Poisson spaces

Journal of Functional Analysis, Dec 1, 2004

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special cl... more The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W Ã-algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.

Research paper thumbnail of Hamiltonian Hopf Bifurcation with Symmetry

Archive for Rational Mechanics and Analysis, May 1, 2002

In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf... more In this paper we study the appearance of branches of relative periodic orbits in Hamiltonian Hopf bifurcation processes in the presence of compact symmetry groups that do not generically exist in the dissipative framework. The theoretical study is illustrated with several examples.

Research paper thumbnail of The momentum map in Poisson geometry

American Journal of Mathematics, 2009

Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on it... more Every action on a Poisson manifold by Poisson diffeomorphisms lifts to a Hamiltonian action on its symplectic groupoid which has a canonically defined momentum map. We study various properties of this momentum map as well as its use in reduction.

Research paper thumbnail of Asymptotic and Lyapunov stability of Poisson equilibria

arXiv (Cornell University), Apr 26, 2004

This paper includes results centered around three topics, all of them related with the nonlinear ... more This paper includes results centered around three topics, all of them related with the nonlinear stability of equilibria in Poisson dynamical systems. Firstly, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account certain asymptotically stable behavior that may occur in the Poisson category. This method is adapted to Poisson systems obtained via a reduction procedure and we show in examples that the kind of stability that we propose is appropriate when dealing with the stability of the equilibria of some constrained systems. Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick et al. [Paal02].

Research paper thumbnail of The Geometric Nature of the Flaschka Transformation

Communications in Mathematical Physics, Mar 20, 2017

We show that the Flaschka map, originally introduced to analyze the dynamics of the integrable To... more We show that the Flaschka map, originally introduced to analyze the dynamics of the integrable Toda lattice system, is the inverse of a momentum map. We discuss the geometrical setting of the map and apply it to the generalized Toda lattice systems on semisimple Lie algebras, the rigid body system on Toda orbits, and to coadjoint orbits of semidirect products groups. In addition, we develop an infinite-dimensional generalization for the group of area preserving diffeomorphisms of the annulus and apply it to the analysis of the dispersionless Toda lattice PDE and the solvable rigid body PDE.

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

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.

Research paper thumbnail of On the geometry of saddle point algorithms

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

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

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

There exist three main approaches to reduction associated with canonical Lie group actions on a s... more There exist three main approaches to reduction associated with canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux et al. In this case, it will be shown that the three reduced spaces mentioned above do not coincide, in general. More specifically, the Marsden-Weinstein reduced spaces are not symplectic but Poisson and their symplectic leaves are given by the optimal reduced spaces. Foliation reduction produces a symplectic reduced space whose Poisson quotient by a certain Lie group associated to the group of symmetries of the problem equals the Marsden-Weinstein reduced space. We illustrate these constructions with concrete examples, special emphasis being given to the reduction of a magnetic cotangent bundle of a Lie group in the situation when the magnetic term ensures the non-existence of the momentum map for the lifted action. The precise relation of the cylinder valued momentum map with group valued momentum maps for Abelian Lie groups is also given.

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

arXiv (Cornell University), Oct 20, 2003

The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integral... more The Banach Poisson geometry of multi-diagonal Hamiltonian systems having infinitely many integrals in involution is studied. It is shown that these systems can be considered as generalizing the semi-infinite Toda lattice which is an example of a bidiagonal system, a case to which special attention is given. 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. Action-angle variables for the Toda system are constructed.

Research paper thumbnail of Curvature of the Virasoro-Bott group

arXiv (Cornell University), Jan 26, 1998

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.

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

This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Fr... more This paper shows that the left-invariant geodesic flow on the symplectic group relative to the Frobenius metric is an integrable system that is not contained in the Mishchenko-Fomenko class of rigid body metrics. This system may be expressed as a flow on symmetric matrices and is bi-Hamiltonian. This analysis is extended to cover flows on symmetric matrices when an isomorphism * Research partially supported by the NSF. † Research partially supported by the California Institute of Technology and NSF. ‡ Research partially supported by the Swiss NSF. 1 Introduction 2 with the symplectic Lie algebra does not hold. The two Poisson structures associated with this system, including an analysis of its Casimirs, are completely analyzed. Since the system integrals are not generated by its Casimirs it is shown that the nature of integrability is fundamentally different from that exhibited in the Mischenko-Fomenko setting.

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

Journal of Differential Equations, Jul 1, 2005

This paper includes results centered around three topics, all of them related with the nonlinear ... more This paper includes results centered around three topics, all of them related with the nonlinear stability of equilibria in constrained dynamical systems. First, we prove an energy-Casimir type sufficient condition for stability that uses functions that are not necessarily conserved by the flow and that takes into account the asymptotically stable behavior that may occur in certain constrained systems, such as Poisson and Leibniz dynamical systems. Second, this method is specifically adapted to Poisson systems obtained via a reduction procedure and we show in examples that the kind of stability that we propose is appropriate when dealing with the stability of the equilibria of some constrained mechanical systems. Finally, we discuss two situations in which the use of continuous Casimir functions in stability studies is equivalent to the topological stability methods introduced by Patrick et al. (Arch. Rational Mech. Anal., 2004, preprint arXiv:math.DS/0201239v1, to appear).

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

The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the categor... more The category of Banach Lie-Poisson spaces is introduced and studied. It is shown that the category of W *-algebras can be considered as one of its subcategories. Examples and applications of Banach Lie-Poisson spaces to quantization and integration of Hamiltonian systems are given. The relationship between classical and quantum reduction is discussed.

commutes. By a theorem of Wigner, the automorphism » is of the form =(p) = UpU*, where U is a unitary or anti-unitary operator on M. Due to the hypothesis that im K is linearly dense in 6, if such an automorphism © exists, it is necessarily unique. It is natural to interpret © as the quantization of o. Denote by Aut(h!(M)) the linear Poisson isomorphisms of h'(M). The set of all Poisson diffeomorphisms o for which a  » as above exists, form a subgroup Diffx(P, {-,-}) of the Poisson diffeomorphism group Diff(P, {-,-}) of P. The map

Research paper thumbnail of Banach Lie-Poisson Spaces

Banach Lie-Poisson Spaces

WORLD SCIENTIFIC eBooks, 2005

Research paper thumbnail of Integration on Manifolds

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.