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>>
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.
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.
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.
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.
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>>
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.
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.
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.
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.
Uploads
Papers by Tudor Ratiu