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Tudor S. Ratiu

Publications and source records attributed to Tudor S. Ratiu.

At least 55 records · Page 3Linked to original sources
arXiv2010

Exact geometric theory of dendronized polymer dynamics

Dendronized polymers consist of an elastic backbone with a set of iterated branch structures (dendrimers)attached at every base point of the backbone. The conformations of such molecules depend on the elastic deformation of the backbone and the branches, as well as on nonlocal (e.g., electrostatic, or Lennard-Jones) interactions between the elementary molecular units comprising the dendrimers and/or backbone. We develop a geometrically exact theory for the dynamics of such polymers, taking into account both local (elastic) and nonlocal interactions. The theory is based on applying symmetry reduction of Hamilton's principle for a Lagrangian defined on the tangent bundle of iterated semidirect products of the rotation groups that represent the relative orientations of the dendritic branches of the polymer. The resulting symmetry-reduced equations of motion are written in conservative form.

nlin.CD↗
arXiv2010

Applying Hodge theory to detect Hamiltonian flows

We prove that when Hodge theory survives on non-compact symplectic manifolds, a compact symplectic Lie group action having fixed points is necessarily Hamiltonian, provided the associated almost complex structure preserves the space of harmonic one-forms. For example, this is the case for complete Kähler manifolds for which the symplectic form has an appropriate decay at infinity. This extends a classical theorem of Frankel for compact Kähler manifolds to complete non-compact Kähler manifolds.

math.SG↗
arXiv2009

Lagrange-Poincare field equations

The Lagrange-Poincare equations of classical mechanics are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The Kelvin-Noether theorem is formulated in this context. Applications to the isoperimetric problem, the Skyrme model for meson interaction, metamorphosis image dynamics, and molecular strands illustrate various aspects of the theory.

nlin.CD↗
arXiv2009

Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids

The Lagrangian and Hamiltonian structures for an ideal gauge-charged fluid are determined. Using a Kaluza-Klein point of view, the equations of motion are obtained by Lagrangian and Poisson reductions associated to the automorphism group of a principal bundle. As a consequence of the Lagrangian approach, a Kelvin-Noether theorem is obtained. The Hamiltonian formulation determines a non-canonical Poisson bracket associated to these equations.

math-ph↗
arXiv2009

Affine Lie-Poisson Reduction, Yang-Mills magnetohydrodynamics, and superfluids

This paper develops the theory of affine Lie-Poisson reduction and applies this process to Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids. As a consequence of this approach, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples.

math-ph↗
arXiv2009

The Geometric Structure of Complex Fluids

This paper develops the theory of affine Euler-Poincaré and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids, spin glasses, microfluids, and liquid crystals. As a consequence of the Lagrangian approach, the variational formulation of the equations is determined. On the Hamiltonian side, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples.

math-ph↗
arXiv2009

The momentum map in Poisson geometry

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.

math.SG↗
arXiv2009

Dynamics of charged molecular strands

Euler-Poincare equations are derived for the dynamical folding of charged molecular strands (such as DNA) modeled as flexible continuous filamentary distributions of interacting rigid charge conformations. The new feature is that the equations of motion for the dynamics of such molecular strands are nonlocal when the screened Coulomb interactions, or Lennard-Jones potentials between pairs of charges are included. These nonlocal dynamical equations are derived in the convective representation of continuum motion by using modified Euler-Poincare and Hamilton-Pontryagin variational formulations that illuminate the various approaches within the framework of symmetry reduction of Hamilton's principle for exact geometric rods. In the absence of nonlocal interactions, the equations recover the classical Kirchhoff theory of elastic rods in the spatial representation. The motion equations in the convective representation are shown to be affine Euler-Poincare equations relative to a certain cocycle. This property relates the geometry of the molecular strands to that of complex fluids. An elegant change of variables allows a direct passage from the infinite dimensional point of view to the covariant formulation in terms of Lagrange-Poincare equations. In another revealing perspective, the convective representation of the nonlocal equations of molecular strand motion is transformed into quaternionic form.

nlin.AO↗
arXiv2006

Openness and convexity for momentum maps

The purpose of this paper is finding the essential attributes underlying the convexity theorems for momentum maps. It is shown that they are of 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 "Lokal-global-Prinzip" 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 non-compact manifolds.

math.SG↗
arXiv2006

Metric convexity in the symplectic category

We introduce an extension of the standard Local-to-Global Principle used in the proof of the convexity theorems for the momentum map to handle closed maps that take values in a length metric space. This extension is used to study the convexity properties of the cylinder valued momentum map introduced by Condevaux, Dazord, and Molino and allows us to obtain the most general convexity statement available in the literature for momentum maps associated to a symplectic Lie group action.

math.SG↗
arXiv2006

The restricted Grassmannian, Banach Lie-Poisson spaces, and coadjoint orbits

We investigate some basic questions concerning the relationship between the restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using universal central extensions of Lie algebras, we find that the restricted Grassmannian is symplectomorphic to symplectic leaves in certain Banach Lie-Poisson spaces, and the underlying Banach space can be chosen to be even a Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established. Several pathological properties of the restricted algebra are pointed out.

math.DG↗
arXiv2006

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

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 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.

math-ph↗
arXiv2006

On the symmetry breaking phenomenon

We investigate the problem of symmetry breaking in the framework of dynamical systems with symmetry on a smooth manifold. Two cases will be analyzed: general and Hamiltonian dynamical systems. We give sufficient conditions for symmetry breaking in both cases.

math.DS↗
arXiv2005

Symmetry and symplectic reduction

This encyclopedia article briefly reviews without proofs some of the main results in symplectic reduction. The article recalls most the necessary prerequisites to understand the main results, namely, group actions, momentum maps, and coadjoint orbits, among others.

math.SG↗
arXiv2005

Poisson reduction

This encyclopedia article briefly reviews without proofs some of the main results in Poisson reduction. The article recalls most the necessary prerequisites to understand the main results.

math.SG↗
arXiv2005

Cotangent bundle reduction

This encyclopedia article briefly reviews without proofs some of the main results in cotangent bundle reduction. The article recalls most the necessary prerequisites to understand the main results.

math.SG↗
arXiv2005

Singular cosphere bundle reduction

This paper studies singular contact reduction for cosphere bundles at the zero value of the momentum map. A stratification of the singular quotient, finer than the contact one and better adapted to the bundle structure of the problem, is obtained. The strata of this new stratification are a collection of cosphere bundles and coisotropic or Legendrian submanifolds of their corresponding contact components.

math.SG↗