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The Resource Geometry, Mechanics, and Dynamics : The Legacy of Jerry Marsden

Geometry, Mechanics, and Dynamics : The Legacy of Jerry Marsden

Label
Geometry, Mechanics, and Dynamics : The Legacy of Jerry Marsden
Title
Geometry, Mechanics, and Dynamics
Title remainder
The Legacy of Jerry Marsden
Creator
Contributor
Subject
Language
eng
Summary
This book illustrates the broad range of Jerry Marsden's mathematical legacy in areas of geometry, mechanics, and dynamics, from very pure mathematics to very applied, but always with a geometric perspective. Each contribution develops its material from the viewpoint of geometric mechanics beginning at the very foundations, introducing readers to modern issues via illustrations in a wide range of topics. The twenty refereed papers contained in this volume are based on lectures and research performed during the month of July 2012 at the Fields Institute for Research in Mathematical Sciences, in a program in honor of Marsden's legacy. The unified treatment of the wide breadth of topics treated in this book will be of interest to both experts and novices in geometric mechanics. Experts will recognize applications of their own familiar concepts and methods in a wide variety of fields, some of which they may never have approached from a geometric viewpoint. Novices may choose topics that interest them among the various fields and learn about geometric approaches and perspectives toward those topics that will be new for them as well
Member of
Cataloging source
MiAaPQ
Literary form
non fiction
Nature of contents
dictionaries
Series statement
Fields Institute Communications Ser.
Series volume
v.73
Geometry, Mechanics, and Dynamics : The Legacy of Jerry Marsden
Label
Geometry, Mechanics, and Dynamics : The Legacy of Jerry Marsden
Link
http://libproxy.rpi.edu/login?url=https://ebookcentral.proquest.com/lib/rpi/detail.action?docID=2095797
Publication
Copyright
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Carrier category
online resource
Carrier category code
cr
Carrier MARC source
rdacarrier
Color
multicolored
Content category
text
Content type code
txt
Content type MARC source
rdacontent
Contents
  • Preface to Fields Volume -- Contents -- A Global Version of the Koon-Marsden Jacobiator Formula -- 1 Introduction -- 2 Nonholonomic Systems -- 2.1 The Hamiltonian Viewpoint -- 2.2 The Nonholonomic Bracket -- 2.3 The Jacobiator Formula -- 3 The Koon-Marsden Adapted Coordinates -- 4 The Coordinate Version of the Jacobiator Formula -- 4.1 Interpretation of the Adapted Coordinates -- 4.2 The Jacobiator in Adapted Coordinates -- 5 Example: The Snakeboard -- References -- Geometry of Image Registration: The Diffeomorphism Group and Momentum Maps -- 1 Introduction -- 1.1 Outline of the Notes -- 1.2 Image Registration with LDDMM -- 1.3 Anatomical Shape and Function -- 1.3.1 Alzheimer's Disease and the Shape of Subcortical Structures -- 1.4 Analysis of Longitudinal Data -- Longitudinal Study of the Shape of Hippocampi -- 1.5 Propagation of Anatomical Information -- Automatic Labelling via Ontologies -- Patient-Specific Models for Atrial Fibrillation McDowell2012 -- 1.6 Other Applications -- 2 Geometry of Matching Problems -- 2.1 Abstract Formulation -- 2.2 The Adjoint Action -- 2.3 The Coadjoint Action -- 2.4 Variations of the Flow -- 2.5 The Momentum Map -- 2.6 Momentum Maps in Geometric Mechanics -- 2.7 Tangent and Cotangent Lifted Actions -- 2.8 The -Map -- 2.9 Derivative of the Matching Energy -- 2.10 Image Matching -- 2.11 Conservation of Momentum -- 2.12 Differentiating Ad and Ad -- 2.13 The Euler-Poincaré Equation -- 2.14 The EPDiff Equation -- 2.15 Momentum Map for Image Matching -- 2.16 Evolution Equations on TV -- 2.17 Evolution Equations for Image Matching -- 2.18 Matching via Initial Momentum -- 2.19 Interpretation via Riemannian Geometry -- 2.20 Riemannian Geometry on V -- 3 Existence of Solutions for Image Registration -- References -- Multisymplectic Geometry and Lie Groupoids -- 1 Introduction
  • 2 Poisson Structures and Symplectic Groupoids -- 3 Multisymplectic Structures -- 4 Multisymplectic Groupoids and Their Infinitesimal Versions -- 4.1 The Case k=1 -- 5 Descriptions of k-Poisson Structures -- 6 Some Examples and Final Remarks -- References -- The Topology of Change: Foundations of Probabilitywith Black Swans -- 1 Introduction -- 2 The Mathematics of Uncertainty -- 3 Rare Events and Change -- 4 New Axioms for Probability Theory: The Topology of Change -- 5 Existence and Representation Theorems -- 6 Heavy Tails and Families of Purely Finitely Additive Measures -- 6.1 Contrasting Monotone Continuity and the Topology of Change -- 6.2 Heavy Tails -- 6.3 The Family PA of Purely Finitely Additive Measures on R -- 7 The Axiom of Choice and Godel's Incompleteness Theorem -- Appendix -- Example: A Probability that is Biased Against Frequent Events -- Example: The Dual Space L{u221E} Consists of Countably Additive and Finitely Additive Measures -- Example: A Finitely Additive Measure that is Not Countably Additive -- References -- Chaos in the Kepler Problem with Quadrupole Perturbations -- 1 Introduction -- 2 The Melnikov Conditions -- 3 Final Remarks -- References -- Groups of Diffeomorphisms and Fluid Motion: Reprise -- 1 Introduction -- 2 Construction of Equations -- 3 Function Spaces -- 4 Proof of Well-Posedness -- 5 Further Remarks -- References -- Dual Pairs for Non-Abelian Fluids -- 1 Introduction -- Dual Pairs of Momentum Maps -- Plan of the Paper -- 2 The EPAut Equations and Momentum Maps -- 2.1 The Automorphism Group and Euler-Poincaré Equations -- Euler-Poincaré Equations, EPDiff, and EPAut -- 2.2 A Pair of Momentum Maps for the EPAut Equations -- Review of the EPDiff Case -- Momentum Maps for the EPAut Equations -- 2.2.1 Left Action Momentum Map -- 2.2.2 Right Action Momentum Map -- 3 The Dual Pair Property of the EPAut Momentum Maps
  • 3.1 Cotangent Lifted Actions -- 3.1.1 Left Cotangent Action -- 3.1.2 Right Cotangent Action -- 3.2 Transitivity Results -- 4 The Incompressible EPAut Equation and Momentum Maps -- 4.1 The Group of Volume Preserving Automorphisms -- 4.2 Review of the Ideal Fluid Case -- 4.2.1 The Right Action Momentum Map -- 4.2.2 The Left Action Momentum Map -- 4.3 A Pair of Momentum Maps for the EPAutvol Equations -- 4.3.1 Left Action Momentum Map -- 4.3.2 Right Action Momentum Map -- 4.3.3 The Pair of Momentum Maps -- 4.4 Yang-Mills Phase Space -- 4.4.1 Left Action Momentum Map -- 4.4.2 Right Action Momentum Map -- 5 The Dual Pair Property of the EPAutvol Momentum Maps -- 6 Conclusion and Future Works -- References -- The Role of SE(d)-Reduction for Swimming in Stokesand Navier-Stokes Fluids -- 1 Introduction -- 1.1 Main Contributions -- 1.2 Background -- 1.3 Conventions and Notation -- 2 Limit Cycles -- 3 Relative Limit Cycles -- 4 Lagrange-d'Alembert Formalism -- 5 Fluid-Structure Interaction -- 5.1 Navier Stokes Fluids in the Lagrange-d'Alembert Formalism -- 5.2 Solids -- 5.3 Fluid-Solid Interaction -- 5.4 Reduction by Frame Invariance -- 6 Asymptotic Behavior -- 7 Swimming -- 7.1 Analytic Concerns and Approximate Relative Limit Cycles -- 8 Conclusion and Future Work -- References -- Lagrangian Mechanics on Centered Semi-direct Products -- 1 Introduction -- 1.1 Background -- 1.2 Main Contributions -- 1.3 A Motivating Example -- 2 A Centered Semi-direct Product Theory -- 2.1 Preliminary Material on Lie Groups -- 2.1.1 Group Actions -- 2.1.2 Adjoint and Coadjoint Operators -- 2.2 Centered Semi-direct Products -- 3 Euler-Poincaré Theory -- 4 Examples -- 4.1 A Toy Example -- 4.2 An Isotropy Group of a Second Order Jet Groupoid -- 5 Conclusion -- References -- Vortices on Closed Surfaces -- 1 Introduction and Main Results
  • 1.1 Marsden and Weinstein: Vortices as Coadjoint Orbits -- 1.2 Green Functions -- 1.3 Contributions of Jerry Marsden to Vortex Systems -- 1.4 Vortex History on a Capsule -- 1.5 Point Vortices on Curved Surfaces: Current Status -- 1.6 Bose-Einstein Condensates on Surfaces -- 1.7 Organization of the Paper -- 2 Main Theorems -- 3 Outline of the Proofs -- 3.1 Theorem 1 : Main Theorem -- 3.2 Theorem 4: Conformal Transformations of the Domain -- 3.3 Proof of Kimura's Conjecture on Dipole Motion -- 4 Are Green Functions Computable? -- 4.1 Genus 1 Surfaces -- 4.2 Green's Function on Flat Tori -- Constancy of Their Associated Robin's Function -- 4.3 Genus {u2265}2 Constant Curvature and Other Canonical Metrics -- 4.4 Discrete Computational Geometry -- 5 Examples -- 5.1 Surfaces of Revolution: Momentum Map and Reduction -- 5.1.1 Cone -- 5.1.2 Torus -- 5.1.3 Catenoid -- 5.2 Vortex Pair Equations for Metrics g on S2 -- 5.3 Triaxial Ellipsoid -- 6 Heuristics for the Laplace Beltrami Green Function -- 6.1 Prandtl-Batchelor Theorem -- 6.2 Riemann Surfaces: Three Point Green Functions -- 6.3 When the Total Vorticity Vanishes: Divisors -- 7 Suggestions for Research -- 7.1 Numerical Experimentation and Visualizations -- 7.2 Contour Dynamics -- 7.3 Physical Experiments -- 7.4 Symplectomorphism Between S S and T*S and Kimura's Question -- 7.5 Elaboration on Kimura's Conjecture -- 7.6 Expansions for Green and Batman Functions Near the Diagonal -- 7.7 Integrability and Chaos: Vortex Dynamics in the Large -- 7.8 Higher Dimensions -- 7.9 Domains in a Riemann Surface: Schottky-Klein Prime Functions -- 7.10 Schottky Doubles -- 7.11 Blow Up and Regularization of Collisions -- 7.12 Topological Methods: Compactifications -- 7.13 Relation with Teichmuller Theory -- 7.14 Dynamics of Markers Around a Primary -- 7.15 Quantization of Vortex Systems -- 7.16 Hard Analysis Questions
  • 7.16.1 Thin Domains -- 7.16.2 Properties of Robin Functions -- 7.16.3 Core Energy Desingularization -- 8 Final Remarks -- 8.1 Comparison with Hally's Work Hally -- 8.2 What We Believe is New in This Paper -- Appendix: Complex, Metric and Symplectic Geometry of Two-Dimensional Hydrodynamics -- Perfect Flows in R2 -- Complex Geometry of Surfaces -- Meromorphic Differentials and Direction Fields -- Hodge Theory in 2 Dimensions -- Perfect Flows on Surfaces: Vorticity and Poisson's Equation -- References -- The Geometry of Radiative Transfer -- 1 Introduction -- 2 A Modern Formulation of Radiative Transfer Theory -- 2.1 Derivation -- 2.2 Cosphere Bundle Reduction for Radiative Transfer -- 2.3 Radiative Transfer as a Lie-Poisson System -- 3 Some Connections to the Classical Formulations -- 3.1 Classical Radiometry -- 3.2 Radiative Transfer and Geometrical Optics -- 4 Discussion and Open Questions -- 5 Conclusion -- References -- A Soothing Invisible Hand: Moderation Potentials in Optimal Control -- 1 Introduction -- 2 Constants Matter: Moderation Incentives -- 3 Affine Nonlinear Control Systems and Ellipsoidal Admissible Control Regions -- 3.1 Special Cases: Ü= 1, Ü{u2192}0, and 1Ü= p -- 4 Moderation Potentials and the Synthesis Problem -- 5 Vertical Take-Off Projectile with Controlled Velocity -- 5.1 Ü= 1 2, p = 2, and Constant or -- 5.2 Ü= 1, p = 2, and Constant or -- References -- The Local Description of Discrete Mechanics -- 1 Introduction -- 2 Groupoids and Discrete Mechanics -- 2.1 Lie Groupoids -- 2.2 Discrete Euler-Lagrange Equations -- 2.3 Discrete Poincaré-Cartan Sections -- 2.4 Discrete Lagrangian Evolution Operator -- 2.5 Discrete Legendre Transformations -- 2.6 Discrete Regular Lagrangians -- 3 Discrete Euler-Lagrange Equations: Symmetric Neighborhoods -- 3.1 Symmetric Neighborhoods -- 3.2 Local Coordinate Expressions of Structural Maps
  • 3.3 Invariant Vector Fields
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1 online resource (506 pages)
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online
Isbn
9781493924417
Media category
computer
Media MARC source
rdamedia
Media type code
c
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unknown sound
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