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Geometric Mechanics

Author: Darryl D Holm
Publisher: World Scientific Publishing Company
ISBN: 1911298658
Size: 43.38 MB
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See also GEOMETRIC MECHANICS — Part II: Rotating, Translating and Rolling (2nd Edition) This textbook introduces the tools and language of modern geometric mechanics to advanced undergraduates and beginning graduate students in mathematics, physics and engineering. It treats the fundamental problems of dynamical systems from the viewpoint of Lie group symmetry in variational principles. The only prerequisites are linear algebra, calculus and some familiarity with Hamilton's principle and canonical Poisson brackets in classical mechanics at the beginning undergraduate level. The ideas and concepts of geometric mechanics are explained in the context of explicit examples. Through these examples, the student develops skills in performing computational manipulations, starting from Fermat's principle, working through the theory of differential forms on manifolds and transferring these ideas to the applications of reduction by symmetry to reveal Lie–Poisson Hamiltonian formulations and momentum maps in physical applications. The many Exercises and Worked Answers in the text enable the student to grasp the essential aspects of the subject. In addition, the modern language and application of differential forms is explained in the context of geometric mechanics, so that the importance of Lie derivatives and their flows is clear. All theorems are stated and proved explicitly. The organisation of the first edition has been preserved in the second edition. However, the substance of the text has been rewritten throughout to improve the flow and to enrich the development of the material. In particular, the role of Noether's theorem about the implications of Lie group symmetries for conservation laws of dynamical systems has been emphasised throughout, with many applications. Contents: Fermat's Ray Optics:Fermat's principleHamiltonian formulation of axial ray opticsHamiltonian form of optical transmissionAxisymmetric invariant coordinatesGeometry of invariant coordinatesSymplectic matricesLie algebrasEquilibrium solutionsMomentum mapsLie–Poisson bracketsDivergenceless vector fieldsGeometry of solution behaviourGeometric ray optics in anisotropic mediaTen geometrical features of ray opticsNewton, Lagrange, Hamilton and the Rigid Body:NewtonLagrangeHamiltonRigid-body motionSpherical pendulumLie, Poincaré, Cartan: Differential Forms:Poincaré and symplectic manifoldsPreliminaries for exterior calculusDifferential forms and Lie derivativesLie derivativeFormulations of ideal fluid dynamicsHodge star operator on ℝ3Poincaré's lemma: Closed vs exact differential formsEuler's equations in Maxwell formEuler's equations in Hodge-star form in ℝ4Resonances and S1 Reduction:Dynamics of two coupled oscillators on ℂ2The action of SU(2) on ℂ2Geometric and dynamic S1 phasesKummer shapes for n:m resonancesOptical travelling-wave pulsesElastic Spherical Pendulum:Introduction and problem formulationEquations of motionReduction and reconstruction of solutionsMaxwell-Bloch Laser-Matter Equations:Self-induced transparencyClassifying Lie–Poisson Hamiltonian structures for real-valued Maxwell–Bloch systemReductions to the two-dimensional level sets of the distinguished functionsRemarks on geometric phasesEnhanced Coursework:Problem formulations and selected solutionsIntroduction to oscillatory motionPlanar isotropic simple harmonic oscillator (PISHO)Complex phase space for two oscillatorsTwo-dimensional resonant oscillatorsA quadratically nonlinear oscillatorLie derivatives and differential formsExercises for Review and Further Study:The reduced Kepler problem: Newton (1686)Hamiltonian reduction by stagesℝ3 bracket for the spherical pendulumMaxwell–Bloch equationsModulation equationsThe Hopf map2:1 resonant oscillatorsA steady Euler fluid flowDynamics of vorticity gradientThe C Neumann problem (1859) Readership: Advanced undergraduate and graduate students in mathematics, physics and engineering; non-experts interested in geometric mechanics, dynamics and symmetry.

Geometric Mechanics Part Ii

Author: Darryl D. Holm
Publisher: World Scientific
ISBN: 9781848167773
Size: 15.87 MB
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Advanced undergraduate and graduate students in mathematics, physics and engineering.

Geometric Mechanics And Symmetry

Author: Darryl D. Holm
Publisher: Oxford University Press
ISBN: 0199212902
Size: 80.32 MB
Format: PDF
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Geometric Mechanics and Symmetry is a friendly and fast-paced introduction to the geometric approach to classical mechanics, suitable for a one- or two- semester course for beginning graduate students or advanced undergraduates. It fills a gap between traditional classical mechanics texts and advanced modern mathematical treatments of the subject.The modern geometric approach illuminates and unifies manyseemingly disparate mechanical problems from several areas of science and engineering. In particular, the book concentrates on the similarities between finite-dimensional rigid body motion and infinite-dimensional systems such asfluid flow. The illustrations and examples, together with a large number of exercises, both solved and unsolved, make the book particularly useful.

Introduction To Dynamical Systems And Geometric Mechanics

Author: Jared M. Maruskin
Publisher: Solar Crest Publishing LLC
ISBN: 0985062711
Size: 48.83 MB
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Introduction to Dynamical Systems and Geometric Mechanics provides a comprehensive tour of two fields that are intimately entwined: dynamical systems is the study of the behavior of physical systems that may be described by a set of nonlinear first-order ordinary differential equations in Euclidean space, whereas geometric mechanics explores similar systems that instead evolve on differentiable manifolds. In the study of geometric mechanics, however, additional geometric structures are often present, since such systems arise from the laws of nature that govern the motions of particles, bodies, and even galaxies. In the first part of the text, we discuss linearization and stability of trajectories and fixed points, invariant manifold theory, periodic orbits, Poincar maps, Floquet theory, the Poincar -Bendixson theorem, bifurcations, and chaos. The second part of the text begins with a self-contained chapter on differential geometry that introduces notions of manifolds, mappings, vector fields, the Jacobi-Lie bracket, and differential forms. The final chapters cover Lagrangian and Hamiltonian mechanics from a modern geometric perspective, mechanics on Lie groups, and nonholonomic mechanics via both moving frames and fiber bundle decompositions. The text can be reasonably digested in a single-semester introductory graduate-level course. Each chapter concludes with an application that can serve as a springboard project for further investigation or in-class discussion.

Geometric Mechanics

Author: Richard Talman
Publisher: John Wiley & Sons
ISBN: 352761141X
Size: 43.67 MB
Format: PDF, Kindle
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For physicists, mechanics is quite obviously geometric, yet the classical approach typically emphasizes abstract, mathematical formalism. Setting out to make mechanics both accessible and interesting for non-mathematicians, Richard Talman uses geometric methods to reveal qualitative aspects of the theory. He introduces concepts from differential geometry, differential forms, and tensor analysis, then applies them to areas of classical mechanics as well as other areas of physics, including optics, crystal diffraction, electromagnetism, relativity, and quantum mechanics. For easy reference, the author treats Lagrangian, Hamiltonian, and Newtonian mechanics separately -- exploring their geometric structure through vector fields, symplectic geometry, and gauge invariance respectively. Practical perturbative methods of approximation are also developed. This second, fully revised edition has been expanded to include new chapters on electromagnetic theory, general relativity, and string theory. 'Geometric Mechanics' features illustrative examples and assumes only basic knowledge of Lagrangian mechanics.

Introduction To Mechanics And Symmetry

Author: J.E. Marsden
Publisher: Springer Science & Business Media
ISBN: 0387217924
Size: 57.77 MB
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A development of the basic theory and applications of mechanics with an emphasis on the role of symmetry. The book includes numerous specific applications, making it beneficial to physicists and engineers. Specific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering. This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available.

Emergence Of The Quantum From The Classical Mathematical Aspects Of Quantum Processes

Author: De Gosson Maurice A
Publisher: World Scientific
ISBN: 1786344165
Size: 71.66 MB
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The emergence of quantum mechanics from classical world mechanics is now a well-established theme in mathematical physics. This book demonstrates that quantum mechanics can indeed be viewed as a refinement of Hamiltonian mechanics, and builds on the work of George Mackey in relation to their mathematical foundations. Additionally when looking at the differences with classical mechanics, quantum mechanics crucially depends on the value of Planck's constant h. Recent cosmological observations tend to indicate that not only the fine structure constant α but also h might have varied in both time and space since the Big Bang. We explore the mathematical and physical consequences of a variation of h; surprisingly we see that a decrease of h leads to transitions from the quantum to the classical. Emergence of the Quantum from the Classical provides help to undergraduate and graduate students of mathematics, physics and quantum theory looking to advance into research in the field. Contents: Hamiltonian MechanicsHamilton–Jacobi TheoryMatter Waves, Schrödinger's Equation, and Bohm's Theory The MetatronUncertainties and Quantum BlobsQuantum States and the Density MatrixVarying Planck's ConstantAppendices: The Symplectic GroupThe Metaplectic RepresentationBorn–Jordan QuantizationTwisted Product and Convolution Readership: Undergraduate and graduate students of mathematics and physics, interested in analysis and differential equations, probability and statistics, geometry and topology and mathematical physics. Keywords: Classical Mechanics;Hamiltonian Mechanics;Quantum Mechanics;Planck's Constant;Mathematical Physics;Algebraic GeometryReview:0

Mathematical Methods Of Classical Mechanics

Author: V.I. Arnol'd
Publisher: Springer Science & Business Media
ISBN: 1475720637
Size: 75.29 MB
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This book constructs the mathematical apparatus of classical mechanics from the beginning, examining basic problems in dynamics like the theory of oscillations and the Hamiltonian formalism. The author emphasizes geometrical considerations and includes phase spaces and flows, vector fields, and Lie groups. Discussion includes qualitative methods of the theory of dynamical systems and of asymptotic methods like averaging and adiabatic invariance.