The Principle of Least Action in Geometry and Dynamics

The Principle of Least Action in Geometry and Dynamics

Author: Karl Friedrich Siburg

Publisher: Springer Science & Business Media

Published: 2004-05-17

Total Pages: 148

ISBN-13: 9783540219446

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New variational methods by Aubry, Mather, and Mane, discovered in the last twenty years, gave deep insight into the dynamics of convex Lagrangian systems. This book shows how this Principle of Least Action appears in a variety of settings (billiards, length spectrum, Hofer geometry, modern symplectic geometry). Thus, topics from modern dynamical systems and modern symplectic geometry are linked in a new and sometimes surprising way. The central object is Mather’s minimal action functional. The level is for graduate students onwards, but also for researchers in any of the subjects touched in the book.


The Principle of Least Action

The Principle of Least Action

Author: Alberto G. Rojo

Publisher: Cambridge University Press

Published: 2018-03-29

Total Pages: 269

ISBN-13: 0521869021

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This text brings history and the key fields of physics together to present a unique technical discussion of the principles of least action.


Galileo Unbound

Galileo Unbound

Author: David D. Nolte

Publisher: Oxford University Press

Published: 2018-07-12

Total Pages: 384

ISBN-13: 0192528505

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Galileo Unbound traces the journey that brought us from Galileo's law of free fall to today's geneticists measuring evolutionary drift, entangled quantum particles moving among many worlds, and our lives as trajectories traversing a health space with thousands of dimensions. Remarkably, common themes persist that predict the evolution of species as readily as the orbits of planets or the collapse of stars into black holes. This book tells the history of spaces of expanding dimension and increasing abstraction and how they continue today to give new insight into the physics of complex systems. Galileo published the first modern law of motion, the Law of Fall, that was ideal and simple, laying the foundation upon which Newton built the first theory of dynamics. Early in the twentieth century, geometry became the cause of motion rather than the result when Einstein envisioned the fabric of space-time warped by mass and energy, forcing light rays to bend past the Sun. Possibly more radical was Feynman's dilemma of quantum particles taking all paths at once — setting the stage for the modern fields of quantum field theory and quantum computing. Yet as concepts of motion have evolved, one thing has remained constant, the need to track ever more complex changes and to capture their essence, to find patterns in the chaos as we try to predict and control our world.


An Introduction to the Calculus of Variations

An Introduction to the Calculus of Variations

Author: L.A. Pars

Publisher: Courier Corporation

Published: 2013-12-10

Total Pages: 358

ISBN-13: 0486165957

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Clear, rigorous introductory treatment covers applications to geometry, dynamics, and physics. It focuses upon problems with one independent variable, connecting abstract theory with its use in concrete problems. 1962 edition.


Geometric Mechanics and Symmetry

Geometric Mechanics and Symmetry

Author: Darryl D. Holm

Publisher: Oxford University Press

Published: 2009-07-30

Total Pages: 537

ISBN-13: 0199212902

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A graduate level text based partly on lectures in geometry, mechanics, and symmetry given at Imperial College London, this book links traditional classical mechanics texts and advanced modern mathematical treatments of the subject.


The Principle of Least Action in Geometry and Dynamics

The Principle of Least Action in Geometry and Dynamics

Author: Karl Friedrich Siburg

Publisher: Springer

Published: 2004-04-30

Total Pages: 135

ISBN-13: 3540409858

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New variational methods by Aubry, Mather, and Mane, discovered in the last twenty years, gave deep insight into the dynamics of convex Lagrangian systems. This book shows how this Principle of Least Action appears in a variety of settings (billiards, length spectrum, Hofer geometry, modern symplectic geometry). Thus, topics from modern dynamical systems and modern symplectic geometry are linked in a new and sometimes surprising way. The central object is Mather’s minimal action functional. The level is for graduate students onwards, but also for researchers in any of the subjects touched in the book.


Mathematical Methods of Classical Mechanics

Mathematical Methods of Classical Mechanics

Author: V.I. Arnol'd

Publisher: Springer Science & Business Media

Published: 2013-04-09

Total Pages: 530

ISBN-13: 1475720637

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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.


Lectures on Symplectic Geometry

Lectures on Symplectic Geometry

Author: Ana Cannas da Silva

Publisher: Springer

Published: 2004-10-27

Total Pages: 240

ISBN-13: 354045330X

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The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and clarifications have been made, and the layout has been improved.