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Collecting together contributed lectures and mini-courses, this
book details the research presented in a special semester titled
"Geometric mechanics - variational and stochastic methods" run in
the first half of 2015 at the Centre Interfacultaire Bernoulli
(CIB) of the Ecole Polytechnique Federale de Lausanne. The aim of
the semester was to develop a common language needed to handle the
wide variety of problems and phenomena occurring in stochastic
geometric mechanics. It gathered mathematicians and scientists from
several different areas of mathematics (from analysis, probability,
numerical analysis and statistics, to algebra, geometry, topology,
representation theory, and dynamical systems theory) and also areas
of mathematical physics, control theory, robotics, and the life
sciences, with the aim of developing the new research area in a
concentrated joint effort, both from the theoretical and applied
points of view. The lectures were given by leading specialists in
different areas of mathematics and its applications, building
bridges among the various communities involved and working jointly
on developing the envisaged new interdisciplinary subject of
stochastic geometric mechanics.
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.
We consider quantum dynamical systems (in general, these could be
either Hamiltonian or dissipative, but in this review we shall be
interested only in quantum Hamiltonian systems) that have, at least
formally, a classical limit. This means, in particular, that each
time-dependent quantum-mechanical expectation value X (t) has as i
cl Ii -+ 0 a limit Xi(t) -+ x1 )(t) of the corresponding classical
sys- tem. Quantum-mechanical considerations include an additional
di- mensionless parameter f = iiiconst. connected with the Planck
constant Ii. Even in the quasiclassical region where f~ 1, the dy-
namics of the quantum and classicalfunctions Xi(t) and XiCcl)(t)
will be different, in general, and quantum dynamics for expectation
val- ues may coincide with classical dynamics only for some finite
time. This characteristic time-scale, TIi., could depend on several
factors which will be discussed below, including: choice of
expectation val- ues, initial state, physical parameters and so on.
Thus, the problem arises in this connection: How to estimate the
characteristic time- scale TIi. of the validity of the
quasiclassical approximation and how to measure it in an
experiment? For rather simple integrable quan- tum systems in the
stable regions of motion of their corresponding classical phase
space, this time-scale T" usually is of order (see, for example,
[2]) const TIi. = p,li , (1.1) Q where p, is the dimensionless
parameter of nonlinearity (discussed below) and a is a constant of
the order of unity.
Mathematics of Planet Earth (MPE) was started and continues to be
consolidated as a collaboration of mathematical science
organisations around the world. These organisations work together
to tackle global environmental, social and economic problems using
mathematics.This textbook introduces the fundamental topics of MPE
to advanced undergraduate and graduate students in mathematics,
physics and engineering while explaining their modern usages and
operational connections. In particular, it discusses the links
between partial differential equations, data assimilation,
dynamical systems, mathematical modelling and numerical simulations
and applies them to insightful examples.The text also complements
advanced courses in geophysical fluid dynamics (GFD) for
meteorology, atmospheric science and oceanography. It links the
fundamental scientific topics of GFD with their potential usage in
applications of climate change and weather variability. The
immediacy of examples provides an excellent introduction for
experienced researchers interested in learning the scope and
primary concepts of MPE.
Mathematics of Planet Earth (MPE) was started and continues to be
consolidated as a collaboration of mathematical science
organisations around the world. These organisations work together
to tackle global environmental, social and economic problems using
mathematics.This textbook introduces the fundamental topics of MPE
to advanced undergraduate and graduate students in mathematics,
physics and engineering while explaining their modern usages and
operational connections. In particular, it discusses the links
between partial differential equations, data assimilation,
dynamical systems, mathematical modelling and numerical simulations
and applies them to insightful examples.The text also complements
advanced courses in geophysical fluid dynamics (GFD) for
meteorology, atmospheric science and oceanography. It links the
fundamental scientific topics of GFD with their potential usage in
applications of climate change and weather variability. The
immediacy of examples provides an excellent introduction for
experienced researchers interested in learning the scope and
primary concepts of MPE.
See also GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd
Edition) This textbook introduces modern geometric mechanics to
advanced undergraduates and beginning graduate students in
mathematics, physics and engineering. In particular, it explains
the dynamics of rotating, spinning and rolling rigid bodies from a
geometric viewpoint by formulating their solutions as coadjoint
motions generated by Lie groups. The only prerequisites are linear
algebra, multivariable calculus and some familiarity with
Euler-Lagrange variational principles and canonical Poisson
brackets in classical mechanics at the beginning undergraduate
level.The book uses familiar concrete examples to explain
variational calculus on tangent spaces of Lie groups. Through these
examples, the student develops skills in performing computational
manipulations, starting from vectors and matrices, working through
the theory of quaternions to understand rotations, then
transferring these skills to the computation of more abstract
adjoint and coadjoint motions, Lie-Poisson Hamiltonian
formulations, momentum maps and finally dynamics with nonholonomic
constraints.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. Many worked examples of adjoint and
coadjoint actions of Lie groups on smooth manifolds have also been
added and the enhanced coursework examples have been expanded. The
second edition is ideal for classroom use, student projects and
self-study.
See also GEOMETRIC MECHANICS - Part I: Dynamics and Symmetry (2nd
Edition) This textbook introduces modern geometric mechanics to
advanced undergraduates and beginning graduate students in
mathematics, physics and engineering. In particular, it explains
the dynamics of rotating, spinning and rolling rigid bodies from a
geometric viewpoint by formulating their solutions as coadjoint
motions generated by Lie groups. The only prerequisites are linear
algebra, multivariable calculus and some familiarity with
Euler-Lagrange variational principles and canonical Poisson
brackets in classical mechanics at the beginning undergraduate
level.The book uses familiar concrete examples to explain
variational calculus on tangent spaces of Lie groups. Through these
examples, the student develops skills in performing computational
manipulations, starting from vectors and matrices, working through
the theory of quaternions to understand rotations, then
transferring these skills to the computation of more abstract
adjoint and coadjoint motions, Lie-Poisson Hamiltonian
formulations, momentum maps and finally dynamics with nonholonomic
constraints.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. Many worked examples of adjoint and
coadjoint actions of Lie groups on smooth manifolds have also been
added and the enhanced coursework examples have been expanded. The
second edition is ideal for classroom use, student projects and
self-study.
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.
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.
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