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This book is written with the belief that classical mechanics, as a
theoretical discipline, possesses an inherent beauty, depth, and
richness that far transcends its immediate applications in
mechanical systems. These properties are manifested, by and large,
through the coherence and elegance of the mathematical structure
underlying the discipline, and are eminently worthy of being
communicated to physics students at the earliest stage possible.
This volume is therefore addressed mainly to advanced undergraduate
and beginning graduate physics students who are interested in the
application of modern mathematical methods in classical mechanics,
in particular, those derived from the fields of topology and
differential geometry, and also to the occasional mathematics
student who is interested in important physics applications of
these areas of mathematics. Its main purpose is to offer an
introductory and broad glimpse of the majestic edifice of the
mathematical theory of classical dynamics, not only in the
time-honored analytical tradition of Newton, Laplace, Lagrange,
Hamilton, Jacobi, and Whittaker, but also the more
topological/geometrical one established by Poincare, and enriched
by Birkhoff, Lyapunov, Smale, Siegel, Kolmogorov, Arnold, and Moser
(as well as many others).
This textbook is mainly for physics students at the advanced
undergraduate and beginning graduate levels, especially those with
a theoretical inclination. Its chief purpose is to give a
systematic introduction to the main ingredients of the fundamentals
of quantum theory, with special emphasis on those aspects of group
theory (spacetime and permutational symmetries and group
representations) and differential geometry (geometrical phases,
topological quantum numbers, and Chern-Simons Theory) that are
relevant in modern developments of the subject. It will provide
students with an overview of key elements of the theory, as well as
a solid preparation in calculational techniques.
This book is a translation of an authoritative introductory text
based on a lecture series delivered by the renowned differential
geometer, Professor S S Chern in Beijing University in 1980. The
original Chinese text, authored by Professor Chern and Professor
Wei-Huan Chen, was a unique contribution to the mathematics
literature, combining simplicity and economy of approach with depth
of contents. The present translation is aimed at a wide audience,
including (but not limited to) advanced undergraduate and graduate
students in mathematics, as well as physicists interested in the
diverse applications of differential geometry to physics. In
addition to a thorough treatment of the fundamentals of manifold
theory, exterior algebra, the exterior calculus, connections on
fiber bundles, Riemannian geometry, Lie groups and moving frames,
and complex manifolds (with a succinct introduction to the theory
of Chern classes), and an appendix on the relationship between
differential geometry and theoretical physics, this book includes a
new chapter on Finsler geometry and a new appendix on the history
and recent developments of differential geometry, the latter
prepared specially for this edition by Professor Chern to bring the
text into perspectives.
This book is a translation of an authoritative introductory text
based on a lecture series delivered by the renowned differential
geometer, Professor S S Chern in Beijing University in 1980. The
original Chinese text, authored by Professor Chern and Professor
Wei-Huan Chen, was a unique contribution to the mathematics
literature, combining simplicity and economy of approach with depth
of contents. The present translation is aimed at a wide audience,
including (but not limited to) advanced undergraduate and graduate
students in mathematics, as well as physicists interested in the
diverse applications of differential geometry to physics. In
addition to a thorough treatment of the fundamentals of manifold
theory, exterior algebra, the exterior calculus, connections on
fiber bundles, Riemannian geometry, Lie groups and moving frames,
and complex manifolds (with a succinct introduction to the theory
of Chern classes), and an appendix on the relationship between
differential geometry and theoretical physics, this book includes a
new chapter on Finsler geometry and a new appendix on the history
and recent developments of differential geometry, the latter
prepared specially for this edition by Professor Chern to bring the
text into perspectives.
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