The intended
reader of
this
book is a
graduate student beginning a doctoral program in physics or a closely
related subject, who wants to understand the physical and mathematical
foundations of analytical mechanics and the relation of classical
mechanics to relativity and quantum theory.
The book's distinguishing feature is the introduction of extended
Lagrangian and Hamiltonian methods that treat time as a transformable
coordinate, rather than as the universal time parameter of traditional
Newtonian physics. This extended theory is introduced in Part II, and
is used for the more advanced topics such as covariant mechanics,
Noether's theorem, canonical transformations, and Hamilton--Jacobi
theory.
The obvious motivation for this extended approach is its consistency
with special relativity. Since time is allowed to transform, the
Lorentz transformation of special relativity becomes a canonical
transformation. At the start of the twenty-first century, some hundred
years after Einstein's 1905 papers, it is no longer acceptable to use
the traditional definition of canonical transformation that excludes
the Lorentz transformation. The book takes the position that special
relativity is now a part of standard classical mechanics and should be
treated integrally with the other, more traditional, topics. Chapters
are included on special relativistic spacetime, fourvectors, and
relativistic mechanics in fourvector notation. The extended Lagrangian
and Hamiltonian methods are used to derive manifestly covariant forms
of the Lagrange, Hamilton, and Hamilton--Jacobi equations.
In addition to its consistency with special relativity, the use of time
as a coordinate has great value even in pre-relativistic physics. It
could have been adopted in the nineteenth century, with mathematical
elegance as the rationale. When an extended Lagrangian is used, the
generalized energy theorem (sometimes called the Jacobi-integral
theorem), becomes just another Lagrange equation. Noether's theorem,
which normally requires an longer proof to deal with the intricacies of
a varied time parameter, becomes a one-line corollary of Hamilton's
principle. The use of extended phase space greatly simplifies the
definition of canonical transformations. In the extended approach (but
not in the traditional theory) a transformation is canonical if and
only if it preserves the Hamilton equations. Canonical transformations
can thus be characterized as the most general phase-space
transformations under which the Hamilton equations are form invariant.
This is also a book for those who study analytical mechanics as a
preliminary to a critical exploration of quantum mechanics. Comparisons
to quantum mechanics appear throughout the text, and classical
mechanics itself is presented in a way that will aid the reader in the
study of quantum theory. A chapter is devoted to linear vector
operators and dyadics, including a comparison to the bra-ket notation
of quantum mechanics. Rotations are presented using an operator
formalism similar to that used in quantum theory, and the definition of
the Euler angles follows the quantum mechanical convention. The
extended Hamiltonian theory with time as a coordinate is compared to
Dirac's formalism of primary phase-space constraints. The chapter on
relativistic mechanics shows how to use covariant Hamiltonian theory to
write the Klein--Gordon and Dirac wave functions. The chapter on
Hamilton--Jacobi theory includes a discussion of the closely related
Bohm hidden variable model of quantum mechanics.
The reader is assumed to be familiar with ordinary three-dimensional
vectors, and to have studied undergraduate mechanics and linear
algebra. Familiarity with the notation of modern differential geometry
is not assumed. In order to appreciate the advance that the
differential-geometric notation represents, a student should first
acquire the background knowledge that was taken for granted by those
who created it. The present book is designed to take the reader up to
the point at which the methods of differential geometry should properly
be introduced---before launching into phase-space flow, chaotic motion,
and other topics where a geometric language is essential.
Each chapter in the text ends with a set of exercises, some of which
extend the material in the chapter. The book attempts to maintain a
level of mathematical rigor sufficient to allow the reader to see
clearly the assumptions being made and their possible limitations. To
assist the reader, arguments in the main body of the text frequently
refer to the mathematical appendices, collected in Part III, that
summarize various theorems that are essential for mechanics. I have
found that even the most talented students sometimes lack an adequate
mathematical background, particularly in linear algebra and
many-variable calculus. The mathematical appendices are designed to
refresh the reader's memory on these topics, and to give pointers to
other texts where more information may be found.
This book can be used in the first year of a doctoral physics program
to provide a necessary bridge from undergraduate mechanics to advanced
relativity and quantum theory. Unfortunately, such bridge courses are
sometimes dropped from the curriculum and replaced by a brief classical
review in the graduate quantum course. The risk of this is that
students may learn the recipes of quantum mechanics but lack knowledge
of its classical roots. This seems particularly unwise at the moment,
since several of the current problems in theoretical physics---the
development of quantum information technology, and the problem of
quantizing the gravitational field, to name two---require a fundamental
rethinking of the quantum-classical connection. Since progress in
physics depends on researchers who understand the foundations of
theories and not just the techniques of their application, it is hoped
that this text may encourage the retention or restoration of
introductory graduate analytical mechanics courses.
Oliver
Davis Johns
San Francisco, California
April 2005
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