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Introduction to
Mechanics and Symmetry
A Basic Exposition of Classical Mechanical Systems
Second Edition
Jerrold E. Marsden
and
Tudor S. Ratiu
Last modi
ed on 15 July 1998
v
To Barbara and Lilian for their love and support
........................... 15 July 1998—18h02 ...........................
vi
........................... 15 July 1998—18h02 ...........................
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Preface
Symmetry and mechanics have been close partners since the time of the
founding masters: Newton, Euler, Lagrange, Laplace, Poisson, Jacobi, Ha-
milton, Kelvin, Routh, Riemann, Noether, Poincar
e, Einstein, Schr
odinger,
Cartan, Dirac, and to this day, symmetry has continued to play a strong
role, especially with the modern work of Kolmogorov, Arnold, Moser, Kir-
illov, Kostant, Smale, Souriau, Guillemin, Sternberg, and many others. This
book is about these developments, with an emphasis on concrete applica-
tions that we hope will make it accessible to a wide variety of readers,
especially senior undergraduate and graduate students in science and en-
gineering.
The geometric point of view in bined with solid analy-
sis has been a phenomenal ess in linking various diverse areas, both
within and across standard disciplinary lines. It has provided both insight
into fundamental issues in mechanics (such as variational and Hamiltonian
structures in continuum mechanics,
uid mechanics, and plasma physics)
and provided useful tools in speci
c models such as new stability and bifur-
cation criteria using the energy-Casimir and energy-momentum methods,
new numerical codes based on geometrically exact update procedures and
variational integrators, and new reorientation techniques in control theory
and robotics.
Symmetry was already widely used in mechanics by the founders of the
subject, and has been developed considerably in recent times in such di-
verse phenomena as reductio