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Geometric Mechanics: Part II: Rotating, Translating and Rolling

Darryl D. Holm
Imperial College Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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This is the second edition of a work previously reviewed here. It remains a very good introduction to the subject at an intermediate level. The large variety of examples — including applications to optics and fluid mechanics as well as to particle and rigid body dynamics — distinguishes this book from comparable ones. The second edition has substantially the same organization, but there is new material that includes a more extensive treatment of Noether’s theorem and its implications, an expanded treatment of reduction by symmetry for the spherical pendulum and more examples from classical dynamics in the appendix of “enhanced coursework”.

Part 2 has also been revised. It now incorporates more material on Euler-Poincaré systems, a discussion of coquaternions, new examples of adjoint and coadjoint Lie group actions, and an expanded section on momentum maps (an important topic considered more briefly in Part 1).

This is a solid treatment of the subject presented attractively and well. It would be a stretch for many undergraduates, but could work well for a guided reading course or for special projects.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

  • Galileo
  • Newton, Lagrange, Hamilton and the Rigid Body
  • Quaternions
  • Adjoint and Coadjoint Actions
  • The Special Orthogonal Group SO(3)
  • Adjoint and Coadjoint Semidirect-Product Group Actions
  • Euler–Poincaré and Lie–Poisson Equations on SE(3)
  • Heavy Top Equations
  • The Euler–Poincaré Theorem
  • Lie–Poisson Hamiltonian Form of a Continuum Spin Chain
  • Momentum Maps
  • Round, Rolling Rigid Bodies
  • Geometrical Structure of Classical Mechanics
  • Lie Groups and Lie Algebras
  • Enhanced Coursework
  • Poincaré's 1901 Paper