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Geometric Mechanics: Part I: Dynamics and Symmetry

Darryl D. Holm
Imperial College Press
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

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.

  • Fermat's Ray Optics:
    • Fermat's principle
    • Hamiltonian formulation of axial ray optics
    • Hamiltonian form of optical transmission
    • Axisymmetric invariant coordinates
    • Geometry of invariant coordinates
    • Symplectic matrices
    • Lie algebras
    • Equilibrium solutions
    • Momentum maps
    • Lie–Poisson brackets
    • Divergenceless vector fields
    • Geometry of solution behaviour
    • Geometric ray optics in anisotropic media
    • Ten geometrical features of ray optics
  • Newton, Lagrange, Hamilton and the Rigid Body:
    • Newton
    • Lagrange
    • Hamilton
    • Rigid-body motion
    • Spherical pendulum
  • Lie, Poincaré, Cartan: Differential Forms:
    • Poincaré and symplectic manifolds
    • Preliminaries for exterior calculus
    • Differential forms and Lie derivatives
    • Lie derivative
    • Formulations of ideal fluid dynamics
    • Hodge star operator on ℝ3
    • Poincaré's lemma: Closed vs exact differential forms
    • Euler's equations in Maxwell form
    • Euler's equations in Hodge-star form in ℝ4
  • Resonances and S1 Reduction:
    • Dynamics of two coupled oscillators on ℂ2
    • The action of SU(2) on ℂ2
    • Geometric and dynamic S1 phases
    • Kummer shapes for n:m resonances
    • Optical travelling-wave pulses
  • Elastic Spherical Pendulum:
    • Introduction and problem formulation
    • Equations of motion
    • Reduction and reconstruction of solutions
  • Maxwell-Bloch Laser-Matter Equations:
    • Self-induced transparency
    • Classifying Lie–Poisson Hamiltonian structures for real-valued Maxwell–Bloch system
    • Reductions to the two-dimensional level sets of the distinguished functions
    • Remarks on geometric phases
  • Enhanced Coursework:
    • Problem formulations and selected solutions
    • Introduction to oscillatory motion
    • Planar isotropic simple harmonic oscillator (PISHO)
    • Complex phase space for two oscillators
    • Two-dimensional resonant oscillators
    • A quadratically nonlinear oscillator
    • Lie derivatives and differential forms
  • Exercises for Review and Further Study:
    • The reduced Kepler problem: Newton (1686)
    • Hamiltonian reduction by stages
    • 3 bracket for the spherical pendulum
    • Maxwell–Bloch equations
    • Modulation equations
    • The Hopf map
    • 2:1 resonant oscillators
    • A steady Euler fluid flow
    • Dynamics of vorticity gradient
    • The C Neumann problem (1859)