Book Review
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ure and Applied Geophysics
Book Review ‘‘Geometric Mechanics. Part II: Rotating, Translating and Rolling’’, by Darryl D. Holm, Second Edition, Imperial College Press; 2011; ISBN-13: 978-1-84816-778-0 (pbk), USD 31.00 ANDRZEJ ICHA1 The second Part of Prof. Holm’s book is a masterfully written, very interesting theoretical mechanics text, covering the essence of the relationship between two reductions: Lagrangian symmetry reduction and Hamiltonian reduction. The first leads to the EulerPoincare0 equation, whereas the second gives a Lie– Poisson equation. The book comprises twelve chapters and four appendices. Chapter 1 introduces a principle of Galilean relativity and explains the following notions: a uniformly moving reference frame, the Galilean transformations, the Galilean group, the special Euclidean group (SE(3)), the matrix representation of SE(3), Lie algebra of SE(3), etc. For example, the rigid motion in R3 refers to a smoothly altering sequence of changes of reference frame along a time-dependent path in SE(3). Chapter 2 concisely describes Newtonian, Lagrangian and Hamiltonian approaches to a freely rotating rigid body. A short discussion of a rigid body rotating in a field with a quadratic potential using Manakov’s method is also presented. Chapters 3 and 4 continue the study of freely rotating motion. Because of the close relation of quaternions with rigid-body rotations, the basic philosophy of operating with quaternions is described first. Next, the application of quaternions in Kepler’s problem is presented. Then, the quaternionic conjugation is explained (Cayley-Klein parameters, Hopf’s
1 Pomeranian Academy in Słupsk, Institute of Mathematics, ul. Arciszewskiego 22d, 76-200 Słupsk, Poland. E-mail: [email protected]
fibration, etc). The chapter also introduces the notion of coquaternions in the context of complexified mechanics. Chapter 4, in turn, is intended to provide a comprehensive understanding of the Cayley–Klein dynamics for the rigid body, with a particular emphasis on the Heisenberg Lie group (that is the simplest non-commutative Lie group). In order to study the rotations and translations in R3 in more detail, Lie groups and their actions are needed. Chapter 5 defines the adjoint and coadjoint actions of SO(3) (SO(3) denotes the special orthogonal group). These actions, defined on the appropriate groups studied in chapter 6, provide adequate instruments to derive the Euler-Poincare´ equations. In this context, one should read a translation of Poincare´’s 1901 fundamental two-page paper, placed in Appendix D. In chapter 7 a brief derivation of the Euler-Poincare´ equations is performed. Then, Kelvin–Noether theorem for this equation for SE(3) is formulated and proved. Subsequently, the Kirchoff equations for the motion of an ellipsoidal underwater body, based on vector Euler-Poincare´ equations are rederived. Some remarks about the dynamics of the heavy-top prepares the reader for chapter 8. The heavy-top refers to a rigid body which moves around a fixed point in a gravitational field.
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