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**The general problem of the stability of motion. With a biography of Lyapunov by V. I. Smirnov and a bibliography of Lyapunov’s works by J. F. Barrett. Transl. and ed. by A.T. Fuller.**
*(English)*
Zbl 0786.70001

London: Taylor & Francis. ix, 270 p. (1992).

There is no doubt that the pioneering work of A. M. Lyapunov on the stability of motion is of paramount importance for qualitative dynamics, a field most popular in our days in connection with deterministic chaos and related ideas. Lyapunov presented his results in a few articles and, most comprehensively, in 1892 as a memoir that served as his doctoral dissertation at Moscow University. The book under review presents an English translation of this memoir, thereby making it available to a much wider audience than the Russian original (Kharkov, 1892; JFM 24.0876.02) and a French translation [Toulouse Ann. (2) 9, 203–474 (1907; JFM 38.0738.07)] did. The same English translation was published already in 1992 as a special issue of the “International Journal of Optimal Control”. (Lyapunov’s memoir should not be confused with the content of the well- known English book A. M. Lyapunov: “Stability of motion” New York, Academic Press (1966).) The latter is a translation of an 1893 paper, less than one-third of the 1892 memoir in length, that chiefly deals with an exceptional case not treated in the memoir.)

In 1892 Lyapunov was an experienced teacher and researcher, 35 years of age, and his main ideas on the stability of motion were fully developed. He subdivided his memoir into three chapters. Chapter I is the most important part since it presents, for a system of first order ordinary differential equations, the now famous criteria for “Lyapunov stability” in terms of “Lyapunov exponents” and in terms of “Lyapunov functions”. Chapters II and III utilize these general results for cases where the coefficients of the linearized system are constant and periodic, respectively. Whereas these two chapters contain some sections on differential equations of a specific analytic nature, no explicit examples and no applications to physics are given. In particular, and rather surprisingly, the relevance of the results presented to celestial mechanics is not advertised. The exposition is fairly detailed, containing proofs of all relevant results. Although Lyapunov’s way of expression is certainly not the most elegant from a modern point of view, the text is not very difficult to read. In some cases, where the terminology is not in agreement with modern use or where it is a little bit more difficult to follow Lyapunov’s line of thought, a translator’s note is inserted for the reader’s convenience.

The book ends with a biography of A. M. Lyapunov and with a list of his publications containing 46 scientific papers (apart from lecture notes, translations, obituaries etc.). The biography is the English translation of a 1948 text by V. I. Smirnov. It gives a good idea of Lyapunov’s scientific activities, centered upon St. Petersburg and Kharkov, but also of his personal course of life from his birth in 1857 until his suicide in 1918.

The book will be very useful to everyone interested in nonlinear dynamics, in particular to those with an additional interest in historical aspects of mathematics and mechanics.

In 1892 Lyapunov was an experienced teacher and researcher, 35 years of age, and his main ideas on the stability of motion were fully developed. He subdivided his memoir into three chapters. Chapter I is the most important part since it presents, for a system of first order ordinary differential equations, the now famous criteria for “Lyapunov stability” in terms of “Lyapunov exponents” and in terms of “Lyapunov functions”. Chapters II and III utilize these general results for cases where the coefficients of the linearized system are constant and periodic, respectively. Whereas these two chapters contain some sections on differential equations of a specific analytic nature, no explicit examples and no applications to physics are given. In particular, and rather surprisingly, the relevance of the results presented to celestial mechanics is not advertised. The exposition is fairly detailed, containing proofs of all relevant results. Although Lyapunov’s way of expression is certainly not the most elegant from a modern point of view, the text is not very difficult to read. In some cases, where the terminology is not in agreement with modern use or where it is a little bit more difficult to follow Lyapunov’s line of thought, a translator’s note is inserted for the reader’s convenience.

The book ends with a biography of A. M. Lyapunov and with a list of his publications containing 46 scientific papers (apart from lecture notes, translations, obituaries etc.). The biography is the English translation of a 1948 text by V. I. Smirnov. It gives a good idea of Lyapunov’s scientific activities, centered upon St. Petersburg and Kharkov, but also of his personal course of life from his birth in 1857 until his suicide in 1918.

The book will be very useful to everyone interested in nonlinear dynamics, in particular to those with an additional interest in historical aspects of mathematics and mechanics.

Reviewer: Volker Perlick (Berlin)

### MSC:

70-02 | Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems |

70K20 | Stability for nonlinear problems in mechanics |

34D20 | Stability of solutions to ordinary differential equations |

01A70 | Biographies, obituaries, personalia, bibliographies |

34D15 | Singular perturbations of ordinary differential equations |

01A55 | History of mathematics in the 19th century |