Analytical Mechanics

858 Pages·2002·17.21 MB·English

by  A. I. Lurie (auth.)

Foundations of Engineering Mechanics
A. I. Lurie
Springer-Verlag Berlin Heidelberg GmbH
A. I. Lurie
Analytical Mechanics
Translated by A. Belyaev
With 92 Figures
Springer
Series Editors:
Vladimir 1. Babitsky J. Wittenberg
Department of Mechanical Engineering Institut fur Technische Mechanik
Loughborough University Universităt Karlsruhe (TH)
LEU 3TU Loughborough, Leicestershire KaiserstraBe 12
Great Britain 76128 Karlsruhe I Germany
Author:
A. 1. Lurie t
Translator:
A. Belyaev
State Technical University
of St. Petersburg
Polytekhnicheskaya 29
195251 St. Petersburg
Russia
ISBN 978-3-642-53650-2 ISBN 978-3-540-45677-3 (eBook)
DOI 10.1007/978-3-540-45677-3
Lurie, A.I.:
Analytical Mechanics / A.I. Lurie; translated by A. Belyaev. p. cm. - Berlin; Heidelberg; New York;
Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002
(Foundations of engineering mechanics)
Includes bibIiographical references and index.
ISBN 978-3-642-53650-2
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Anatolii I. Lurie
This book was written by a great Rw;sian scholar and teacher, A.I. Lurie,
in the period when his talent flourished.
Anatolii Isakovich Lurie was born in 1901 in Mogilev. In 1918 he grad
uated from a high school (gymnasium), and was admitted to the Faculty
of Physics and Mechanics of the SaintrPetcrsburg Polytechnic In,o.;titutc,
named after Peter the Great, where he has been working ever since. In
1939 he was conferred the degree of Doctor of Science. He headed the De
partment of Theoretical Mechanics through the period from 1936 to 1941,
and from 1944 to 1977 he was the Head of the Department of Dynamics and
Strength of Machines (which was renamed as the Department of Mechanics
and Control Processes in 1960). A.I. Lurie was a Corresponding Member
of the USSR Academy of Sciences, Division of Mechanics and Control Pro
cesses. He was a member of the Presidium of the National Committee for
Theoretical and A pplied Mechanics and a member of the National Commit
tee for Automatic Control. A.I. Lurie was a member of the Editorial Boards
of the renowned Russian journals" Applied Mathematics and Mechanics",
and "Mechanics of Solids".
His scientific activity, lasting for more than half a century, has brought
remarkable achievements. He wrote a number of magnificent books:
1. Nikolai, E.L and Lurie, A.I. Vibrations of the Frame-type Foundations.
Leningrad, Moscow, Gosstroyizdat, 1933,83 pp.
2. Loitsianskii, L.G. and Lurie, A.l. Theoretical Mechanics. In three vol
umes. Leningrad, Moscow, GMTI, 1934.
3. Lurie, A.I. Statics of Thin-walled Elastic Shells. Moscow, Gostekhiz
dat, 1947, 252 pp_
4
4. Lurie, A.1. Some Nonlinear Problems of the Theory of Automatic
Control. Moscow, Gostekhizdat, 1951, 216 pp.
5. Lurie, A.1. Operational Calculus and its Application to the Problems
in Mechanics. Moscow, GITTL, 1951, 432 pp.
6. Lurie, A.1. Three-dimensional Problems of the Theory of Elasticity.
Moscow, GITTL, 1955. 492 pp.
7. Loitsianskii, L.G. and Lurie, A.1. A Course in Theoretical Mechanics.
In two volumes (5th edition). Moscow, GITTL, 1955, 380 pp., 596 pp.
8. Lurie, A I. Analytical Mechanics. Moscow, Nauka, 1961,824 pp.
9. Lurie, A.I. Theory of Elasticity. Moscow, Nauka, 1970, 940 pp.
10. Lurie, A.I. Nonlinear Theory of Elasticity. Moscow, Nauka, 1980,
512 pp.
The last book was written when A.I. Lurie was already seriously ill. He
did not live to see the proofs, nor did he see the original Russian edition
of the book. It has been translated into English by his son K.A. Lurie and
published by North Holland Publishers in 1990.
The original style of Lurie's scientific work manifested itself already in his
early publications; this is the ability to establish strong bonds between the
achievements of classical mechanics and the needs of modern technology.
His books are unparalleled by a number of practical applications. A. I. Lurie
became an ardent promoter of the so-called direct, or invariant, vector (and
later tensor) calculus. It is now difficult to imagine that once the relations
in theoretical mechanics were expressed and written in the cumbersome
coordinate form!
The work by A.I. Lurie in the field of application of operational calculus
to the study of stability of mechanical systems with distributed parameters
brought him a great fame. This study, as well as his direct contacts with
mathematicians stimulated research in the field of distribution of the roots
of quasi-polynomials.
Probably the greatest resonance in the world scientific community was
produced by Lurie's work on the theory of absolute stability of control
systems. The very statement of the problem was pioneering, as well as the
application of the Lyapunov function method to its solution. These results
initiated an enormous flow of scientific literature.
Professor Lurie is also the author of a number of articles and books on the
theory of elasticity. He devoted the last fifteen years of his life exclusively
to those problems. The typical feature of this works was its focus towards
obtaining analytical results. He did not pay any attention to numerical
methods that became so popular nowadays.
Professor Lurie was an extraordinary person. He was an attentive and re
spectfullistener, but his interest sharpened when a colleague demonstrated
his own scientific ideas. This feature was especially attractive for the young
researchers and lecturers who wanted his opinion. His study was always full
of visitors seeking his advice, his review of papers, or simply his support.
He worked hard through all of his life, writing books, giving lectures, re-
5
viewing papers. He disliked and even might be hostile to the idle, though
possibly talented people.
In the spring of 1979 Professor Lurie underwent a serious surgery. It
took him the whole summer to recover after it. In September he came back
from Moscow. He looked fine. He said to me (I was already acting as the
Head of his Chair): "I am going to read my favorite "Theory of Elasticity"
course". I tried to object to this, and offered to read his lectures as well
as mine, to stimulate him to relax. He reacted rather sharply and insisted
on reading his own course. However, he was able to continue only until
October. In November, he gave up saying that it was too difficult. He died
on 12 February 1980. He was 78 yeas old.
I hope that the English-speaking reader will enjoy" Analytical Mechan
ics" by A.I. Lurie. A good and talented pen.;on can write only a good book!
Professor Vladimir A. Palmov,
Head of Lurie's Chair
Preface
According to established tradition, courses on analytical mechanics include
general equations of motion of holonomic and non-holonomic systems, vari
ational principles, theory of canonical transformations, canonical equations
and theory of their integration (the Hamilton-Jacobi theorem), integral in
variants, theory of last multiplier and others. The fundamental laws of
mechanics are taken for granted and are not subject to discussion.
The present book is concerned with those issues of the above listed sub
jects which, in the author's opinion, are most closely related to engineering
problems.
Application of the methods of analytical mechanics to non-trivial prob
lems at the very stage of constructing the equations requires detailed knowl
edge of the issues that are normally only briefly touched upon. With this
perspective considerable attention is paid to ways of introducing the gener
alised coordinates, the theory of finite rotation, methods of calculating the
kinetic energy, the energy of accelerations, the potential energy of forces
of various nature, and the resisting forces. These introductory chapters,
which have to some extent independent significance, are followed by those
on methods of constructing differential equations of motion for holonomic
and non-holonomic systems in various forms. In these chapters the issues of
their interrelations, determination of the constraint forces and some prob
lems of analytical statics are discussed as well. It is thought useful to include
geometric considerations of the motion of a material system as motion of
the representative point in Riemannian space. Further this approach is ap
plied to the problems of perturbation theory. A special chapter is devoted to
the dynamics of relative motion illustrated by numerous applied problems.
8
This is followed by the study of canonical equations, canonical transfor
mation and the prohlem of integration. The last chapter deals with the
Hamilton-Ostrogradsky principle, the principle of least action by Lagrange
and the theory of the perturbation of trajectories.
General methods are explained for particular examples, some of which
are not devoid of interest in our opinion. These examples include the prob
lem of motion of a rigid body on a moving base, motion of a rigid body
with a cavity filled by fluid, the problem of rocket motion, application of
the Hamilton-Ostrogradsky principle to systems with distributed mass and
many others. Special attention is given to problems associated with the
perturbed motion of Earth satellites.
The examples analysed in the present book confirm the significance of
the methods of analytical mechanics for a wide range of applications which
was one of the primary aims of the author. In considering the examples
attention is paid to the statement of the problem and construction of the
equations of motion whilst their integration and analysis of results occupy
less space.
To facilitate reading, the book is provided with appendices to which
the reader is referred for the basic notion from matrix theory and tensor
analysis.
Equation numbering is as follows. The first number in parentheses indi
cates the Chapter, the second - the Section, whilst the third - the equation
number in the Section. When a cross-reference is made within the same
Section only the last number is used. Both second and third numbers are
used for a cross-reference within the same Chapter. The complete number
appears when an equation from another Chapter is referred to.
The list of references contains the most important sources, a detailed list
not being the objective of the present book.
Some parts of the book are based on the lecture courses on analytical
mechanics and vibration theory taught by the author for more than twenty
years at the Faculty of Physics and Mechanics of the Leningrad Polytech
nic Institute!. However the author hopes that students and researchers in
various fields of engineering will find this book useful.
Professor D.R. Merkin, who kindly consented to edit the book, gave a
great number of valuable suggestions to the author. A great assistance in
preparation of the manuscript and drawings was provided by A.K. Gibyan
skaya and K.A. Lurie. It is the author's pleasure to express his deep grati
tude to these people.
ITranslator's note: now the State Technical University of St. Petersburg