In this post, we will see another book from the Little Mathematics Library titled The Shortest Lines Variational Problems by Lyusternik.

About the book

A pamphlet based on lectures read by the author to Moscow University’s schools mathematics club. Deals in an elementary way with a number of variational problems, such as finding the shortest curve uniting two points on a given surface. Suitable for readers with “O” level mathematics.

This book is an attempt to examine from the elementary
point of view a number of so-called variational problems.
These problems deal with quantities dependent on a curve,
and a curve for which this quantity is either maximum or
minimum is sought. Such are, for example, problems in
which it is required to find the shortest of all the curves connecting two points on a surface, or among all the closed curves of a given length on a plane it is necessary to find that one which bounds the maximum area, and so on.

The material of this book was basically presented by the author in his lectures at a school mathematical circle of the
Moscow State University. The contents of the first lecture
(Secs. I.1-III.3) in the main coincides with the contents of
Geodesic Lines, published by the author in 1940. Only the knowledge of elementary mathematics is required to master this course. Moreover, the first chapters are quite elementary. Others while not requiring special knowledge demand a greater aptitude for mathematical perusal and meditation.

The entire material of this book may be considered as an
elementary introduction to the calculus of variations (a
branch of mathematics dealing with problems of finding the
functional minimum or maximum). The calculus of variations does not enter into the first concentric cycle of the higher mathematics course that is studied, for example, in
technical colleges. In our opinion, however, for a student
who begins to study higher mathematics, it is not useless
to look further ahead.

The book was translated from Russian by Yuri Ermolyev and was published in 1983.

A big thank you to @4evercla6 for this and two more books from LML series. See the comment in the LML taking stock post. We will see them in the next couple of posts.

Now only The Euler Characteristic by Yu. A. Shaskin remains from the list!

You can get the book here.

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Contents

Introduction 7

LECTURE 1

Chapter I. Shortest Lines on Simple Surfaces 9

I.1. Shortest Lines on Polyhedral Surfaces 9
I.2. Shortest Lines on a Cylindrical Surface 14
I.3. Shortest Lines on a Conical Surface 22
I.4. Shortest Lines on a Spherical Surface 30

Chapter II. Some Properties of Plane and Space Curves and Associated Problems 38

II.1. Tangent and Normals to Plane Curves and Associated Problems 38
II.2. Some Information on the Theory of Plane and Space Curves 42

Chapter III. Geodesic Lines 49

III.1. Bernoulli’s Theorem on Geodesic Lines 49
III.2. Additional Remarks Concerning Geodesic Lines 54
III.3. Geodesic Lines on Surfaces of Revolution 58

LECTURE 2

Chapter IV. Problems Associated with the Potential Energy of Stretched Threads 61

IV.1. Motion of Lines that Does Not Change Their Length 61
IV.2. Evolutes and Involutes 67
IV.3. Problems of Equilibrium of a System of Elastic Threads 68

Chapter V. The Isoperimetric Problem 73

V.1. Curvature and Geodesic Curvature 73
V.2. An Isoperimetric Problem 76

Chapter VI. Fermat’s Principle and Its Corollaries 82

VI.1. Fermat’s Principle 82
VI.2. The Refraction Curve 85
VI.3. The Problem of Brachistochrone 89
VI.4. The Catenary and the Problem of the Minimal Surface of Revolution 92
VI.5. Interrelation Between Mechanics and Optics 101