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

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Dear Dima,

God bless you! Thank you very much.

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All credits to @4evercla6

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