Mathematician and mechanic Joseph Louis Lagrange (Lagrange J.L., 01/25/1736 - 04/10/1813) was born in Turin (Sardinia, Italy) into a family military treasurer. The parents were wealthy people, but the father, having indulged in dubious speculation, lost his fortune. J. Lagrange believed that this circumstance was an incentive for good study. He studied at Royal Artillery School, where he showed exceptional math skills and even before graduating from school at the age of 17, he began teach in it mathematics. Some of his students were his classmates, and some were older than him. In 1754 at age 18 he became a professor of mathematics at this school. With a group of his students he organized scientific society, which was later transformed into Turin Academy of Sciences. The first volume of the works of this academy was published in 1759. In 1759, on the recommendation of L., who had a very high opinion of the mathematical works of J. Lagrange, he was elected to Berlin Academy of Sciences, and in 1766, also on the recommendation, became president this academy. He held this position for 21 years until 1787. In 1772, J. Lagrange was elected member of the Paris Academy of Sciences, and in 1787, after the death of the Prussian King Frederick II in 1786, he moved to Paris and began to lecture: from 1795 - at the Normal School, and from 1797 - at the Polytechnic.

During the Berlin period, J. Lagrange wrote his famous "Analytical mechanics", the first edition of which was published in Paris in 1788. In this work mechanics problems were solved on the basis of the principle of possible displacements and the principle, using the concepts introduced by Lagrange "generalized forces" And "generalized coordinates". Let us note, however, that the basic ideas of the principle of possible movements can be found in a letter dated 1725 from I. Bernoulli to P. . (More details about “Analytical Mechanics” -)

In the preface to the first edition of this book, J. Lagrange writes: “There are no drawings in this book. The methods studied in it do not require either geometric constructions or mechanical reasoning; they require only geometric operations, subject to a correct and uniform course. Lovers of analysis will be pleased to see that mechanics is becoming a new branch of analysis, and will be grateful to me for such an expansion of its field.”

In 1771 Lagrange studied cantilever beam bending constant cross-section, loaded at the free end with a force, based on the integration of an exact differential equation, researched bent axes of compressed rods after loss of stability, and also stability of a hinged rod. He put the problem of the most advantageous shape of the rod outline from the point of view of the least weight. J. Lagrange also carried out important research on the calculus of variations, mathematical analysis, number theory, algebra, differential equations, mathematical cartography and astronomy. The complete works of J. Lagrange were published from 1867 to 1894. and consisted of 14 volumes.

Napoleon highly valued Lagrange and awarded him title of count and appointed member of the House of Peers (senator).

Submitting your good work to the knowledge base is easy. Use the form below

Students, graduate students, young scientists who use the knowledge base in their studies and work will be very grateful to you.

Posted on http://www.allbest.ru/

MINISTRYEDUCATIONANDSCIENCERF

FEDERALSTATEBUDGETEDUCATION

INSTITUTIONHIGHERPROFESSIONALEDUCATION

"NORTH CAUCASIANSTATE

HUMANITIES AND TECHNOLOGYACADEMY"

MIDDLE PROFESSIONALCOLLEGE

In mathematics.

On the topic: Lagrange Joseph Louis

Completed:

Student of PSO group 1-2

Baboeva Leila

Introduction

1. First achievements

2. Berlin period

3. Years of the French Revolution

4. Last years and death

5. Works of Joseph Louis Lagrange

6. Interesting facts

Conclusion

References

Introduction

In our time, we must not forget about the great scientific figures who gave impetus to the development of science. It was they who laid the foundation for enormous enrichment in various fields of activity. It follows that the significance of their works and achievements is quite great, since it is these achievements that we apply to this day, which cannot be irrelevant in our time.

The purpose of this essay is to study the biography and scientific activities of the French mathematician, astronomer and mechanic Joseph Louis Lagrange. It is necessary to consider his achievements and evaluate his contribution to science.

In accordance with the purpose of our study, the following tasks were set:

collect, study and systematize theoretical material on the research topic;

study the life and work of a mathematician;

present the main achievements of Joseph Louis Lagrange;

indicate the significance of his works and achievements;

consider interesting facts;

When writing this work, magazines and books from various publications provided great assistance.

I chose this topic because I am interested not only in the biography of the famous mathematician, but also in his works. This topic is quite extensive. In this essay I will begin by examining the biography of Joseph Louis Lagrange. Next we will consider the works of this great mathematician.

1. First achievements

Lagrange's father, at one time the military treasurer of Sardinia, was married to Maria Theresa Gro, the only daughter of a wealthy doctor from Cambiano (a place near Turin in Italy), and had 11 children with her. Of these, only the youngest, Joseph Louis, born on January 25, 1736, did not die in infancy. His father was a wealthy man, but also an incorrigible businessman, and when Joseph Louis was ready to assume his rights as sole heir, there was nothing left to inherit. Due to his family's financial difficulties, he was forced to start an independent life early. Lagrange later recalled this misfortune as one of the most fortunate events that happened to him: “If I had inherited a fortune, I probably would not have had to connect my fate with mathematics.”

Lagrange's early school interests focused on ancient languages. His father wanted his son to become a lawyer, and therefore sent him to the University of Turin. In connection with the study of classics, he early became acquainted with the geometric works of Euclid and Archimedes. But the latter did not seem to make a strong impression on him. Then the young Lagrange came across an essay by Halley (a friend of Newton) about the advantages of analysis over the synthetic geometric methods of the ancient Greeks. He was captivated and converted to a new faith, sensing his true calling. In an incredibly short time, he mastered, completely independently, everything that had been done in analysis by that time, and at the age of 16 he began teaching mathematics at the Artillery School in Turin. Thus began his activity, one of the most striking in the history of mathematics.

From the very beginning, Lagrange was an analyst, not a geometer. His analytical treatment of mechanics marks the first complete break with the tradition of the ancient Greeks. Newton, his contemporaries and immediate successors constantly used drawings to help them study problems in mechanics. Lagrange gave preference to analysis. This feature of his thinking was clearly revealed in Analytical Mechanics, conceived as a 19-year-old boy in Turin, but published in Paris only in 1788, when Lagrange was 52 years old. “You will not find any drawings in this book,” he wrote in the preface. Lagrange showed that greater flexibility and incomparably greater power are achieved if general analytical methods are used from the very beginning.

In 1755, Lagrange was appointed teacher of mathematics at the Royal Artillery School in Turin, where, despite his youth, he enjoyed the reputation of an excellent teacher. The young professor lectured to students who were all older than him. Soon, among the most capable, he organized a scientific society, which then grew into the Turin Academy of Sciences. The first volume of the academy's proceedings was published in 1759, when Lagrange was 23 years old. Lagrange himself presented here an article on the maxima and minima of the calculus of variations. With the help of this calculus, Lagrange unified mechanics and, as Hamilton said, created “a kind of scientific poem.”

In the same Turin volume, Lagrange takes another big step forward: he applies analysis to the theory of probability, and significantly advances beyond Newton in the mathematical theory of sound. At the age of 23, Lagrange was recognized as equal to the greatest mathematicians of the century - Euler and Bernoulli.

Euler always generously assessed the work of other scientists. When 19-year-old Lagrange sent Euler some of his works, the famous mathematician immediately recognized their merits and encouraged the brilliant aspiring scientist. 4 years later, Lagrange communicated to Euler a genuine method for solving isoperimetric problems in the calculus of variations, which had eluded Euler's semi-geometric methods for many years. But instead of hastening to print the solution he had been looking for for many years, Euler postpones it until Lagrange can publish it first - "so as not to deprive you of one particle of the glory that you deserve."

To this we can add that Euler achieved the election of Lagrange as a foreign member of the Berlin Academy of Sciences (October 2, 1759), despite his unusually young age - 23 years. This official recognition abroad was a great help for Lagrange at home.

Euler and d'Alembert, partly for personal reasons, were eager to see their brilliant young friend as court mathematician in Berlin. After lengthy negotiations, they achieved their goal.

A devoted friend and generous admirer of Lagrange, d'Alembert encouraged his humble young friend to pursue difficult and important problems. He also forced Lagrange to take prudent care of his health, although his own health was not strong. To d'Alembert's letters, Lagrange briefly replied that he felt excellent and was working like crazy. But in the end he paid for it. In this respect, Lagrange's activity is similar to Newton's. By middle age, prolonged concentration on problems of primary importance had dulled Lagrange's enthusiasm, and although his mind remained powerful, he became indifferent to mathematics. Fortunately for mathematics, Lagrange's black depression, with its inevitable consequence - the conviction that no human knowledge is worth striving for - was still 20 glorious years away from the time Euler and d'Alembert plotted to bring Lagrange into the fold. Berlin.

In 1759, Lagrange published works on mechanics and the calculus of variations, for the first time applied analysis to probability theory, and developed the theory of oscillations and acoustics.

In 1762, Lagrange gives the first description of the general solution of the variational problem. It was not clearly justified and was met with harsh criticism. Euler in 1766 gave a strict justification for variational methods and subsequently supported Lagrange in every possible way.

Among the problems that Lagrange worked on before coming to Berlin was the problem of the libration of the Moon, an example of the famous three-body problem. Why is the Moon always facing the Earth with one side and at the same time there are some small incomprehensible irregularities in its movement. For solving the problem of libration of the Moon: in this case, three bodies are the Earth, the Sun, the Moon, mutually attracting each other in inverse proportion to the square of the distances between their centers of gravity. Lagrange was awarded the Grand Prize of the Paris Academy of Sciences in 1764 - he was then only 28 years old. Encouraged by this brilliant success, the Academy proposed an even more difficult problem, and Lagrange again received the prize in 1766. This was a six-body problem, the material for which was the Jupiter system (Sun, Jupiter and four satellites known at that time). A complete mathematical solution is beyond our capabilities, but by using approximate methods, Lagrange made significant progress in explaining the observed irregularities.

This kind of application of Newtonian theory was of greatest interest to Lagrange throughout his active life. In 1772 he again received the Paris Prize for his work on the three-body problem, and in 1774 and 1778 he achieved similar success in connection with his work on the motion of the Moon and the perturbations of comets.

On November 6, 176, at the invitation of the Prussian king Frederick the Second, Lagrange moved to Berlin (also on the recommendation of D'Alembert and Euler). Frederick the Great, “the greatest king of Europe,” as he “modestly” styled himself, welcomed Lagrange to Berlin, declaring that he considered it an honor to have “the greatest mathematician” at his court. The latter, in any case, was true. Lagrange became the director of the physics and mathematics department of the Berlin Academy of Sciences and for twenty years filled its Memoirs with his outstanding works, one after another. He was not required to give lectures.

2. Berlin period

The Berlin period (1766 - 1787) was the most fruitful in the life of Joseph Louis. Lagrange's innate dislike of discussion served him well in Berlin. In this he differed favorably from Euler, who

rushed from one religious or philosophical debate to another. Lagrange, pressed into a corner by arguments and urged to answer, always sincerely prefaced his opinion with the statement: “I don’t know.” But when his beliefs were touched, he knew how to stand up for them, displaying both inspiration and logic. Here he performed important work in algebra and number theory, including rigorous proofs of several of Fermat's assertions and Wilson's theorem.

Soon after settling in Berlin, Lagrange summoned one of his young relatives, maternal cousin Victoria Conti, from Turin, and in 1767 married her. The marriage turned out to be happy. But soon the wife fell ill for a long time. Lagrange, forgetting about sleep, looked after her. In 1783, when she died, his heart was broken. He found solace in his work: “My studies boiled down to me calmly and quietly working out mathematics.”

In 1767, Lagrange published his memoir “On the Solution of Numerical Equations” and then a number of additions to it. It dealt with general issues of solvability of algebraic equations. At that time, for the first time in mathematics, a finite group of substitutions appeared. Lagrange suggested that not all equations above the 4th degree are solvable in radicals. A rigorous proof of this fact and specific examples of such equations were given by Abel in 1824-1826, and general conditions for solvability were found by Galois in 1830-1832.

In 1772, Lagrange was elected a foreign member of the Paris Academy of Sciences.

After the death of Frederick the Great (August 17, 1786), indignation against non-Prussians and the ensuing indifference to science made Berlin an unsuitable place of residence for Lagrange and his fellow foreigners associated with the academy, he began to seek resignation. It was allowed to him on the condition that he would send articles to the Berlin Academy for several years, to which Lagrange agreed. He gladly accepted the invitation of Louis XVI to continue his mathematical research in Paris as a member of the French Academy. Upon his arrival in Paris in 1787, he was received with great honor by the royal family and also by the academy. He was given a comfortable apartment in the Louvre, where he lived until the revolution.

At the age of 50, Lagrange felt that he was exhausted. It was a classic case of nervous exhaustion caused by prolonged and excessive overwork. The Parisians found in him a kind and benevolent interlocutor, but not a master of minds. He said that his enthusiasm had burned out and that he had lost his taste for mathematics. A copy of “Analytical Mechanics” (“Mecanique analytique”) lay unopened on his desk for two years, which became the pinnacle of Lagrange’s scientific activity.

Hamilton called this masterpiece a "scientific poem." In this work, generalized coordinates were introduced, the principle of least action was developed, and for the first time since Archimedes, a monograph on mechanics did not contain a single drawing, which Lagrange was especially proud of. Tired of everything connected with mathematics, Lagrange turned to philosophy, the evolution of thinking, the history of religion, the general theory of languages, medicine and botany. Fascinated by this strange mixture, he surprised his friends with his extensive knowledge and insightful mind on issues far from mathematics. He foresaw that in the future the best minds of mankind would show the greatest interest in chemistry, physics and the natural sciences, and he considered mathematics to be finished or, at least, entering a period of decline. Fortunately, Lagrange lived long enough to see the healthy beginning of the great work of Gauss, the first of a galaxy of great mathematicians - Abel, Galois, Cauchy and others.

3. Years of the French Revolution

In the early years of the Revolution, friends urged Lagrange to return to Berlin, but he refused to leave Paris, saying that he preferred to stay and see the “experiment” in full. Neither he nor his friends foresaw the period of terror, and when it came, Lagrange bitterly regretted remaining until it was too late to escape.

The revolution destroyed Lagrange's apathy. The revolutionaries' grandiose plans to remake humanity and change human nature did not impress Lagrange. But when his friend the chemist Lavoisier, who was a tax farmer, went to the guillotine, Lagrange expressed his indignation at the stupidity of the execution with the words: “They will only need one moment for his head to fall, but perhaps a hundred years will not be enough for a head like this to appear.” to her". Although almost the entire creative life of Lagrange passed under the patronage of royalty, his sympathies were not on the side of the royalists, but they did not belong to the revolutionaries either. Lagrange was treated tolerantly. He was granted a “pension” by a special decree, and when inflation reduced this pension to almost zero, he was appointed a member of the Invention Committee, then the Coinage Committee, to give him the opportunity to exist. Lagrange also worked on the development of the metric system of weights and measures and a new calendar. Lagrange's most important activity during the Revolution was his leading participation in improving the metric system of weights and measures. It was only thanks to Lagrange's irony and common sense that the number 12 was not chosen as a base instead of the number 10.

The "advantages" of the number 12 are obvious, and they are continued to be advanced to this day in impressive treatises by zealous propagandists who differ only by a hair from those who seek to square the circle. The number taken instead of the number 10 of our number system would be a hexagonal plug of a pentagonal hole.” To bring home to the defenders of the number 12 the absurdity of such a solution, Lagrange proposed the number 11 as even better, since any prime number underlying the number system determines its advantage that all fractions end up with the same denominator. The shortcomings of this proposal are numerous and obvious enough to anyone who has comprehended division with abbreviations. The commission saw the essence of the issue and kept the number 10.

Despite all this interesting activity, Lagrange was still lonely and prone to losing his nerve. He was rescued from the twilight state between life and death at the age of 56 by a girl, the daughter of his friend, the astronomer Lemonnier. She was touched by Lagrange's unfortunate fate and married him. The marriage turned out to be ideal. Of all his successes, he valued most highly the fact that he had found such a caring and devoted companion as his young wife.

In 1795, the Normal School was established, Lagrange became its professor of mathematics. When the École Normale closed and the famous École Polytechnique was founded (1797), Lagrange drew up a plan for its mathematics course and became its first professor. He had to give lectures to poorly prepared students. Adapting to the level of knowledge of his students, Lagrange led them through arithmetic and algebra to analysis, himself sounding more like a student than a professor. The greatest mathematician of the century became a great teacher of mathematics, training the fierce young cohort of Napoleonic military engineers. Having gone much beyond the elementary level, he developed new mathematics before the eyes of his students, and soon they themselves took part in its development. Lagrange gave an account of analysis without using Leibniz's "infinitesimals" and Newton's specific concept of limit. His own theory was published in two ores: “The Theory of Analytic Functions” (1797) and “Lectures on the Calculus of Functions” (1801).

The importance of these works lay in the fact that they gave Cauchy and other scientists the impetus for a rigorous construction of analysis.

The French paid honor to Lagrange. The scientist, who was Marie Antoinette's favorite, now became the idol of the people who sentenced her to death. When by decree of the Convention it was decided to expel from France all those born outside its borders, a special exception to this rule was made for Lagrange. His fame was so great that in 1796, when France annexed Piedmont, Talleyrand was ordered to pay a visit to Lagrange’s father, who still lived in Turin, and inform him: “Your son, of whom Piedmont, which gave birth to him, and France, which owns it, are proud, honors his a genius to all humanity." When Napoleon turned to civil affairs between his military campaigns, he often talked with Lagrange about philosophical questions and the role of mathematics in the state and showed exceptional respect for his calm and never dogmatic interlocutor.

Lagrange's calmness concealed a caustic wit that flared up unexpectedly on occasion. He once said: "These astronomers are strange people; they don't believe a theory until it agrees with their observations." Even sincere veneration of Newton is not without a faint admixture of the same gentle irony: “How lucky Newton was that in his time the system of the world still remained undiscovered.”

During these years, Lagrange published two of his important works - “The Theory of Analytical Functions (“Theorie des fonctions analytiques”, 1797) and “On the solution of numerical equations” (“De la resolution des equations numeriques”, 1798) - where he summarized everything, what was known about these issues in his time, and the new ideas and methods contained in them were developed in the works of mathematicians of the 19th century. In 1801, Lectures on the Calculus of Functions were published.

4. Last years and death

Lagrange's last scientific effort was the revision and expansion of Analytical Mechanics for a second edition. His former strength completely returned to him, although he was already over 70. Remembering his previous habits, he worked incessantly, but only established that his body did not obey the boyar’s mind. Lagrange's disease, which he knew would lead to death, did not disturb his serenity; He lived his whole life as philosophers like to live, indifferent to their fate.

2 days before Lagrange's death, Monge and other friends came to him, knowing that he was dying and wanted to tell them something about his life. They found him temporarily recovered, except for memory loss.

"I want to die, yes," I want to die and I find pleasure in it... I did my job, I achieved some fame in mathematics. I never hated anyone, I didn't do anything wrong... "He died early in the morning April 10, 1813, at the age of 78. Buried in the Pantheon.

5. Works of Joseph Louis Lagrange

Lagrange's works on mathematics, astronomy and mechanics comprise 14 volumes. He managed to successfully develop many important issues of mathematical analysis. Lagrange gave a very practical formula for expressing the remainder term of the Taylor series, a formula for finite increments and an interpolation formula, and introduced the method of multipliers for solving the problem of finding conditional extrema.

In algebra, he developed a theory, a generalization of which is the Galois theory, found a method for approximate calculation of the roots of an algebraic equation using continued fractions, a method for separating the roots of an algebraic equation, a method for eliminating a variable from a system of equations, and decomposing the roots of an equation into the so-called Lagrange series. In number theory, using improper fractions, he solved indefinite equations of the second degree with two unknowns, and developed the theory of quadratic forms.

In the field of differential equations, Lagrange developed the theory of singular solutions and the method of varying arbitrary constants for solving linear differential equations. Based on the basic laws of dynamics, he indicated two basic forms of differential equations of motion of a non-free system, which are now known as Lagrange equations of the first kind, and derived equations in generalized coordinates - Lagrange equations of the second kind.

Particularly characteristic of Lagrange, in comparison with his closest predecessors and contemporaries, was the creation of extensive theoretical concepts that combined a number of problems, statements and individual methods. Colossal new material was collected and systematized, requiring further generalization. Lagrange stood out for the “perfection of the analytical method” (the words of the famous mathematician Fourier), special elegance, conciseness, and at the same time generality of presentation, which became the distinctive features of the French mathematical school.

6. Interesting facts

Lagrange expressed his opinion about the power of the mind with the words: “If you want to see a truly great mind, visit Newton’s office, in which he decomposed sunlight and discovered the system of the world.”

“Laplace and Lavoisier were members of the commission as soon as it was formed, but after 3 months they were removed from it during the “purge” along with some other scientists. Lagrange remained the chairman of the commission. “I don’t understand why they left me,” - - he remarked, not realizing that his silence saved him not only his position, but also his head.

“Noticing Lagrange absorbed in carelessness at a musical evening, someone asked him why he loved music. “I love it because,” Lagrange answered, “that it isolates me. I hear the first three bars; on the fourth I don’t distinguish anything.” “I indulge in my thoughts, and nothing distracts me, this is how I solved more than one difficult problem.”

Even his sincere reverence for Newton is not without a faint admixture of the same gentle irony. “Newton,” he declared, “is undoubtedly an incomparable genius, but we must agree that he is also the happiest of geniuses: only once can the system of the world be discovered.” And again: “How lucky Newton was that in his time the system of the world still remained undiscovered.”

“...among those who most effectively extended the limits of our knowledge, Newton and Lagrange possessed to the highest degree the happy art of discovering new data that constitute the essence of knowledge...” Laplace wrote about Lagrange.

Lagrange's name is included in the list of the greatest scientists of France, placed on the first floor of the Eiffel Tower.

Named in his honor: a crater on the Moon, streets in Paris and Turin, many scientific concepts and theorems in mathematics, mechanics and astronomy.

Joseph Lagrange mathematician astronomer

Conclusion

And in conclusion, we can say that Joseph Louis Lagrange is a very talented person and developed in all directions. Having studied the biography, scientific activities and achievements of the mathematician Joseph Louis Lagrange, we can conclude that the scientist made an invaluable contribution to the development of science. He gave new directions for the study of yet undiscovered areas of knowledge.

The work also examined the main achievements of Joseph Louis Lagrange. Another issue identified in our work is the significance of his works and achievements. In addition, interesting facts from the life of the great mathematician were considered.

In the course of writing the essay, its goal was achieved - the biography and scientific activities of the French mathematician, astronomer and mechanic Joseph Louis Lagrange were studied, his achievements were examined and his contribution to science was assessed.

The main works are works on mathematical analysis, calculus of variations, algebra, number theory, differential equations and mechanics. Lagrange's works "Analytical Mechanics", "Treatise on the Solution of Numerical Equations of All Degrees", "Theory of Analytical Functions", "Lectures on the Calculus of Functions" were published.

In mathematical analysis, Lagrange derived a number of formulas and introduced the method of multipliers to solve the problem of finding conditional extrema. In the field of differential equations and algebra, he developed theories for solutions to all kinds of problems and equations.

The structure of the abstract is determined by its purpose and objectives.

This work is of interest to undergraduate and graduate students of physics and mathematics faculties, teachers, as well as people involved in the exact sciences.

References

1. Joseph Louis Lagrange. 1736 -- 1936. Sat. articles for the 200th anniversary of his birth. M. - L., 1937 [p. 231-232].

2. Lagrange J.L. Analytical mechanics. M. - L., 1950 [p. 12, 14].

3. Bell E.T. Creators of mathematics. M.: Education, 1979, chapter 10.

4. History of mathematics, edited by A. P. Yushkevich in three volumes, M.: Nauka. Volume 111: Mathematics of the XV111th century. (1972) [p. 350].

5. Tyulina N. A. Joseph Louis Lagrange: 1736 -- 1813. M.: Book House "Librocom", 2010, Series: Physico-mathematical heritage [p.224]

6. Website: http//mathem.hl.ru/lagranzh.html

Posted on Allbest.ru

...

Similar documents

Analysis of the role of mathematics in assessing the quantitative and spatial relationships of objects in the real world. Interpretation and justification of the mathematical theorems of Fermat, Rolle, Lagrange, Cauchy and L'Hopital. Review of the biography, activities and works of great mathematicians.

course work, added 04/08/2013


Application of the Lagrange function in convex and linear programming. The simplest problem of Boltz and the classical calculus of variations. Using the Euler-Lagrange equation to solve the isoperimetric problem. Boundary conditions for finding constants.

course work, added 01/16/2013


The childhood and adolescence of Andrei Kolmogorov - a Soviet mathematician, one of the founders of modern probability theory. Student years of A.N. Kolmogorov, his development in science. Scientific and pedagogical activity of a scientist, recognition of merit.

abstract, added 03/17/2014


Proof of the existence and uniqueness of the Lagrange interpolation polynomial. The concept of Lagrangian coefficients. Methods for specifying the slopes of an interpolating cubic spline, its use for approximating functions over large intervals.

presentation, added 10/29/2013


Application of Lagrange's theorem in solving problems. Its use in solving inequalities and equations, when finding the number of roots of some equation. Solving problems using the monotonicity condition. Relationships between increasing or decreasing functions.

abstract, added 03/14/2013


Famous Ukrainian mathematician Mikhail Fillipovich Kravchuk. Biography. Entering the scientific mathematical environment. Practical application of his works. Teaching activities. The last years of life: repression, reasons for arrest, death in the camp.

test, added 11/18/2007


Advantages of Lagrange equations and their applications. Classification of connections within a mechanical system. Possible movements of the mechanical system and the number of degrees of freedom. Application of Lagrange equations of the second kind to the study of a mechanical system.

course work, added 08/21/2009


A method for solving a problem in which the coefficients a[i] are determined by directly solving the system - the method of undetermined coefficients. Newton's interpolation formula and its variants. Construction of a Lagrange interpolation polynomial for a given function.

laboratory work, added 11/16/2015


Rolle's theorem and its proof, structure and geometric meaning. The essence of Lagrange's mean value theorem and the use of the results of Rolle's theorem in it. Reflection and generalization of Lagrange’s work in Cauchy’s theorem, the method of its proof.

abstract, added 08/15/2009


Determination of absolute and relative errors of approximate numbers. Estimation of result errors. Interpolation and extrapolation of data, Lagrange and Newton interpolation polynomials, their main characteristics and comparative description.

] Translation from French by V.S. Gokhman. Edited and with notes by L.G. Loitsyansky and A.I. Lurie. Second edition.
(Moscow - Leningrad: Gostekhizdat, 1950. - Classics of natural science. Mathematics, mechanics, physics, astronomy)
Scan, processing, Djv format: mor, 2010

• TABLE OF CONTENTS:
  From the publisher (1).
  Author's preface to the second edition (9).
  STATICS
  Section one. On various principles of statics (17).
  Section two. The general statics formula for the equilibrium of any system of forces and the method of applying this formula (48).
  Section three, General properties of equilibrium of a system of bodies, derived from the previous formula (68).
  § I. Equilibrium properties of a free system with respect to translational motion (69).
  § II. Properties of equilibrium with respect to rotational motion (72).
  § III. On the addition of rotational motions around various axes and moments about these axes (83).
  § IV. Properties of equilibrium in relation to the center of gravity (90).
  § V. Equilibrium properties related to maximum and minimum (95).
  Section four. A simpler and more general method of applying the equilibrium formula given in section two (105).
  § I. Method of multipliers (106).
  § II. Application of the same method to the equilibrium formula for solid bodies, all points of which are under the influence of some forces (112).
  § III. The analogy between the problems under consideration and the problems of maximum and minimum (122).
  Section five. Resolution of various problems of statics (147).
  Chapter one. On the equilibrium of several forces applied to the same point, on the addition and decomposition of forces (147).
  § I. On the equilibrium of a body or point under the influence of several forces (149).
  § II. On the addition and expansion of forces (153).
  Chapter two. On the equilibrium of several forces applied to a system of bodies considered as points and connected to each other by threads or rods (159).
  § I. On the equilibrium of three or more bodies fixed on an inextensible thread or on an extensible thread capable of contracting (160).
  § II. On the balance of three or more bodies fixed on an inflexible and rigid rod (173).
  § III. On the equilibrium of three or more bodies fixed on an elastic rod (180).
  Chapter three. About the equilibrium of a thread, all points of which are under the influence of some forces, and which is considered as flexible or inflexible, or elastic, and at the same time - tensile or inextensible (184).
  § I. On the equilibrium of a flexible and inextensible thread (185).
  § II. On the balance of a flexible and at the same time stretchable and contractible thread or surface (197).
  § III. On the equilibrium of an elastic thread or plate (203).
  § IV. On the equilibrium of a rigid thread of a given shape (215).
  Chapter Four. On the equilibrium of a solid body of finite size and any shape, all points of which are under the influence of any forces (227).
  Section six. On the principles of hydrostatics (234).
  Section seven. On the equilibrium of incompressible fluids (243).
  § I. On the equilibrium of a liquid in a very narrow tube (243).
  § II. Derivation of the general laws of equilibrium of incompressible fluids from the properties of the particles that make them up (250).
  § III. On the equilibrium of a free liquid mass with a solid body covered by it (269).
  § IV. On the equilibrium of incompressible fluids contained in vessels (278).
  Section eight. On the equilibrium of compressible and elastic fluids (281).
  DYNAMICS
  Section one. On the various principles of dynamics (291).
  Section two. General formula of dynamics for the motion of a system of bodies under the influence of any forces (321).
  Section three. General properties of motion derived from the previous formula (332).
  § I. Properties concerning the center of gravity (332).
  § II. Properties of areas (338).
  § III. Properties relating to rotations caused by impulses (349).
  § IV. Properties of fixed axes of rotation of a free body of any shape (357).
  § V. Properties associated with living force (369).
  §VI. Properties relating to the least action (379).
  Section four. Differential equations for solving all problems of dynamics (390).
  Section five. A general approximate method for solving dynamics problems based on the variation of arbitrary constants (412).
  § I. Derivation of the general relationship between variations of arbitrary constants from the equations given in the previous section (413).
  § II. Derivation of the simplest differential equations for determining variations of arbitrary constants arising from disturbing forces (419).
  § III. Proof of an important property of a quantity expressing living force in a system under the influence of disturbing forces (432).
  Section six. On small vibrations of any system of bodies (438).
  § I. General solution to the problem of small oscillations of a system of bodies around their equilibrium points (438).
  § II. On the oscillations of a system of linearly located bodies (461).
  § III. Application of the above formulas to the vibrations of a stretched string loaded with several bodies, and to the vibrations of an inextensible thread loaded with any number of weights and fixed at both ends or only at one of them (477).
  § IV. About the vibrations of sounding strings, considered as stretched strings, loaded with an infinitely large number of small loads located infinitely close to each other; on the discontinuity of arbitrary functions (495).
  ADDITIONS
  I. L. Poinsot - On the main position of Lagrange’s “Analytical Mechanics” (525).
  II. P.G. Lejeune-Dirichlet - On the stability of equilibrium (537).
  III. J. Bertrand - On the equilibrium of an elastic thread (540).
  IV. J. Bertrand - On the figure of a liquid mass in rotational motion (544).
  V. J. Bertrand - On the equation that Lagrange recognized as impossible (547).
  VI. J. Bertrand - On the differential equations of mechanics and the form that can be given to their integrals (549).
  VII. J. Bertrand - On Poisson's theorem (566).
  VIII. G. Darboux - On infinitesimal oscillations of a system of bodies (574).
  Notes from the editors of the Russian translation (583).

Many researchers believe that Joseph Lagrange is not a French, but an Italian mathematician. And they hold this opinion not without reason. After all, the future explorer was born in Turin in 1736. During his baptism, the boy was named Giuseppe Ludovico. His father held a high political position in the governing apparatus of Sardinia, and also belonged to the noble class. The mother came from a wealthy family of a doctor.

Family of a future mathematician

Therefore, at first, the family into which Joseph Louis Lagrange was born was quite wealthy. But the father of the family was an inept, and yet very stubborn businessman. Therefore, they were soon on the verge of ruin. In the future, Lagrange expresses a very interesting opinion about this life circumstance that befell his family. He believes that if his family had continued to live a rich and prosperous life, then perhaps Lagrange would never have had the chance to connect his fate with mathematics.

The book that changed your life

The eleventh child of his parents was Joseph Louis Lagrange. His biography, even in this respect, can be called successful: after all, all his other brothers and sisters died in early childhood. Lagrange's father was disposed to ensure that his son received an education in the field of jurisprudence. Lagrange himself was not against it at first. At first he studied at the College of Turin, where he was very fascinated by foreign languages ​​and where the future mathematician first became acquainted with the works of Euclid and Archimedes.

However, that fateful moment comes when Lagrange first comes across Galileo’s work entitled “On the Advantages of the Analytical Method.” Joseph Louis Lagrange became incredibly interested in this book - perhaps it was it that turned his entire future destiny upside down. Almost instantly, for the young scientist, jurisprudence and foreign languages ​​remained in the shadow of mathematical science.

According to some sources, Lagrange studied mathematics independently. According to others, he attended classes at the Turin School. Already at the age of 19 (and according to some sources - at 17) Joseph Louis Lagrange was teaching mathematics at the university. This was due to the fact that the best students in the country at that time had the opportunity to teach.

First work: in the footsteps of Leibniz and Bernoulli

So, from this time on, mathematics became Lagrange's main field. In 1754 his first study was published. The scientist formatted it in the form of a letter to the Italian scientist Fagnano dei Toschi. However, Lagrange makes a mistake here. Without a supervisor and preparing on his own, he subsequently discovers that his research has already been carried out. The conclusions he drew were those of Leibniz and Johann Bernoulli. Joseph Louis Lagrange even feared accusations of plagiarism. But his fears turned out to be completely unfounded. And great achievements awaited the mathematician.

Introduction to Euler

In 1755-1756, the young scientist sent several of his developments to the famous one, which he highly appreciated. And in 1759, Lagrange sent him another very important study. It was devoted to methods for solving isoperimetric problems that Euler had been struggling with for many years. The experienced scientist was very pleased with the discoveries of young Lagrange. He even refused to publish some of his developments in this area until Joseph Louis Lagrange published his own work.

In 1759, thanks to Euler's proposal, Lagrange became a foreign member of the Berlin Academy of Sciences. Here Euler showed a little cunning: after all, he really wanted Lagrange to live as close to him as possible, and thus the young scientist was able to move to Berlin.

Work and overwork

Lagrange was engaged not only in research in the fields of mathematics, mechanics and astronomy. He also created a scientific community, which later became the sciences of Turin. But the price for the fact that Joseph Louis Lagrange developed a huge number of theories in precise fields and became at that time the greatest mathematician and astronomer in the world was attacks of depression.

Constant overwork started to take its toll. Doctors in 1761 declared: they were not going to be responsible for Lagrange’s health unless he moderated his research ardor and stabilized his work schedule. The mathematician did not show self-will and listened to the recommendations of doctors. His health has stabilized. But depression did not leave him until the end of his life.

Research in Astronomy

In 1762, the Paris Academy of Sciences announced an interesting competition. To participate in it, it was necessary to provide work on the topic of the movement of the Moon. And here Lagrange shows himself as an astronomer researcher. In 1763, he sent his work on the libration of the Moon for consideration by the commission. And the article itself arrives at the Academy shortly before the arrival of Lagrange himself. The fact is that the mathematician had a trip to London, during which he became seriously ill and was forced to stop in Paris.

But even here Lagrange found great benefit for himself: after all, in Paris he was able to meet another great scientist - D'Alembert. In the capital of France, Lagrange receives a prize for his research on the libration of the Moon. And the scientist was awarded another prize - two years later he was awarded for research into two satellites of Jupiter.

High post

In 1766, Lagrange returned to Berlin and received an offer to become president of the Academy of Sciences and head of its physics and mathematics department. Many Berlin scientists very cordially welcomed Lagrange into their society. He managed to establish strong friendly ties with mathematicians Lambert and Johann Bernoulli. But there were also ill-wishers in this society. One of them was Castillon, who was three decades older than Lagrange. But after some time their relationship improved. Lagrange married Castiglione's cousin named Vittoria. However, their marriage was childless and unhappy. The often ill wife died in 1783.

Scientist's Ledger

In total, the scientist spent more than twenty years in Berlin. Lagrange’s “Analytical Mechanics” is considered the most productive work. This study was written at the time of his maturity. There are only a few great scientists among whose heritage there would be such a fundamental work. Analytical Mechanics is comparable to Newton's Principia and also to Huygens' Pendulum Clock. It also formulated the famous “Lagrange Principle”, the more complete name of which is the “D’Alembert-Lagrange Principle”. It belongs to the field of general equations of dynamics.

Moving to Paris. Sunset of life

In 1787 Lagrange moved to Paris. He was completely satisfied with the work in Berlin, but this had to be done for the reason that the position of foreigners in the city gradually worsened after the death of Frederick II. A royal audience was held in Paris in Lagrange's honor, and the mathematician even received an apartment in the Louvre. But at the same time he begins to have a serious attack of depression. In 1792, the scientist married for the second time, and now the union turned out to be happy.

At the end of his life, the scientist produces many more works. The last work he planned to undertake was the revision of Analytical Mechanics. But the scientist failed to do this. On April 10, 1813, Joseph Louis Lagrange died. His quotes, especially one of the last, characterize his entire life: “I did my job... I never hated anyone or did harm to anyone.” The death of the scientist, like his life, was calm - he left with a sense of accomplishment.

LAGRANGE JOSEPH LOUIS

(1736 – 1813)

"Lagrange - the majestic pyramid of mathematical sciences."

Napoleon Bonaparte

Joseph Louis Lagrange is generally considered a French mathematician, although some Italian sources, in principle, not unreasonably, write about him as an Italian. The fact is that the future scientist was born on January 25, 1736 in Turin and at baptism received the name Giuseppe Lodovico. His father, Giuseppe Francesco Lodovico Lagrange, was a nobleman and at one time even held the high post of treasurer of Sardinia. Mother, Maria Theresa Gro, came from the family of a wealthy doctor. Thus, the parents of Joseph Louis (hereinafter we will use his French name) initially had substantial capital. However, Giuseppe Lagrange was an incorrigible and unsuccessful businessman. Soon he went bankrupt. Subsequently, Lagrange believed that this circumstance had a very favorable effect on his fate. About the capital lost by his father, he wrote without any regret: “If I had inherited a fortune, I probably would not have had to connect my fate with mathematics.”

Joseph Louis became the eleventh child of the Lagrange couple, but all of his brothers and sisters died at an early age. His father wanted to give Joseph Louis a legal education, and at first the boy was quite pleased with this choice. While studying at the College of Turin, he became interested in ancient languages ​​and became acquainted with the works of Euclid and Archimedes. But then he accidentally came across Halley’s work “On the Advantages of the Analytical Method,” which greatly interested the future scientist and actually turned his fate around. At one point, ancient languages ​​faded into the background, and jurisprudence was forgotten. From now on, mathematics completely captured Lagrange's interests. According to some sources, Joseph Louis studied this science on his own, others claim that he began attending classes at the Turin Royal Artillery School. This discrepancy is apparently due to the fact that already at the age of 19 (and according to some sources, at the age of seventeen) Lagrange taught mathematics at school. In those days, the best students in many educational institutions taught some courses.

One way or another, since then mathematics has become the main field of activity of Joseph Louis Lagrange. On July 23, 1754, his first work was published. It was written in the form of a letter sent to the famous Italian mathematician Fagnano dei Toschi. True, the lack of a supervisor and independent preparation played a cruel joke on the young scientist. Having already published the work, he learned that his results were not original (similar conclusions were made by Johann Bernoulli and Leibniz), and he was even afraid that he would be accused of plagiarism. Fortunately, Lagrange’s fears turned out to be in vain, and the first serious achievements were not long in coming. In 1755–1756, Joseph Louis sent Euler several articles, which were highly appreciated by the venerable scientist. In 1759, the young scientist sent his illustrious colleague another very important work, in which he outlined a method for solving isoperimetric problems, the search for which the famous mathematician struggled for many years. Euler was very happy and did not even publish his own article, partially containing similar results, until Lagrange published a message about his method - “so as not to deprive you of a single particle of the glory that you deserve.” On October 2, 1759, at the suggestion of Euler, Lagrange was elected a foreign member of the Berlin Academy of Sciences. This was not without some cunning, however, quite worthy and understandable: Euler really wanted to see the young and talented scientist in Berlin.

It should be noted that Lagrange did not limit himself to teaching and his own research; he also took up organizational activities. Gathering young mathematicians, he created a scientific society, which later grew into the Royal Academy of Sciences of Turin. The first volume of the Academy's proceedings was published in 1759. Naturally, Lagrange became the main author in this and subsequent collections. His works were published on various problems of mathematics and physics: a voluminous work on the theory of sound propagation, a large article on the calculus of variations, which became the most important step towards the development of this branch of mathematics, works on the application of the calculus of variations in physics, integral calculus, etc.

Lagrange, who by that time could easily be called one of the most outstanding mathematicians in the world, continued to work enthusiastically and intensely. And soon the overwork that had become habitual made itself felt. The scientist paid for his achievements with severe bouts of depression. In 1761, his doctors announced that they refused to be responsible for Lagrange's health unless he took a long rest and followed the regime. Joseph Louis did not become stubborn, and over time his health improved, although bouts of depression still appeared throughout his life.

In 1762, the Paris Academy of Sciences announced a competition for the best work on the movement of the Moon. The following year, Lagrange sent his paper on the libration of the Moon for consideration by the Academy. The article arrived in Paris shortly before the author's arrival. The fact is that in November 1763, Lagrange went on a long journey: he was supposed to accompany the Marquis Caraccioli, an ambassador from Naples, who had previously worked in Turin, and was now assigned to London. However, Joseph Louis never made it to London - he became seriously ill in Paris, and had to abandon a further trip. But every cloud has a silver lining: in France, Lagrange met D’Alembert. The venerable scientist wrote about his young colleague: “Monsieur Lagrange from Turin stayed here for six weeks. He is very seriously ill and needs: no, not financial support, the Marquis of Caraccioli, sent to England, made sure that he did not lack anything, he needs signs of attention from his homeland... In his person Turin has a treasure , the value of which he may not realize.”

In Paris, Lagrange received a prize awarded for his work on libration. He returned to Turin only at the beginning of 1765. Two years later, the scientist received another prize for research into the motion of Jupiter's satellites.

In 1766, Leonhard Euler left Prussia. On the advice of D'Alembert and Euler himself, Frederick II invited Lagrange to Berlin, where he was offered the post of president of the Academy of Sciences and director of its physics and mathematics department. As the monarch himself “modestly” put it in his letter, “the greatest king of Europe would like to have at his court the greatest mathematician of Europe.” In Berlin, most scientists greeted Lagrange very cordially. He became friends with Lambert and Johann Bernoulli. But there were also those who were not happy to see, in their opinion, too young a scientist in the high position of head of the Academy. One of these ill-wishers was Castiglion, who was more than thirty years older than the Turin and believed that he had taken his place. But relations between scientists soon improved, and in connection with events very distant from science: a year after arriving in Berlin, Lagrange married Castiglione’s cousin Vittoria Conti. True, this marriage was childless and, in general, unhappy. A few years after the wedding, Vittoria fell ill. For many years, Lagrange, whose health also left much to be desired, cared for his wife, who died in 1783.

Lagrange served in the service of Frederick the Great for 20 years. This period of the scientist’s life was incredibly fruitful. He wrote about 150 works for the Turin, Berlin and Paris academies. Among them were important works on algebra and number theory, solving partial differential equations, probability theory, and mechanics. Separately, three articles on astronomy on the topics of competitions announced by the Paris Academy should be mentioned. All three received awards. In addition, in Berlin, Lagrange created the fundamental work “Analytical Mechanics”, which became one of the main ones in his life. It’s amazing that he conceived this treatise as a 19-year-old boy. In “Analytical Mechanics,” Lagrange not only summarized the achievements in this field since the time of Newton, but also actually created classical analytical mechanics in the form of the doctrine of general differential equations of motion of arbitrary material systems. The author based all statics on a “general formula”, which is the principle of possible movements. The dynamics were based on a “general formula”, including the principle of possible movements and D’Alembert’s principle.

Analytical Mechanics was published in Paris, where Lagrange moved in 1787. He constantly received invitations from various educational institutions and scientific institutions, especially from Italy. They wanted to see Joseph Louis both at home in Turin and in Naples, offering him a high position at the Naples Academy. But the scientist was satisfied with the work in Berlin, where he was relieved of his teaching load. However, after the death of Frederick II, the situation of foreigners in Prussia deteriorated sharply. Therefore, the offer to move to France and become a member of the Paris Academy, without the obligation to teach, came in very handy. In France, the scientist was greeted very cordially: he was awarded a royal audience and received an apartment in the Louvre, where he lived until the start of the French Revolution. But the move to France coincided with another long attack of melancholy, from which even the long-awaited publication of “Analytical Mechanics” could not bring Lagrange out.

Lagrange accepted the revolution calmly, but the changes it caused brought the scientist out of his state of apathy, and he began to work again. In 1790, Lagrange became a member of the Academy of Sciences committee for standardizing the system of weights and measures. It was he who insisted on adopting the decimal rather than duodecimal number system. And in 1792, Joseph Louis got married; his second wife was Françoise Lemonnier, the daughter of one of his colleagues at the Academy. This marriage became very happy and finally cured Lagrange from bouts of depression.

Joseph Louis was not interested in politics, but in 1793 she herself intervened in his fate. Firstly, in August the Academy of Sciences was disbanded; only the Committee for Standardization of Weights and Measures continued its work. Secondly, in September a law was passed according to which all foreigners, under threat of arrest, had to leave France, and their property was subject to confiscation. Lagrange was about to leave, but he was saved by the intervention of Lavoisier. Fortunately, in the future Lagrange no longer had any serious misunderstandings with the French government: he enjoyed well-deserved respect and honor.

In 1795, the scientist became a professor at the newly established Normal School, and after the creation of the famous Polytechnic School in 1797, he headed the department of mathematics there. The courses taught by Lagrange were published in several works: “The Theory of Analytic Functions” (1797), “On the Solution of Numerical Equations” (1798) and “Lectures on the Calculus of Functions” (1801–1806). These works played an important generalizing role and in many ways became the starting point in the work of many mathematicians (Cauchy, Jacobi, Weierstrass). In 1806 and 1808, Lagrange published two more important works on the theory of planetary motion. In 1810, the scientist began a complete revision and preparation for the reissue of Analytical Mechanics. He failed to complete this work. On April 10, 1813, Joseph Louis Lagrange died.