Differential equations

Date: 2015-10-07; view: 487.

I. Scan the article to find out:

- who was the pioneer in differential equations

- what was the harder part of the theory of differential equations

- whatWilliam Rowan Hamilton in Ireland and Jacobi in Germany showed

Another field that developed considerably in the 19th century was the theory of differential equations. The pioneer in this direction once again was Cauchy. Above all, he insisted that one should prove that solutions do indeed exist; it is not a priori obvious that every ordinary differential equation has solutions. The methods that Cauchy proposed for these problems fitted naturally into his program of providing rigorous foundations for all the calculus. The solution method he preferred, although the less general of his two approaches, worked equally well in the real and complex cases. It established the existence of a solution equal to the one obtainable by traditional power series methods using newly developed techniques in his theory of functions of a complex variable.

The harder part of the theory of differential equations concerns partial differential equations, those for which the unknown function is a function of several variables. In the early 19th century there was no known method of proving that a given second- or higher-order partial differential equation had a solution, and there was not even a method of writing down a plausible candidate. In this case progress was to be much less marked. Cauchy found new and more rigorous methods for first-order partial differential equations, but the general case eluded treatment.

An important special case was successfully prosecuted, that of dynamics. Dynamics is the study of the motion of a physical system under the action of forces. Working independently of each other, William Rowan Hamilton in Ireland and Jacobi in Germany showed how problems in dynamics could be reduced to systems of first-order partial differential equations. From this base grew an extensive study of certain partial differential operators. These are straightforward generalizations of a single partial differentiation (∂/∂x) to a sum of the form

where the a's are functions of the x's. The effect of performing several of these in succession can be complicated, but Jacobi and the other pioneers in this field found that there are formal rules which such operators tend to satisfy. This enabled them to shift attention to these formal rules, and gradually an algebraic analysis of this branch of mathematics began to emerge.

The most influential worker in this direction was the Norwegian, SophusLie. Lie, and independently Wilhelm Killing in Germany, came to suspect that the systems of partial differential operators they were studying came in a limited variety of types. Once the number of independent variables was specified (which fixed the dimension of the system), a large class of examples, including many of considerable geometric significance, seemed to fall into a small number of patterns. This suggested that the systems could be classified, and such a prospect naturally excited mathematicians. After much work by Lie and Killing and later by the French mathematician Élie-Joseph Cartan, they were classified. Initially, this discovery aroused interest because it produced order where previously the complexity had threatened chaos and because it could be made to make sense geometrically. The realization that there were to be major implications of this work for the study of physics lay well in the future.

Vocabulary to remember:

ordinary differential equation - обыкновенноедифференциальноеуравнение

complexcase - случайкомплекснойпеременной

powerseries - степенной ряд

partialdifferentialequation - дифференциальное уравнение в частных производных

partialdifferentialoperator - оператор в частных производных

partial differentiation - определениечастнойпроизводной

II. Translate the attributive groups in the sentences:

1.…method of proving that a given second- or higher-order partial differential equation had a solution…

2. Cauchy found new and more rigorous methods for first-order partial differential equations

III. Read the article in details and answer the questions:

1.Doesevery ordinary differential equation have solutions?

2. What methods did Cauchy propose for these problems?

3. What did his solution method establish?

4. What was the harder part of the theory of differential equations?

5. Why was it hard?

6. What was Cauchy's contribution to it?

7. What special case was successfully prosecuted?

8. What did William Rowan Hamilton in Ireland and Jacobi in Germany show?

9. What was their contribution to the study of partial differential equations?

10. What did Sophus Lie and Wilhelm Killing realise?

11. What did this suggest?

12. Why did this discovery arouse interest?