Kurt Otto Friedrichs
 |
Kurt O. Friedrichs (1901–1982) was a noted mathematician. He was the co-founder of the Courant Institute at New York University and recipient of the National Medal of Science. There is a student prize named after Friedrichs at NYU.
This Biographical Memoir if from the NATIONAL ACADEMY OF SCIENCES and was written by BY CATHLEEN MORAWETZhttp://en.wikipedia.org/wiki/Cathleen_Synge_Morawetz
Kurt Otto Friedrichs was born in Kiel, Germany, on September 28, 1901, but moved before his school days to Düsseldorf. He came from a comfortable background, his father being a well-known lawyer. Between the views of his father, logical and, on large things, wise, and the thoughtful and warm affection of his mother, Friedrichs grew up in an intellectual atmosphere conducive to the study of mathematics and philosophy. Despite being plagued with asthma, he completed the classical training at the local gymnasium and went on to his university studies in Düsseldorf. Following the common German pattern of those times, he spent several years at different universities. Most strikingly for a while he studied the philosophies of Husseri and Heidegger in Freiburg. He retained a lifelong interest in the subject of philosophy but eventually decided his real bent was in mathematics. He chose to complete his studies in Göttingen, the mecca of mathematics in the 1920s. There he met Richard Courant, director of the Institute of Mathematics, who admired enormously his talent but found him somewhat unworldly. In fact, Friedrichs’s childhood asthma had prevented him from participating in the activities where children naturally socialize and he was painfully shy. Courant and Friedrichs enjoyed a lifelong friendship that included a great deal of mathematical stimulation, cooperation and interaction, a lot of practical advice from Courant, and a fair dose from Friedrichs of his logical approach to life and his special values. On the basis of my own observations for over twenty-five years, they dealt with each other’s idiosyncrasies in a remarkably comfortable way.
Friedrichs stayed in Göttingen for five years. During this time he completed his first paper clarifying the logical significance of Einstein’s general covariance postulate (1927,1) and then wrote his dissertation on boundary and eigenvalue problems for elastic plates (1927,2). This was followed by a paper with Hans Lewy (1927,3) on initial value problems for linear hyperbolic partial differential equations.
Those first three papers demonstrate most of Friedrichs’s lifetime in mathematics—the first on the fundamental laws of the nature of matter, the second on applied mathematics viewed through analysis, and the third on basic theorems of wave propagation.
The paper on hyperbolic partial differential equations led naturally to what turned out to be one of the best known and most used results of mathematics of the time (1928,2). In investigating whether difference schemes for a time-varying partial differential equation like the wave or heat equation yield a good approximate solution, Courant, Friedrichs, and Lewy were led to consider the stability of the difference scheme. In a simple difference scheme every partial derivative like df/dx is replaced by a difference of the approximating function at two values of x divided by the difference of “step” in the values of x. The three authors made the remarkable discovery that the steps in time could not be chosen arbitrarily but had to be smaller than some constant times the steps in the space variable. For the wave equation that constant was the reciprocal of the speed of propagation. For other equations there may be many such speeds or one may have a different kind of propagation, but there is always a limitation of space step (x) and time step (t). But the basic idea comes from this paper, and the constants are all known as CFL numbers. There is scarcely a talk or a paper on modeling phenomena governed by so-called explicit difference schemes where this number does not come up.
Friedrichs was interested, however, in proving existence theorems for partial differential equations by letting the mesh size and time step become vanishing small. The modeling aspect was a side product that only became important after World War II when one could compute something useful on a large computer. In later life, when Friedrichs was pressed to say something about the important role of computer modeling in applied mathematics to which he had made such a fundamental contribution, he simply would not bite.
After Friedrichs completed his dissertation and two years of an assistantship, according to Constance Reid’s fascinating obituary’ for the Intelligencer, Courant advised his shy young friend that in the severe competition for positions at German universities Friedrichs would need some special advantage and therefore he should become an applied mathematician. Thus, Friedrichs followed Theodore v. Karman to Aachen where v. Karman had become the first professor of aeronautical engineering. That was an exciting time in aerodynamics in Germany. First, it was a new and eminently practical subject, not like the exotic theories of quantum mechanics and relativity. Second, the interdiction of the building of airplanes in Germany in accordance with the Versailles treaty made the understanding of aerodynamics from a theoretical point of view a matter of vital concern to the nation. Friedrichs returned to Göttingen two years later with a deep knowledge of aerodynamics, which was to serve him later in America.
But his mathematical interest had shifted to the modern theory of operators in Hilbert spaces. He even rewrote a paper (1934) couched in classical language so that it was in von Neumann’s “new” abstract language. He solved several problems in spectral theory and used the new method to solve the initial value problem for hyperbolic equations with only energy integrals. This led some years later to the idea of weak solutions, a concept that is the backbone of the modern theory of linear and some nonlinear equations.
In 1931 Friedrichs was called to Braunschweig to the Technische Hochschule as a full professor, a rare recognition in prewar Germany for a man of thirty. But the Hitler era was about to begin. After five years of increasing political difficulties at Braunschweig, a visit to Courant who had emigrated to New York, and most important, after meeting Nellie Bruell, his future wife, Friedrichs saw clearly that he too would have to emigrate, which he did in 1937.
Once again under Courant’s influence, Friedrichs started dancing on what he liked to call his “other foot”—namely, doing applied mathematics. On the whole, until the end of the Second World War his main contributions were in fluid dynamics with Courant and in elasticity with J. J. Stoker.
When I first met him in 1946, Friedrichs was celebrating the end of the years of war effort in applied mathematics by teaching an innovative course in topology. But he and Courant were also finishing their basic and still used book on compressible fluid dynamics (1948,1). I was selected to edit it mainly for English and thus I came to know Friedrichs’s meticulous, very careful, and correct, but at times slightly pedantic, approach to the subject. Since English was after all my native language, I had some trouble with Friedrichs’s desire to find a rule (mostly from Fowler’s) to follow. His English was remarkable, but he never quite became idiomatic. Courant enlivened the text considerably, sometimes to Friedrichs’s disappointment at the expense of correctness.
By 1951 his spectral theory work, which he had renewed after 1945, led him into his fundamental work in the quantum theory of fields. He published five monographs, which inspired a number of today’s mathematical physicists, especially in the circle of Olga Ladyzhenskaya and Ludwig Fadeev in Russia. All the time as the spirit moved him he would do a basic piece for applied mathematics or a big chunk of writing for the famous Courant-Hilbert2 volume 2, completed in German in 1937 but essentially rewritten in English by many of the faculty of the burgeoning institute that was to become the Courant Institute in the early sixties.
By the 1940s, Friedrichs was recognized as one of the new leaders in American mathematics. He was elected to the National Academy of Sciences in 1959. He received many awards and honorary degrees culminating in the National Medal of Science in 1976, which he received “for bringing the powers of modern mathematics to bear on problems in physics, fluid dynamics and elasticity.”
Despite all the recognition, Friedrichs’s modesty could be overwhelming. When we discussed what should be republished in his Selecta, he kept reiterating about each of his discoveries that someone else had done it better later and why should someone want to read his less effective original presentation. I succeeded in most cases in overruling him.
He was never shy mathematically and was, by the 1950s, much less shy socially. He had a great impact on what Courant’s fledgling institute became and thereby he greatly influenced applied mathematics in America both directly and indirectly.
I must mention one other important influence. While afternoon teas as a fundamental ingredient of the intellectual life of a mathematician had been conceived at the Institute for Advanced Study in Princeton, Friedrichs carried the idea to the surroundings of NYU with his imposing principles. When the secretaries balked at washing the dishes after colloquia and Anneli Lax and I refused to take over, Friedrichs persuaded Courant to hire someone to make tea and coffee and wash up every day of the week. Today the notion that this custom is conducive to producing mathematics has been taken over almost everywhere.
He was a prodigious worker. Nellie shielded him from unnecessary dealings with the outside world, even to providing breakfast on a tray for his most productive early morning labors. But his five children, Walter, Liska, David, Christopher, and Martin, were a source of enormous interest and pleasure, and he meticulously guarded the time he spent with them from being interrupted by some mathematical spasm.
When he died at the age of eighty-one, one of Friedrichs’s wishes was that his last works that dealt with the true way to regard the uncertainty principle and other quantum mechanical concepts should be properly understood and seen correctly from a philosophical point of view.
See also
A SUMMARY OF THE MATHEMATICAL WORK AND PUBLICATIONS OF K. O. FRIEDRICHS
Note: K. O. Friedrichs wrote this document in December 1978 as his own very summary review of his primary work. It was followed by a list of his 150 publications written between 1927-1981 (including 81 journal articles, 36 lecture series, 28 reports and 5 separate books – numerous other publications where also turned into books.)
The main area of my research was the theory of partial differential equations; in particular of equations that represent laws of physics or engineering science (1927)
My first two publications concern an invariant formulation of Newton’s law of gravitation and the existence theory for the equations of elastic plates. (Ph.D. Thesis).
Among other work done in Göttingen in the late nineteen hundred twenties, I mention the work on the initial problem for linear hyperbolic equations (together with Hans Lewy) (1927) and the work done with H. Lewy and R. Courant on partial difference equations. (1928)
One of the points brought out in the latter paper was that one cannot replace a hyperbolic-differential equation by a difference equation in an arbitrary manner and expect that the solutions of the latter equations approach those of the first one. The ratio of the time difference to the space difference must be sufficiently small. This observation (made by H. Lewy) played a considerable role in later years when hyperbolic-differential equations were computed approximately with the aid of difference equations.
Among the subjects of applied mathematics in subsequent years a remark about the possibility of transforming the minimum of an energy integral into the maximum of another such integral may be mentioned (1929). This remark is still on occasion referred to.
My most significant work in applied mathematics, done with J. J. Stoker in the United States, was the work on the nonlinear boundary problem of the buckled plate. The problem concerned a circular plate subject to uniform compressive forces acting on the edge. If these forces are increased beyond a certain value the plate will buckle, as is well known. Our problem was to investigate what happens to the plate when the compressive forces are increased indefinitely. This was done by hand-computing and took many months. One feature found, which was surprising to us, was that eventually the radial stress at and near the center of the plate becomes a tensile one. This computational result (1939) was verified by an asymptotic-analysis. (In a concrete case this result has been verified experimentally some years ago.) (1941)
Somewhat later I published additional papers connected with plate theory. Two of them (1937)(1947), of a purely mathematical character, concern Korn’s inequality (one for two dimensions, one for any dimension). This inequality is needed in the theory of elastic plates with free boundaries.
In another paper the peculiar boundary conditions one may impose at the edge of a plate with a free boundary were derived by an asymptotic analysis, (with R. F. Dressler). (1949) (1950) (1961)
In the years after 1943 I did various investigations on Fluid Dynamics, partly together with R. Courant. This work led to the publication of the book on “Supersonic Flow and Shockwaves,” which appeared in 1948 and was widely used by mathematically minded aerodynamicists.
In a report “On the Mathematical Theory of Deflagration and Detonations” arguments are used which are related to the “boundary layer analysis” introduced by me somewhat earlier. This paper was reprinted in the “Lectures on Combustion Theory”, Courant Institute 1978.
There are various additional papers that belong more or less to applied mathematics, other than fluid dynamics.
In one of these papers (1937) the theory of perturbation of continuous spectra was initiated and later applied to problems of quantum theory. (1948)
In a paper on nonlinear oscillations (with W. Wasow), concerned with asymptotic considerations, the notion of “singular perturbation” was introduced. (1946)
Various asymptotic phenomena in mathematical physics were presented in the Josiah Willard Gibbs lecture given on invitation by the American Mathematical Society. (1955)
Among earlier purely mathematical work was a series of papers (1934)(1935)on the spectral theory of semi bounded linear operators in which von Neumann’s Hilbert space theory is effectively used. In a later paper (1939) on differential operators of the first order the notions of weak and strong solutions are introduced. It is proved, for the case of constant coefficients, that the weak and the strong solutions are identical. For this proof a particular class of transformations, later called mollifiers, is employed.
In a subsequent paper (1944) it is shown that this tool is strong enough to prove the identity of the weak and the strong solutions, also for first order operators with non-constant coefficients.
In all these considerations neither Lebesgue theory nor distribution theory is used.
Perhaps the most significant work in the years after 1937 was on a class of equation that I introduced and called symmetric hyperbolic linear differential equations. (1954) These are systems of the first order; the symmetry involved is that of the matrix of the coefficients of the differentiated terms. I proved the existence of the solution of the initial value problem. Again the treatment was based on general operator theory and again used modifiers. The equations have two striking properties: the eigenvectors associated with the matrix need not be distinct and the initial data need not have infinitely many derivatives but only a finite number. It is this class of equations to which essentially all non-degenerate equations of motion can be reduced.(1947)
A paper which is not immediately related to applied mathematics should be mentioned. It concerns the existence of differential forms on Riemannian manifolds. It is also based on the theory of linear operators. One feature of this work is the use of a basic inequality which is not covariant although the resultant differential forms are covariant. (l955)
The existence theory for “accretive” equations, which may be partly hyperbolic, partly elliptic, and partly of intermediate types, may be regarded as an extension of the previous work on elliptic and hyperbolic equations. An essential feature of this work was that one need not pay any attention to the places at which the type of the equation changes. (1958)
A boundary value problem for pseudo differential operators was treated, with P. D. Lax, in a paper of 1965. Additional material on such operators was presented in Courant Institute lecture notes of 1970.
In a series of five papers on quantum theory of fields an attempt was made to put this theory on a sound mathematical basis. (1951-53)
This attempt had only limited success. Still a few observations that resulted may be of some value. Among them is the observation that there are different, non-equivalent realizations of the basic field operators. These five papers reappeared in book on Mathematical Aspects of the Quantum Theory of Fields. (1953) A somewhat different approach to this theory is presented in a book on Perturbation of Spectra in Hilbert Space (1967-second printing)
There are a number of papers on fluid dynamics of which only few will be mentioned.
One paper, with D. H. Hyers, is on the existence of the solitary wave. (1954)
Another work which appeared in lecture notes is the one on wave motion in magneto fluid dynamics. In particular it is shown that these equations can be written as symmetric-hyperbolic ones. The equations of relativistic magneto fluid dynamics were also formulated; but that they can be put into symmetric-hyperbolic-form was shown only in later work. (1974)
In a note written with P. D. Lax (1971) systems of conservation equations are treated which possess a convex extension. One of the statements in this paper is that such equations can be reduced to symmetric-hyperbolic ones.
A somewhat different treatment of such conservation equations is given by me in 1974. This treatment is applied in particular to the laws of relativistic electro magneto fluid dynamics. (1977) One of the results is the reduction of the equations of relativistic magneto fluid dynamics to symmetric-hyperbolic ones. Another result is the resolution of the controversy about the proper formulation of the energy momentum tensor of relativistic-electro magnetism.
In a paper “On the Notion of State in Quantum Mechanics” it is shown that the future values of unobserved observables can be determined in terms (1979) of their unobserved initial values.