Ken Meyer has been a leading researcher in dynamical systems and celestial mechanics since the 1960s. Based on recent conversations with him, I give here the briefest possible outline of Ken’s academic life, followed by a chronological survey of some of the best and most representative parts of his work.

Kenneth R. Meyer was born May 26, 1937 in Cincinnati, Ohio, and was raised there. He attended Cornell University and finished his bachelors degree in engineering physics in 1960. Returning home, he received masters and PhD degrees in mathematics from the University of Cincinnati (UC) in 1962 and 1964. After graduate school, Ken first took research and teaching positions at Brown University (1964-67), then moved to the University of Minnesota where he became associate professor in 1968. He came back to UC in Cincinnati as full professor in 1972 and remained there, serving as department head for three years and receiving the honorific Charles Phelps Taft professorship in 1984. He retired in 2003, and continues to do research as an emeritus professor at UC. Ken enjoys explaining how he never endured the process of an academic promotion during his career, but was instead hired directly into each successive position.

Ken is married to Carol Meyer, has a son, Karl, and two grandchildren, Max and Charlotte. There is of course much more personal history to tell, but—at Ken’s request—this sketch will focus on his career. Rather than give a commented laundry list of his one hundred or so publications, I’ll concentrate on a few key results and activities. This will lead us through the phases of Ken’s varied career, and introduce some of the people he got to know and the stories that go with them.

RIAS and Brown

During his last year of graduate studies at UC, Ken was a research assistant at the Research Institute for Advanced Studies (RIAS) in Baltimore, Maryland. (In those days, corporations receiving large contracts from the U.S. Government were required to conduct a certain amount of “basic research” and, roughly speaking, RIAS was set up by the Glenn Martin Company to fulfill this purpose.) In its prime, RIAS was a center of mathematical excellence which included researchers such as Solomon Lefschetz, Rudolph Kalman, Harold Kushner, Rodney Driver, Jack Hale, Joseph LaSalle and others. Ken established lasting connections with several leading mathematicians at RIAS. In fact, LaSalle and Lefschetz took such strong interest in his work that Ken counts them as unofficial thesis advisors alongside Archibald Macintyre, his official advisor at UC. Ken says that Macintyre trained him in analysis (especially Bloch’s theorem) and showed him how to do research; LaSalle gave him a problem (on Liapunov stability); and Lefschetz took increasing interest in his work (especially when Ken showed that it corrected an earlier oversight of Lefschetz).

Before leaving RIAS, Ken began work with Polish physicist Wictor Baron on what became his first striking result. At the time, folk wisdom circulating among nuclear engineers said that in models of nuclear reactors, one could ignore the effect of so-called delayed neutrons (neutrons arising from secondary—as opposed to primary—nuclear reactions) because the delayed neutrons had a stabilizing effect. But Baron and Meyer gave a counterexample [1] showing that these neutrons could in fact be destabilizing. This caused nuclear engineers to sit up, take notice, and revise their models accordingly.

When the government mandate for basic research was relaxed in the mid 1960s, the core of researchers in dynamical systems at RIAS moved to Brown University in Providence, Rhode Island, and Ken moved with them as an assistant professor. Jack Hale had become interested in differential delay equations, which at that time were studied using traditional methods from ordinary differential equations. But Ken soon teamed with Jack to permanently alter the direction of research in this area. In their paper [2], they combined dynamical systems methods with functional analysis (evolution operators on Banach spaces) to show how spectral theory and semigroups could be used to get entirely new results.

Minnesota

After moving to the University of Minnesota in 1967, Ken ran into Julian Palmore, a fellow student from his undergraduate days at Cornell. Julian introduced Ken to celestial mechanics and the numerical work of André Deprit and Jacques Henrard, and gave him detailed descriptions of their (and his own) methods for computing periodic orbits in the restricted three body problem. When Ken heard this, he had a sort of epiphany, and responded by writing what he now recalls as one of his favorite papers [3] describing the generic behavior, under parameter variations, of fixed and periodic points in area-preserving maps of the plane.

This was the start of a gold mine of bifurcation problems for Ken (and also for Jacques Henrard and Dieter Schmidt, Ken’s first PhD student). Ken wishes only that better symbolic computation methods had been available at the time—a bigger shovel to dig more gold. At any rate, Julian’s computations led to the first proof, by Ken and Dieter, of what later became known as the Hamiltonian-Hopf bifurcation [4]. In the spring of 1970, at Northwestern University in Evanston, Illinois, a number of young researchers in dynamical systems (including Ken, Clark Robinson, John Franks, Bob Williams, Charles Conley, Joel Robbin and others) held a small research conference that would later be seen as the first “Midwest Dynamical Systems Conference” (MWDSC). This conference series continues today and has grown into a large event, regularly funded by the National Science Foundation. After Clark Robinson, Ken has probably been the most frequent organizer and fundraiser for the MWDSC over the last four decades.

Jacques Henrard with Ken in Namur, Belgium, 1970s.

Meanwhile, in 1971, an important dynamics meeting took place in Salvador da Bahia, Brazil, where Ken presented some original results based on his readings of Steve Smale's recent papers. These included a theorem on regular reduction (now often called Marsden-Weinstein reduction in its most general form), and the whole process taught Ken the perils of including an important result in a conference proceedings [5].

Another colleague of Ken’s at Minnesota was Larry Markus. One day at lunch, the two sat down and outlined what is now one of the most well-known papers for each of them, though it took another half-decade to write up [6]. Ken says that the paper’s fame rests more on its catchy title (“Generic Hamiltonian dynamical systems are neither integrable nor ergodic”) than on its content, while Markus, in a recent e-mail, says tongue-in-cheek that it “shattered the basis of statistical mechanics.”

 

 

 

Cincinnati

When Ken returned to UC in 1972, he joined André Deprit there, and Dieter Schmidt came soon thereafter. Jacques Henrard also visited the department at times. Together with a few others, this group began some of the first forays into symbolic computation, using computers to do mathematics in a rigorous way. Although it’s now routine to use Mathematica or Maple in mathematical research, Ken recalls that some were skeptical of such methods at the time. Ken cites an early paper [7] that encountered initial resistance from referees. Yet many other such papers followed, and symbolic methods are now not only accepted, but essential to research in celestial mechanics and other areas. While Ken was department head at UC and on his way to a meeting of heads in Columbus, Ohio, he had the idea for the first of his satires, in which he used the principle of least action to show that administrators vacillate infinitely often [8]. Other satires followed, some unpublished (but Dieter still has them).

Ken with Dieter Schmidt at the Palomar Observatory in California, 1987.

In 1993 at UC, I showed Ken and Chris McCord a translation of Alain Albouy’s thesis I was working on. I didn’t think much about it at the time, but they returned soon after, showing keen interest in a particular passage and wanting to be sure I had translated it faithfully. Following Steve Smale, Albouy had found a gap in G.D. Birkhoff ’s argument showing that the integral manifolds of the N body problem change only at relative equilibria, and he conjectured that they also changed elsewhere. Together with Ken’s PhD student Quidong Wang, Ken and Chris were able to combine methods from analysis and algebraic topology to show rigorously that Albouy was right and Birkhoff was wrong [9].

Carol Meyer, Ken, and Jack Hale in Florence, Italy, 1993.

Following his retirement, Ken gave up administration, cut back on teaching, but continued research as usual. He was especially happy to connect with Patricia Yanguas and Jesús Palacián of the Public University of Navarre in Pamplona, Spain, two mathematicians who carry on the spirit of Deprit and Henrard using the best symbolic and numerical computation techniques to explore the boundaries of what is known in celestial mechanics. I had the privilege to work with and learn from this team a few years ago when we wrote a joint paper [10]. I hope we’ll have further chances to collaborate in the future, and even more, I hope Ken enjoys a long and fruitful retirement here in his native Cincinnati. But even were he to quit working now, he would already have accomplished more than most of us could in several lifetimes.

 

 

H.S. Dumas, June 2011

 

 

 

References

[1] W. Baron and K.R. Meyer,

Effect of delayed neutrons on the stability of a nuclear power reactor, Nuclear Science and Engineering, 24 (1966) 35–61.

[2] J.K. Hale and K.R. Meyer, A class of functional equations of neutral type, Memoirs of the American Mathematical Society, No. 76, American Mathematical Society, Providence, R.I., 1967.

[3] K.R. Meyer, Generic bifurcation of periodic points, Transactions of the American Mathematical Society, 149 (1970) 95–107.

[4] K.R. Meyer and D.S. Schmidt, Periodic orbits near L4 for mass ratios near the critical mass ratio of Routh, Celestial Mechanics, 4 (1971) 99–109.

[5] K.R. Meyer, Symmetries and integrals in mechanics, in Dynamical Systems (Proceedings of the Symposium at University of Bahia, Salvador, Brazil, 1971) pp. 259–272, Academic Press, New York, 1973.

[6] L. Markus and K.R. Meyer, Generic Hamiltonian dynamical systems are neither integrable nor ergodic, Memoirs of the American Mathematical Society, No. 144, American Mathematical Society, Providence, R.I., 1974.

[7] J. Henrard and K.R. Meyer, Averaging and bifurcation in symmetric systems, SIAM Journal on Applied Mathematics, 32 (1), (1977) 133–145.

[8] K.R. Meyer, An application of Poincaré’s recurrence theorem to academic administration (a satire), American Mathematical Monthly, 88 (1), (1981) 32–33.

[9] C.K. McCord, K.R. Meyer, and Q. Wang, Integral manifolds of the three body problem, Memoirs of the American Mathematical Society 132, no. 628 (1998) 1–91.

[10] P. Yanguas, J. Palacián, K.R. Meyer, and H.S. Dumas, Periodic solutions in Hamiltonian systems, averaging, and the lunar problem, SIAM Journal on Applied Dynamical Systems, 7 (2), (2008) 311–340.

 

Books (co)edited and (co)authored by Ken Meyer

Hamiltonian Dynamical Systems, Contemporary Mathematics, 81 (Ed. with D. Saari), American Mathematical Society, Providence, R.I, 1988.

Computer Aided Proofs in Analysis (Ed. with D.S. Schmidt), IMA Volumes in Mathematics and its Applications, 28, Springer-Verlag, New York, 1991.

Introduction to Hamiltonian Dynamical Systems and the N-Body Problem (with G.R. Hall), Springer-Verlag, New York, 1992.

Twist Mappings and Their Applications (Ed. with R. McGehee), IMA Volumes in Mathematics and its Applications, 44, Springer-Verlag, New York, 1992.

Hamiltonian Dynamical Systems: History, Theory, and Applications (Ed. with H.S. Dumas and D.S. Schmidt), IMA Volumes in Mathematics and its Applications, 63, Springer-Verlag, New York, 1995.

Periodic Solutions of the N-Body Problem, Lecture Notes in Mathematics, No. 1719, Springer-Verlag, New York, 1999.

Introduction to Hamiltonian Dynamical Systems and the N-Body Problem, 2nd Edition (with G.R. Hall and D. Offin), Springer-Verlag, New York, 2009.