A Brief Biographical Sketch of Ken Meyer

Ken Meyer has been a leading researcher in dynamical systems and celestial mechanics since the1960s. Based on recent conversations with him, I give here the briefest possible outline of Ken’sacademic life, followed by a chronological survey of some of the best and most representative partsof his work.

Kenneth R. Meyer was born May 26, 1937 in Cincinnati, Ohio, and was raised there. He attendedCornell 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 1962and 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 departmenthead for three years and receiving the honorific Charles Phelps Taft professorship in 1984. He retiredin 2003, and continues to do research as an emeritus professor at UC. Ken enjoys explaining how henever endured the process of an academic promotion during his career, but was instead hired directlyinto 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 focuson his career. Rather than give a commented laundry list of his one hundred or so publications, I’llconcentrate on a few key results and activities. This will lead us through the phases of Ken’s variedcareer, 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 Institutefor Advanced Studies (RIAS) in Baltimore, Maryland. (In those days, corporations receiving largecontracts 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 asSolomon Lefschetz, Rudolph Kalman, Harold Kushner, Rodney Driver, Jack Hale, Joseph LaSalleand others. Ken established lasting connections with several leading mathematicians at RIAS. Infact, LaSalle and Lefschetz took such strong interest in his work that Ken counts them as unofficialthesis advisors alongside Archibald Macintyre, his official advisor at UC. Ken says that Macintyretrained him in analysis (especially Bloch’s theorem) and showed him how to do research; LaSalle gavehim a problem (on Liapunov stability); and Lefschetz took increasing interest in his work (especiallywhen Ken showed that it corrected an earlier oversight of Lefschetz).

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

When the government mandate for basic research was relaxed in the mid 1960s, the core ofresearchers 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

1

delay equations, which at that time were studied using traditional methods from ordinary differentialequations. 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 (evolutionoperators on Banach spaces) to show how spectral theory and semigroups could be used to getentirely new results.

Minnesota

After moving to the University of Minnesota in 1967, Ken ran into Julian Palmore, a fellow studentfrom his undergraduate days at Cornell. Julian introduced Ken to celestial mechanics and thenumerical work of Andre 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. WhenKen heard this, he had a sort of epiphany, and responded by writing what he now recalls as oneof his favorite papers [3] describing the generic behavior, under parameter variations, of fixed andperiodic points in area-preserving maps of the plane.

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

This was the start of a gold mine of bifurcationproblems for Ken (and also for Jacques Henrard andDieter Schmidt, Ken’s first PhD student). Ken wishesonly that better symbolic computation methods hadbeen available at the time—a bigger shovel to digmore gold. At any rate, Julian’s computations ledto the first proof, by Ken and Dieter, of what laterbecame known as the Hamiltonian-Hopf bifurcation[4].

In the spring of 1970, at Northwestern Universityin Evanston, Illinois, a number of young researchersin dynamical systems (including Ken, Clark Robin-son, John Franks, Bob Williams, Charles Conley, JoelRobbin and others) held a small research conferencethat would later be seen as the first “Midwest Dy-namical Systems Conference” (MWDSC). This con-ference series continues today and has grown into alarge event, regularly funded by the National ScienceFoundation. After Clark Robinson, Ken has proba-bly been the most frequent organizer and fundraiserfor the MWDSC over the last four decades.

Meanwhile, in 1971, an important dynamics meet-ing took place in Salvador da Bahia, Brazil, whereKen presented some original results based on his read-ings of Steve Smale’s recent papers. These includeda theorem on regular reduction (now often called Marsden-Weinstein reduction in its most generalform), and the whole process taught Ken the perils of including an important result in a conferenceproceedings [5].

Another colleague of Ken’s at Minnesota was Larry Markus. One day at lunch, the two satdown and outlined what is now one of the most well-known papers for each of them, though it tookanother 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 statisticalmechanics.”

2

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

Cincinnati

When Ken returned to UC in 1972, he joinedAndre Deprit there, and Dieter Schmidt camesoon thereafter. Jacques Henrard also visitedthe department at times. Together with a fewothers, this group began some of the first foraysinto symbolic computation, using computers todo mathematics in a rigorous way. Althoughit’s now routine to use Mathematica or Maplein mathematical research, Ken recalls that somewere skeptical of such methods at the time. Kencites an early paper [7] that encountered initialresistance from referees. Yet many other suchpapers followed, and symbolic methods are nownot only accepted, but essential to research incelestial mechanics and other areas.

While Ken was department head at UC andon 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 actionto show that administrators vacillate infinitelyoften [8]. Other satires followed, some unpub-lished (but Dieter still has them).

In 1993 at UC, I showed Ken and Chris Mc-Cord a translation of Alain Albouy’s thesis I wasworking on. I didn’t think much about it at thetime, but they returned soon after, showing keen interest in a particular passage and wanting to besure I had translated it faithfully. Following Steve Smale, Albouy had found a gap in G.D. Birkhoff’sargument showing that the integral manifolds of the N body problem change only at relative equilib-ria, and he conjectured that they also changed elsewhere. Together with Ken’s PhD student QuidongWang, Ken and Chris were able to combine methods from analysis and algebraic topology to showrigorously that Albouy was right and Birkhoff was wrong [9].

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

Following his retire-ment, Ken gave up ad-ministration, cut back onteaching, but continued re-search as usual. He was es-pecially happy to connectwith Patricia Yanguas andJesus Palacian of the Pub-lic University of Navarrein Pamplona, Spain, twomathematicians who carryon the spirit of Depritand Henrard using the bestsymbolic and numericalcomputation techniques toexplore the boundaries of

3

what is known in celestial mechanics. I had the privilege to work with and learn from this team afew years ago when we wrote a joint paper [10]. I hope we’ll have further chances to collaboratein the future, and even more, I hope Ken enjoys a long and fruitful retirement here in his nativeCincinnati. But even were he to quit working now, he would already have accomplished more thanmost 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 theAmerican Mathematical Society, No. 76, American Mathematical Society, Providence, R.I., 1967.

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

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

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

[6] L. Markus and K.R. Meyer, Generic Hamiltonian dynamical systems are neither integrable nor er-godic, 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 onApplied Mathematics 32 (1), (1977) 133–145.

[8] K.R. Meyer, An application of Poincare’s recurrence theorem to academic administration (asatire), 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, Memoirsof the American Mathematical Society 132, no. 628 (1998) 1–91.

[10] P. Yanguas, J. Palacian, K.R. Meyer, and H.S. Dumas, Periodic solutions in Hamiltonian sys-tems, 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), AmericanMathematical Society, Providence, R.I, 1988.

4

Computer Aided Proofs in Analysis (Ed. with D.S. Schmidt), IMA Volumes in Mathematics and itsApplications 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 andits Applications 44, Springer-Verlag, New York, 1992.

Hamiltonian Dynamical Systems: History, Theory, and Applications (Ed. with H.S. Dumas andD.S. Schmidt), IMA Volumes in Mathematics and its Applications 63, Springer-Verlag, NewYork, 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. Halland D. Offin), Springer-Verlag, New York, 2009.

5