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Файл:Integrable hamiltonian systems and spectral theory
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J¨urgen Moser
Integrable
Hamiltonian Systems
and Spectral Theory
Moscow Izhevsk
2019
Published by
Regular and Chaotic Dynamics, Moscow-Izhevsk
Universitetskaya, 1, Izhevsk, Russia, 426034
Phone: (7–3412) 50–02–95
Fax: (7–3412) 50–02–95
E-mail: borisov@rcd.ru
Acknowledgement. The publisher is grateful to Springer-Verlag for the permission to reprint the papers included in this volume.
Moser, J¨urgen
INTEGRABLE HAMILTONIAN SYSTEMS AND SPECTRAL THEORY
c
2019 by Regular and Chaotic Dynamics, Moscow-Izhevsk
c
2019 by Institute of Computer Science, Moscow-Izhevsk
All rights reserved. This work may not be translated or copied in whole or in
part without the written permission of the publisher, except for brief excerpts in
connection with reviews or scholarly analysis. Use in connection with any form
of information storage and retrieval, electronic adaptation, computer software, or
by similar or descriptive names, trade names, trademarks, etc., in this publication,
even if the former are not especially identified, is not to be taken as a sign that
such names, as understood by the Trade Marks and Merchandise Marks Act,
may accordingly be used freely by anyone.
ISBN 978-5-4344-0697-0
Printed on acid-free paper
Printed in the Russian Federation
Contents
Curriculum Vitae ............................ 4
Editorial Note .............................. 14
Finitely Many Mass Points on the Line under the Influence of an Expo-
nential Potential — an Integrable System .............. 15
§ 1. Analogue of the Toda Lattice for Finitely Many Mass Points . . . 15
§ 2. Flaschka’s Form of the Differential Equation and Asymptotic
Behavior............................... 17
§3. PartialFractionsandContinuedFractions............. 21
§4. SolutionoftheScatteringProblem................. 27
§5. AssociatedDifferentialEquations ................. 34
Three Integrable Hamiltonian Systems Connected with Isospectral De-
formations .............................. 41
§ 1. Introduction . . ........................... 41
§2. IsospectralDeformations...................... 45
§3. Then-Particle System on the Line with the Inverse Square Potential 47
§4. AsymptoticBehavior,Marchioro’sConjecture .......... 50
§5. ThePeriodicCase—Sutherland’sEquation............ 53
§6. RationalCharacteroftheSolutionof(2.4) ............ 56
§ 7. The Scattering Problem Associated with the Equation of Kac and
VanMoerbeke............................ 60
Various Aspects of Integrable H amiltonian Systems .......... 65
§1. IntegrableHamiltonianSystems .................. 65
§ 2. Examples of Integrable Systems, Isospectral Deformations . . . . 68
§3. ReductionofaHamiltonianSystemwithSymmetries....... 71
§4. TheInverseSquarePotential.................... 80
§5. ExtensionoftheGeodesicFlow .................. 89
§ 6. Geodesics on an Ellipsoid . .................... 96
§7. AnIntegrableSystemontheSphere................102
§ 8. Hill’s Equation ...........................110
4 Contents
Geometry of Quadrics and Spectral Theory .............. 123
§ 1. Introduction . . ...........................123
a. Background . . .......................123
b. Geodesics on an Ellipsoid . . . . . ............124
c. Perturbations of Rank 2 ...................127
d. Hyperelliptic Curve . . ...................129
e. Applications.........................129
f. ConnectionwithM.Reid’sResult[15]...........130
g. FinalRemarks........................130
§2. PerturbationofRank2 .......................131
a. IsospectralManifolds....................131
b. IsospectralDeformations..................133
c. The Action of Gl(2,R) ..................137
d. TraceFormulae .......................138
§3. ConnectionwithConfocalQuadrics ................140
a. Integrals for the Geodesic Flow on the Ellipsoid . ....140
b. IsospectralDeformation ..................143
c. Interpretation of the Eigenvalues and the Frame of L ...145
d. Joachimsthal’sIntegral ...................147
§ 4. The Hyperelliptic Curve . . . ...................148
a. The Isospectral Manifold M(λ) ..............148
b. AnInverseSpectralProblem................152
c. TheSymplecticStructure..................156
d. DegenerateCase ......................159
e. LimitCases.........................160
§5. ExamplesofIntegrableFlows ...................162
a. ConstrainedSystems ....................162
n−1
b. A Mass Point on the Sphere S
: |x| =1under the
Influence of the Force −Ax (C.Neumann[14])......164
c. A Mass Point on the Ellipsoid Q
(x)+1=0under the
0
Influence of the Force −ax (Jacobi[6])..........165
d. Geodesic Flow on the Orthogonal Group (Manakov [8],
Mischenko[11]) ......................167
e. Hill’s Equation (McKean and Trubowitz [9, 10]) . ....168
§6. Appendix ..............................171
Contents 5
Integrable Hamiltonian Systems and Spectral Theory ......... 175
§ 1. Introduction . . ...........................175
§ 2. Classical Integrable Hamiltonian Systems and Isospectral Defor-
mations ...............................179
1. Hamiltoniansystems ....................179
2. Integrals...........................180
3. Perturbationofintegrablesystems.............183
4. Theinversesquarepotential ................184
5. ConstrainedHamiltoniansystems .............186
§ 3. Geodesics on an Ellipsoid and the Mechanical System of
C.Neumann.............................189
1. Geodesic flow on the ellipsoid . . . ............189
2. Confocalquadrics,constructionofintegrals........191
3. Isospectraldeformations ..................193
4. ThemechanicalproblemofC.Neumann..........194
5. The connection between the two systems via the Gauss
mapping...........................195
6. TheRiemannsurface....................199
§ 4. The Schr¨odinger Equation for Almost Periodic Potentials . . . . 202
1. Thespectralproblem....................202
2. Theperiodiccase......................203
3. Almostperiodicpotential..................206
4. Therotationnumber ....................207
5. TheGreen’sfunctionandatraceformula .........208
6. ConnectionwiththeKdVequation.............211
§5. FiniteBandPotentials .......................214
1. Formulationoftheproblem ................214
2. Representation of G(x, x; λ) in terms of partial fractions 215
3. Connectionwiththemechanicalproblem .........217
4. Solutionoftheinverseproblem ..............219
5. Finitegappotentialsasalmostperiodicfunctions.....221
6. The elliptic coordinates on the sphere . . . . . . .....223
7. Alternativechoiceofthebranchpoints ..........224
§6. LimitCases,BargmannPotentials .................225
1. Schwarz–Christoffelmapping...............225
2. Basis for the frequency module . . ............226
3. Stationary solutions and their stability behavior . .....228
4. The flow on the unstable manifold W
) ........229
+(en
6 Contents
5. TheBargmannpotentials..................231
6. A focussing property on S
2
................234
7. N-solitons..........................236
8. Concludingremarks.....................237
Discrete Versions of Some Classical Integrable Systems and Factorization
of Matrix Polynomials ........................ 241
§ 0. Introduction . . ...........................241
§1. TheDiscreteVersionoftheDynamicsofaRigidBody......246
1.1. TheEquationsof«Motion» ................246
T
1.2. The Solution of the Matrix Eq. (6): ω
J −Jω = M . . 249
1.3. IsospectralDeformations..................252
1.4. TheSymplecticGeometryofEq.(6) ...........254
1.5. TheIntegrationoftheDiscreteEulerEquation ......258
1.6. Explicit Formulas for the Discrete Dynamics of the 3-Di-
mensionalRigidBody ...................261
§ 2. The Discrete Dynamics on Stiefel Manifolds and the Heisenberg
ChainwithClassicalSpins.....................264
2.1. The Equation of the Dynamics and Isospectral Deformations265
2.2. Discrete Version of the Neumann System and the Heisen-
hergChainWithClassicalSpins..............266
§ 3. The Billiard Inside an Ellipsoid . . . . . . ............269
3.1. The Splittings and Isospectral Deformations ........270
3.2. Connection Between the Ellipsoidal Billiard and the Dis-
creteNeumannSystem...................272
Curriculum Vitae
Personal Data
Born: July 4, 1928, K¨onigsberg, Germany; U S Citizen
Education
1947 – 1952 Student at the University of G¨ottingen, Germany
1952 Dr. rer. nat. (Ph.D.), University of G¨ottingen
Employment and Professional History
1953 – 1954 Fulbright Fellowship, to visit New York University
1954 – 1955 Assistant (with C. L. Siegel) in G¨ottingen
1955 – 1956 Research Associate, New York University
1956 – 1957 Assistant Professor, New York University
1957 – 1960 Associate Professor, M.I. T. (Massachusetts Institute of Techno-
logy), Cambridge
8 Curriculum Vitae
1960 – 1980 Professor, Courant Institute of Mathematical Sciences, New York
1960 – 1967 Consultant to IBM, Yorktown Heights
1961 – 1963 Sloan Fellowship, to visit USSR
1967 – 1970 Director of the Courant Institute, New York
1980 – 1995 Professor Eidgen¨ossische Technische Hochschule, ETH Z ¨urich,
Switzerland
1983 – 1986 President of the International Mathematical Union (IMU)
1984 – 1995 Director of the Mathematics Research Institute, ETH Z ¨urich
(Forschungsinstitut f¨ur Mathematik – FIM)
1991 – 1997 Obmann der Sektion (reine) Mathematik der Deutschen Akade-
mie der Naturforscher Leopoldina
Memberships
American Mathematical Society
Society for Industrial and Applied Mathematics (SIAM)
International Astronomical Union (IAU) – Consultant
American Academy of Arts and Sciences, Cambridge/MA (1964)
National Academy of Sciences of the USA (1971)
Akademie der Wissenschaften und der Literatur, Mainz, Germany (1981)
The Royal Swedish Academy of Sciences (1981)
Deutsche Akademie der Naturforscher Leopoldina, Halle, Germany (1982)
International Mathematical Union, President (1983 – 1986)
The Finnish Academy of Science and Letters (1987)
Российская академия наук (1994)
Acad´emie des Sciences de la France (1995)
Московское Математическое Общество (1995)
The London Mathematical Society (1996)
Schweizerische Mathematische Gesellschaft (1997)
Awards
George D.Birkhoff Prize in Applied Mathematics (1968)
Craig Watson Medal, National Academy of Sciences, USA (1969)
J. von Neumann Lecture, SIAM, Seattle (1984)
L. E. J. Brouwer Medal, Groningen (1984)
Professor Hon´orario, IMPA, Rio de Janeiro (1989)
Dr. rer. nat. h. c., Ruhr-Universit¨at Bochum (1990)
Doctor h. c., Universit´e Pierre et Marie Curie de Paris (1990)
Curriculum Vitae 9
Georg-Cantor-Medaille, DMV-Tagung Berlin (1990)
Wolf Prize (Wolf Foundation Israel) (1994/95)
Publications
1) St¨orungstheorie des kontinuierlichen Spektrums f¨ur gew¨ohnliche Differentialgle-
ichungen zweiter Ordnung. Math. Ann. 125, 1953, 366–393.
2)¨Uber periodische L¨osungen des restringierten Dreik¨orperproblems, die sich erst
nach vielen Uml¨aufen schliessen. Math. Ann 126, 1953, 325–335.
3)¨Uber periodische L¨osungen kanonischer Differentialgleichungssysteme. Nachrich-
ten der Akademie der Wissenschaften, G¨ottingen, Math. Phys. Kl. IIa, 1953,
23–48.
4) Singular perturbation of eigenvalue problems for linear differential equations of
even order. Comm. Pure Appl. Math. 8, 1955, 251–278.
5) Nonexistence of integrals for canonical systems of differential equations. Comm.
Pure Appl. Math. 8, 1955, 409–436.
6) Stabilit¨atsverhalten kanonischer Differentialgleichungssysteme. Nachr. Akad.
Wiss. G¨ottingen. Math. Phys. Kl. IIa, 1955, 87–120.
7) The resonance lines for the synchroton. Proc. of the CERN Symposium, I, 1956,
290–292.
8) Analytic invariants on an area-preserving mapping near an unstable fixed point.
Comm. Pure Appl. Math. 9, 1956, 673–692.
9) On the generalizations of a theorem of A. Liapounoff. Comm. Pure Appl. Math.,
11, 1958, 257–271.
10) New aspects in the theory of stability of Hamiltonian systems. Comm. Pure Appl.
Math., 11, 1958, 81–114.
11) Stability of the Asteroids. The Astronomical Journal, 63, 1958, 439–443.
12) On the elimination of the irrationality condition and Birkhoff’s concept of complete
stability. Boletin de la Soc. Mat. Mexicana, 1960, 167–175.
13) On the integrability of an area-preserving Cremona mappings near an elliptic fixed
point, Boletin de la Soc. Mat. Mexicana, 1960, 176–180.
14) Remarks on the preceding paper of Louis Howard. Journal of Math. Phys., 37,
1959, 299–304.
15) A new proof of di Giorgi’s theorem concerning the regularity problem for elliptic
differential equations. Comm. Appl. Math., Vol. 13, 1960, 457–468.
10 Curriculum Vitae
16) Bistable system of differential equations. Symposium on the numerical treatment of
ordinary differential equations, integral and integro-differential equations. Proc. of
the Rome Symposium (Sept. 1960), organized by the Prov. Internat. Computation
Centre, Birkh¨auser Verlag, Basel, 1960, 320–329.
17) The order of a singularity in Fuchs’ theory. Math. Zeitschrift, 72, 1960, 379–398.
18) Bistable systems of differential equations with applications to tunnel diode circuits.
IBM Journal of Research and Development, Vol. 5, No.3, 1961, 226–240.
19) A new technique for the construction of solutions for nonlinear differential equa-
tions. Proc. Nat. Acad. of Sci., USA, Vol. 47, No. 11, 1961, 1824–1831.
20) On the regularity problem for elliptic and parabolic differential equations. Pro-
ceedings of an International Conference on Partial Differential Equations and Continuum Mechanics. The Univ. of Wisconsin Press, 1961, 159–169.
21) On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl.
Math., 14, 1961, 577–591.
22) Stability and nonlinear character of ordinary differential equations. Nonlinear
problems. Proc. Symposium, Madison, Wisconsin, April 30 – May 1, 1962, Ed.
Langer. The University of Wisconsin Press 1963, 139–150.
23) On invariant curves of area-preserving mappings of an annulus. Nachr. Akad.
Wiss. G¨ottingen, Math. Phys. Kl. IIa, 1962, 1–20.
24) New results on the stability of periodic motions. Proc. Internat. Congress Math.,
Stockholm 1962, 584–586.
25) Perturbation theory for almost periodic solutions for undamped nonlinear differ-
ential equations. Internat. Symposium on Nonlinear Differential Equations and
Nonlinear Mechanics. Acad. Press, 1963, 71–79.
26) On the differential equations of electrical circuits and the global nature of the
solutions. Internat. Symposium on Nonlinear Diff. Equations and Nonlinear
Mechanics. Acad. Press, 1963, 147–154.
27) Some problems and results in the theory of nonlinear differential equations. Proc.
IBM Scientific Computing Symposium. Dec. 9–11, 1963; Data Proc. Division,
White Plains, N.Y., 1965, 5–16.
28) On invariant manifolds of vector fields and symmetric partial differential equations.
Differential Analysis, Bombay Coll., 1964, 227–236.
29) On the volume elements on a manifold. Transactions of the AMS, 120, No. 2,
1965, 286–294.
30) Some results in the stability of nonlinear networks containing negative resistances.
IEEE Trans. on Circuit Theory. Corresp., Vol. CT-11, No. 1, 1964, 165–167.
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