Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors

Cushman, R.

doi:10.1090/S0002-9947-1978-0478223-8

Examples of nonintegrable analytic Hamiltonian vector fields with no small divisors
HTML articles powered by AMS MathViewer

by R. Cushman

Trans. Amer. Math. Soc. 238 (1978), 45-55

DOI: https://doi.org/10.1090/S0002-9947-1978-0478223-8

PDF | Request permission

Abstract:

Any analytic symplectic diffeomorphism $\Phi$ of a symplectic manifold M is the Poincaré map of a real analytic Hamiltonian vector field ${X_H}$. If $\Phi$ does not have an analytic integral, then ${X_H}$ has no analytic integral which is not a power series in H. Let $M = {{\mathbf {R}}^2}$. If $\Phi$ has a finite contact homoclinic point, then $\Phi$ is nonintegrable. Also Moser’s polynomial mapping is nonintegrable.

References

V. I. Arnol′d and A. Avez, Ergodic problems of classical mechanics, W. A. Benjamin, Inc., New York-Amsterdam, 1968. Translated from the French by A. Avez. MR 0232910

Foundations of mechanics

36

Richard B. Barrar, Isolated fixed points in celestial mechanics, J. Math. Anal. Appl. 36 (1971), 506–517. MR 289862, DOI 10.1016/0022-247X(71)90034-5

Analytic integrable area preserving diffeomorphisms of the closed

disc

Foundations of differential geometry

Jürgen Moser, Nonexistence of integrals for canonical systems of differential equations, Comm. Pure Appl. Math. 8 (1955), 409–436. MR 79171, DOI 10.1002/cpa.3160080308
Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
Jürgen K. Moser, Lectures on Hamiltonian systems, Mem. Amer. Math. Soc. 81 (1968), 60. MR 0230498
Jürgen Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math. 9 (1956), 673–692. MR 86981, DOI 10.1002/cpa.3160090404
S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
Carl Ludwig Siegel, On the integrals of canonical systems, Ann. of Math. (2) 42 (1941), 806–822. MR 5818, DOI 10.2307/1969262
Carl Ludwig Siegel and Jürgen K. Moser, Lectures on celestial mechanics, Die Grundlehren der mathematischen Wissenschaften, Band 187, Springer-Verlag, New York-Heidelberg, 1971. Translation by Charles I. Kalme. MR 0502448, DOI 10.1007/978-3-642-87284-6
E. Zehnder, Homoclinic points near elliptic fixed points, Comm. Pure Appl. Math. 26 (1973), 131–182. MR 345134, DOI 10.1002/cpa.3160260204