# Formal Integrals for a Nonintegrable Dynamical System: Preliminary Report

## Abstract

The Birkhoff-Gustavson normal form has been generated to a high order for the Henon-Heiles model Hamiltonian. This analysis provides approximate integrable classical dynamics for the well-studied stochastic behavior of this system. Accelerated convergence of the series expansion for a second integral of the motion shows divergence of the series only in small regions of phase space, indicating approximate local integrability of the dynamics. The regions of divergence correspond well to the regions where stochastic motion first appears. If the regions of divergence are much smaller than Planck’s constant, one would expect such behavior would be irrelevant with respect to quantum mechanics. This result lends support to the semiclassical quantization methods of Swimm and Delos and of Jaffé and Reinhardt since most of the invariant manifold structure remains intact.

## Keywords

Normal Form Null Space Canonical Transformation Invariant Torus Level Curf## Preview

Unable to display preview. Download preview PDF.

## References

- 1.H. Goldstein, “Classical Mechanics,” Addison-Wesley, Reading, Mass. (1950) is one of many possible introductory treatments of classical dynamics.Google Scholar
- 2.See, for example, J. Ford, Adv. Chem. Phys. 24: 155 (1973).CrossRefGoogle Scholar
- 3.See, for example, M. V. Berry, in “AIP Conference Proceedings,” No. 46, S. Jorna, ed., American Institute of Physics, New York (1978), p. 16.CrossRefGoogle Scholar
- 4.R. T. Swimm and J. B. Delos, J. Chem. Phys, 71: 1706 (1979).MathSciNetCrossRefGoogle Scholar
- 5.C. Jaffé and W. P. Reinhardt, to be published; C. Jaffé, unpublished Ph.D. Thesis, Univ. of Colorado (1979).Google Scholar
- 6.G. D. Birkhoff, “Dynamical Systems,” American Mathematical Society, New York (1927).MATHGoogle Scholar
- 7.See Ref. 1, Ch. 8.Google Scholar
- 8.F. G. Gustavson, Astron. J. 71: 670 (1966).CrossRefGoogle Scholar
- 9.e.g. F. B. Hildebrand, “Methods of Applied Mathematics,” Prentice-Hall, Englewood Cliffs, New Jersey (1956), p. 29.Google Scholar
- 10.MACSYMA is maintained by the Mathlab group, Laboratory for Computer Science, Massachussetts Institute of Technology, 545 Technology Square, Cambridge, Massachussetts 02139.Google Scholar
- 11.For a more complete description of the Poincaré surface of section as well as other figures relating to the Henon-Heiles Hamiltonian see W. P. Reinhardt and C. Jaffé in this volume, and Refs. 2, 3.Google Scholar
- 12.J. R. McDonald, J. Appl. Phys. 35: 3034 (1964).CrossRefGoogle Scholar
- 13.I. C. Percival, Adv. Chem. Phys. 36: 1 (1977).CrossRefGoogle Scholar