Abstract.
We discuss the coarse-grained level density of the Hénon-Heiles system above the barrier energy, where the system is nearly chaotic. We use periodic orbit theory to approximate its oscillating part semiclassically via Gutzwiller’s semiclassical trace formula (extended by uniform approximations for the contributions of bifurcating orbits). Including only a few stable and unstable orbits, we reproduce the quantum-mechanical density of states very accurately. We also present a perturbative calculation of the stabilities of two infinite series of orbits (R n and L m ), emanating from the shortest librating straight-line orbit (A) in a bifurcation cascade just below the barrier, which at the barrier have two common asymptotic Lyapunov exponents χ R and χ L .
Similar content being viewed by others
References
M. Hénon C. Heiles (1964) Astr. J. 69 73 Occurrence Handle10.1086/109234 Occurrence Handle1964AJ.....69...73H
G. H. Walker J. Ford (1969) Phys. Rev. 188 416 Occurrence Handle260978 Occurrence Handle10.1103/PhysRev.188.416 Occurrence Handle1969PhRv..188..416W
M. C. Gutzwiller (1990) Chaos in Classical and Quantum Mechanics Springer New York
M. Brack R. K. Bhaduri (2003) Semiclassical Physics EditionNumber2 Westview Press Boulder
N. Fulton J. Tennyson D. A. Sadovskií B. I. Zhilinskií (1993) J. Chem. Phys. 99 906 Occurrence Handle10.1063/1.465355 Occurrence Handle1993JChPh..99..906S
Brack, M., Meier, P., Tanaka, K.: J. Phys. A32, 331 (1999); Brack, M., Creagh, S. C., Law, J.: Phys. Rev. A57, 788 (1998); Lauritzen, B., Whelan, N. D.: Ann. Phys. (NY) 244, 112 (1995); Brack, M., Bhaduri, R. K., Law, J., Maier, Ch., Murthy, M. V. N.: Chaos 5, 317 (1995); Erratum: Chaos 5, 707 (1995)
Kaidel, J., Brack, M.: Phys. Rev. E70, 016206 (2004); E72, 049903(E) (2005)
J. Kaidel P. Winkler M. Brack (2004) Phys. Rev. E70 066208 Occurrence Handle2004PhRvE..70f6208K
Strutinsky, V. M.: Nucl. Phys. A122, 1 (1968); Brack, M., Pauli, H.-C.: Nucl. Phys. A20, 401 (1973)
M. C. Gutzwiller (1971) J. Math. Phys. 12 343 Occurrence Handle10.1063/1.1665596
Churchill, R. C., Pecelli, G., Rod, D. L.: In: Stochastic Behaviour in Classical and Quantum Hamiltonian Systems (Casati, G., Ford, J., eds.), p. 76. Berlin Heidelberg New York: Springer 1979; Davies, K. T. R., Huston, T. E., Baranger, M.: Chaos 2, 215 (1992); Vieira, W. M., Ozorio de Almeida, A. M.: Physica D90, 9 (1996)
M. Brack (2001) Found. of Phys. 31 209 Occurrence Handle1838907 Occurrence Handle10.1023/A:1017582218587
M. Brack M. Mehta K. Tanaka (2001) J. Phys. A34 8199 Occurrence Handle1873179 Occurrence Handle2001JPhA...34.8199B
H. Schomerus (1998) J. Phys. A31 4167 Occurrence Handle1627383 Occurrence Handle1998JPhA...31.4167S
Fedotkin, S. N., Magner, A. G., Brack, M.: (to be published)
M. Abramowitz I. A. Stegun (1970) Handbook of Mathematical Functions, 9th Printing Dover New York
S. C. Creagh (1996) Ann. Phys. (NY) 248 60 Occurrence Handle0867.58058 Occurrence Handle1393767 Occurrence Handle10.1006/aphy.1996.0051 Occurrence Handle1996AnPhy.248...60C
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brack, M., Kaidel, J., Winkler, P. et al. Level Density of the Hénon-Heiles System Above the Critical Barrier Energy. Few-Body Systems 38, 147–152 (2006). https://doi.org/10.1007/s00601-005-0124-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00601-005-0124-0