### Abstract

We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, an N degrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

Original language | English |
---|---|

Article number | 710274 |

Journal | International Journal of Mathematics and Mathematical Sciences |

Volume | 2011 |

DOIs | |

Publication status | Published - 2011 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*International Journal of Mathematics and Mathematical Sciences*,

*2011*, [710274]. https://doi.org/10.1155/2011/710274

**Geometry of hamiltonian dynamics with conformal Eisenhart metric.** / Sun, Huafei; Peng, Linyu; Sun, Xiao.

Research output: Contribution to journal › Article

*International Journal of Mathematics and Mathematical Sciences*, vol. 2011, 710274. https://doi.org/10.1155/2011/710274

}

TY - JOUR

T1 - Geometry of hamiltonian dynamics with conformal Eisenhart metric

AU - Sun, Huafei

AU - Peng, Linyu

AU - Sun, Xiao

PY - 2011

Y1 - 2011

N2 - We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, an N degrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

AB - We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, an N degrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads.

UR - http://www.scopus.com/inward/record.url?scp=80052690720&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=80052690720&partnerID=8YFLogxK

U2 - 10.1155/2011/710274

DO - 10.1155/2011/710274

M3 - Article

AN - SCOPUS:80052690720

VL - 2011

JO - International Journal of Mathematics and Mathematical Sciences

JF - International Journal of Mathematics and Mathematical Sciences

SN - 0161-1712

M1 - 710274

ER -