Naiara V. de Paulo, Seongchan Kim, Pedro A. S. Salomão, Alexsandro Schneider
{"title":"Transverse foliations for two-degree-of-freedom mechanical systems","authors":"Naiara V. de Paulo, Seongchan Kim, Pedro A. S. Salomão, Alexsandro Schneider","doi":"arxiv-2409.00445","DOIUrl":null,"url":null,"abstract":"We investigate the dynamics of a two-degree-of-freedom mechanical system for\nenergies slightly above a critical value. The critical set of the potential\nfunction is assumed to contain a finite number of saddle points. As the energy\nincreases across the critical value, a disk-like component of the Hill region\ngets connected to other components precisely at the saddles. Under certain\nconvexity assumptions on the critical set, we show the existence of a weakly\nconvex foliation in the region of the energy surface where the interesting\ndynamics takes place. The binding of the foliation is formed by the index-$2$\nLyapunov orbits in the neck region about the rest points and a particular\nindex-$3$ orbit. Among other dynamical implications, the transverse foliation\nforces the existence of periodic orbits, homoclinics, and heteroclinics to the\nLyapunov orbits. We apply the results to the H\\'enon-Heiles potential for\nenergies slightly above $1/6$. We also discuss the existence of transverse\nfoliations for decoupled mechanical systems, including the frozen Hill's lunar\nproblem with centrifugal force, the Stark problem, the Euler problem of two\ncenters, and the potential of a chemical reaction.","PeriodicalId":501155,"journal":{"name":"arXiv - MATH - Symplectic Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Symplectic Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.00445","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the dynamics of a two-degree-of-freedom mechanical system for
energies slightly above a critical value. The critical set of the potential
function is assumed to contain a finite number of saddle points. As the energy
increases across the critical value, a disk-like component of the Hill region
gets connected to other components precisely at the saddles. Under certain
convexity assumptions on the critical set, we show the existence of a weakly
convex foliation in the region of the energy surface where the interesting
dynamics takes place. The binding of the foliation is formed by the index-$2$
Lyapunov orbits in the neck region about the rest points and a particular
index-$3$ orbit. Among other dynamical implications, the transverse foliation
forces the existence of periodic orbits, homoclinics, and heteroclinics to the
Lyapunov orbits. We apply the results to the H\'enon-Heiles potential for
energies slightly above $1/6$. We also discuss the existence of transverse
foliations for decoupled mechanical systems, including the frozen Hill's lunar
problem with centrifugal force, the Stark problem, the Euler problem of two
centers, and the potential of a chemical reaction.