English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For the classical $N$-body problem in $\mathbb{R}^d$ with $d\ge2$,Maderna-Venturelli in their remarkable paper [Ann. Math. 2020] proved theexistence of hyperbolic motions with any positive energy constant, startingfrom any configuration and along any non-collision configuration. Theiroriginal proof relies on the long time behavior of solutions by Chazy 1922 andMarchal-Saari 1976, on the H\"{o}lder estimate for Ma\~{n}\'{e}'s potential byMaderna 2012, and on the weak KAM theory.  We give a new and completely different proof for the above existence ofhyperbolic motions. The central idea is that, via some geometric observation,we build up uniform estimates for Euclidean length and angle of geodesics ofMa\~{n}\'{e}'s potential starting from a given configuration and ending at theray along a given non-collision configuration. Note that we do not need any ofthe above previous studies used in Maderna-Venturelli's proof.  Moreover, our geometric approach works for Hamiltonians$\frac12\|p\|^2-F(x)$, where $F(x)\ge 0$ is lower semicontinuous and decreasesvery slowly to $0$ faraway from collisions. We therefore obtain the existenceof hyperbolic motions to such Hamiltonians with any positive energy constant,starting from any admissible configuration and along any non-collisionconfiguration. Consequently, for several important potentials $F\inC^{2}(\Omega)$, we get similar existence of hyperbolic motions to thegeneralized $N$-body system $\ddot{x} = \nabla_x F(x)$, which is an extensionof Maderna-Venturelli [Ann. Math. 2020].

Existence of hyperbolic motions to a class of Hamiltonians and generalized $N$-body system via a geometric approach