A1 Journal article (refereed)
Existence of Hyperbolic Motions to a Class of Hamiltonians and Generalized N-Body System via a Geometric Approach (2023)
Liu, J., Yan, D., & Zhou, Y. (2023). Existence of Hyperbolic Motions to a Class of Hamiltonians and Generalized N-Body System via a Geometric Approach. Archive for Rational Mechanics and Analysis, 247(4), Article 64. https://doi.org/10.1007/s00205-023-01894-5
JYU authors or editors
Publication details
All authors or editors: Liu, Jiayin; Yan, Duokui; Zhou, Yuan
Journal or series: Archive for Rational Mechanics and Analysis
ISSN: 0003-9527
eISSN: 1432-0673
Publication year: 2023
Publication date: 12/06/2023
Volume: 247
Issue number: 4
Article number: 64
Publisher: Springer
Publication country: Germany
Publication language: English
DOI: https://doi.org/10.1007/s00205-023-01894-5
Publication open access: Not open
Publication channel open access:
Publication is parallel published: https://arxiv.org/abs/2112.07450v3
Abstract
For the classical N-body problem in Rd with d≧2, Maderna–Venturelli in their remarkable paper (Ann Math 192:499–550, 2020) proved the existence of hyperbolic motions with any positive energy constant, starting from any configuration and along any non-collision configuration. Their original proof relies on the long time behavior of solutions by Chazy 1922 and Marchal-Saari 1976, on the Hölder estimate for Mañé’s potential by Maderna 2012, and on the weak KAM theory. We give a new and completely different proof for the above existence of hyperbolic motions. The central idea is that, via some geometric observation, we build up uniform estimates for Euclidean length and angle of geodesics of Mañé’s potential starting from a given configuration and ending at the ray along a given non-collision configuration. Moreover, our geometric approach works for Hamiltonians 12∥p∥2−F(x), where F(x)≧0 is lower semicontinuous and decreases very slowly to 0 faraway from collisions. We therefore obtain the existence of hyperbolic motions to such Hamiltonians with any positive energy constant, starting from any admissible configuration and along any non-collision configuration. Consequently, for several important potentials F∈C2(Ω), we get similar existence of hyperbolic motions to the generalized N-body system x¨=∇xF(x), which is an extension of Maderna–Venturelli [Ann Math 2020].
Keywords: dynamical systems; differential geometry
Contributing organizations
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Ministry reporting: Yes
VIRTA submission year: 2023
JUFO rating: 3