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For the classical N-body problem in R d 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 1 2 ‖ 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 ∈ C 2 (Ω) , we get similar existence of hyperbolic motions to the generalized N-body system x ¨ = ∇ x F (x) , which is an extension of Maderna–Venturelli [Ann Math 2020]. [ABSTRACT FROM AUTHOR]

It is shown that each lobe of the figure-eight orbit is star-shaped, which indicates that the corresponding polar angle is monotone. In general, it is not clear under which assumptions a trajectory in a minimizer is star-shaped. In this paper, we study minimizers connecting two fixed-ends (i.e. the Bolza problem) in the planar three-body problem with two equal masses. We show that if the Jacobi coordinates of the two fixed-ends are both orthogonal, then for any minimizer, both Jacobi vectors describe a star-shaped curve. If the Jacobi coordinates are orthogonal only on one of the fixed-ends and they are in adjacent closed quadrants, then their polar angles have at most one critical point. As an application, we show the existence and prove some geometric properties of two families of periodic orbits. [ABSTRACT FROM AUTHOR]

For each given n ≥ 2 , we study the variational existence of a new spatial periodic orbit in the 2n-body problem. Besides excluding possible collision singularities, the main challenges left are to show that the orbit is not a relative equilibrium and it is spatial. By introducing a new estimate in the collinear central configuration, we can prove that it is not a relative equilibrium in the 2n-body problem. Furthermore, with the help of numerical calculation, the orbit is spatial in the equal-mass four-body problem. [ABSTRACT FROM AUTHOR]