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On the dawn of celestial mechanics
... In order to label them, we further need to distinguish the case φ 0 ∈ 0, arcsin 1−| p| | p| and φ 0 ∈ π − arcsin 1−| p| | p| , π : in the first one (on the left), external (resp., internal) solutions have positive (resp., negative) angular momentum, in the second one (on the right), the picture is reversed. When φ 0 ∈ arcsin 1−| p| | p| , π − arcsin 1−| p| | p| , solutions do not exist at all The proof of Theorem 2.1 relies on a purely geometrical argument, based on the well-known fact (see, for instance, [10,19]) that the trajectories of solutions of (3) lie on ellipses with one focus at the origin and major axis of lenght 1; using polar coordinates (r , φ), such kind of ellipses can be parametrized as ...
We revisit a classical result by Jacobi (J Reine Angew Math 17:68–82, 1837) on the local minimality, as critical points of the corresponding energy functional, of fixed-energy solutions of the Kepler equation joining two distinct points with the same distance from the origin. Our proof relies on the Morse index theorem, together with a characterization of the conjugate points as points of geodesic bifurcation.
... The proof of Theorem 2.1 relies on a purely geometrical argument, based on the well-known fact (see, for instance, [10,19]) that the trajectories of solutions of (3) lie on ellipses with one focus at the origin and major axis of lenght 1; using polar coordinates (r, φ), such kind of ellipses can be parametrized as ...
We revisit a classical result by Jacobi on the local minimality, as critical points of the corresponding energy functional, of fixed-energy solutions of the Kepler equation joining two distinct points with the same distance from the origin. Our proof relies on the Morse index theorem, together with a characterization of the conjugate points as points of geodesic bifurcation.
... For instance, Albers et al. [3], in their study of the planar circular restricted 3-body problem, cf. [18], show that the energy hypersurface for levels slightly above the energy of the first Lagrange point is the contact connected sum of two copies of RP 3 . In this specific case, however, one does not need Theorem 4.1 to find a periodic orbit, for the contact form on the connected sum comes from the standard Weinstein model, so the existence of a periodic orbit is obvious and indeed known classically, cf. ...
We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mild extension of previous results we present existence statements on periodic Reeb orbits on contact manifolds with suitable supporting open books.
In physics and astronomy, Euler's three-body problem is to solve for the motion of a body that is acted upon by the gravitational field of two other bodies. This problem is named after Leonhard Euler (1707-1783), who discussed it in memoirs published in the 1760s. In these publications, Euler found that the parameter that controls the relative distances among three collinear bodies is given by a quintic equation. Later on, in 1772, Lagrange dealt with the same problem, and demonstrated that for any three masses with circular orbits, there are two special constant-pattern solutions, one where the three bodies remain collinear, and the other where the bodies occupy the vertices of two equilateral triangles. Because of their importance, these five points became known as Lagrange points. The quintic equation found by Euler for the relative distances among the collinear bodies was also found later by Lagrange, and because of that, Euler has also been given credit for the discovery of the three collinear Lagrange points. A practical application of the collinear points for satellite location is also presented.
These are notes based on a mini-course at the conference RIEMain in Contact, held in Cagliari, Sardinia, in June 2018. The main theme is the connection between Reeb dynamics and topology. Topics discussed include traps for Reeb flows, plugs for Hamiltonian flows, the Weinstein conjecture, Reeb flows with finite numbers of periodic orbits, and global surfaces of section for Reeb flows. The emphasis is on methods of construction, e.g. contact cuts and lifting group actions in Boothby-Wang bundles, that might be useful for other applications in contact topology.
As low-thrust propulsion technology becomes increasingly popular, orbital estimation for low-thrust spacecraft may become an area of increasing interest. More frequent use of low-thrust propulsion to place satellites in orbit gives more opportunities for collisions and radio frequency interference as these spacecraft travel slowly through altitude ranges. The purpose of this paper is to develop a method for estimation of the osculating orbital elements for low-thrust spacecraft. To overcome the instability of the estimation problem with low-thrust acceleration, we estimate the mean elements instead of osculating elements. By use of the averaging technique, Hudson and Scheeres proposed an analytical model of secular variations of orbital elements under thrust acceleration. The resulting averaged equation has a nice property in which only a finite number of Fourier coefficients of the thrust acceleration appear because of the orthogonality of the trigonometric function. Based on the nonlinear state equation representation for the extended state variables which include not only orbital elements but also unknown Fourier coefficients, mean orbital elements and thrust history are estimated from perturbed observation data of mean orbital elements. Then, the mapping from mean to osculating elements which is derived from the perturbation theory is used to estimate the osculating elements. The proposed method is demonstrated through numerical simulations.
Space Situational Awareness (SSA) has been recognized to be important for safe space activities. As low-thrust propulsion technology becomes increasingly popular, SSA for low-thrust spacecraft may become an area of increasing interest. In this paper, we propose an orbital estimation method to predict the long-term evolution of spacecraft trajectory under unknown low-thrust acceleration. In particular, by the use of the perturbation theory and a nonlinear Kalman filter, long-term variations of orbital elements and thrust accelerations can be estimated from observation data of mean orbital elements. Performance of our method are evaluated for both controlled and uncontrolled orbits.
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