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Description: These notes contain introductory material for the n-body problem, including the equations of motion and some important invariance properties. The ten classical integrals are derived. The solutions of the Kepler, and two body problems are found, and design of conic trajectories is discussed . Several other configurations are studied numerically, including the Sitnikov problem.
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Description: This set of notes describes the method of differential corrections, and some applications. The variational equations and the state transition matrix for the n-body problem are developed. Examples are given which show how to use the method to solve boundary value or `targeting' problems in celestial mechanics. In particular a scheme for computing choreography orbits is presented.
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Description: These notes deal with the general three body problem. The special triangular solutions of Lagrange and collinear solutions of Euler are treated numerically, with emphasis on numerical and dynamical stability. In addition the equations of motion for the three body problem in Jacobi coordinates are developed. Finally some kinematic results for dynamics in rotating coordinate frames are derived.
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Description: These notes are an introduction to the circular restricted three body problem. In this problem two massive bodies orbit one another in circular orbits, and a smaller body moves in their field. The equations of motion and the Jacobi integral are derived. After locating the five libration points (equilibria), the linearized equations about these points are found and studied. The notes also contain numerical studies of the resulting dynamics including the occurrence of ballistic capture.
MatLab Code for Note Set 4 |
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Description: In this set of notes, procedures for computing and analyzing symmetric periodic orbits in the circular restricted three body problem are developed. In addition, the stable and unstable manifolds of such orbits are defined and a method is presented for computing these objects. These methods are applied to halo orbits in the Earth/Sun system. Halo orbits are out-of-plane periodic orbits near the collinear libration points. These orbits have received much attention in recent years as they have properties that make them ideal for astronomical purposes such as solar observatories, or communication posts (say for uninterrupted communication with a mission on the dark side of the moon). We analyze a halo orbit about L1 given by approximate physical data, and use this to compute a one parameter family of halo orbits parameterized by the Jacobi integral. From this we develop a method that allows one to compute such a family of halo orbits near any of the collinear libration points at any value of mu. We apply this method at L2 and find the halo family there. These tools are also applied to the Lyapunov orbits near L1, where we compute the stable and unstable manifolds of these inplane periodic orbits. Symmetric Periodic Orbits in the CRTBP and their Stable and Unstable Manifolds (pdf) MatLab Code for Note Set 5 |
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Description: The frame on the left above shows a halo orbit in the Earth/Moon system, with the branch of it's stable manifold that passes near the Earth. The halo orbit is the thick black ring, and it's stable manifold is the blue tube. Orbits on this tube approach the halo orbit in forward time. The zero velocity surface at the energy level of the halo orbit is shown in red. L1 is the black star in the center of the halo orbit. An orbit which begins near the Sun, passes through the L1 neck, performs a flyby of the Earth, and then returns to the Sun through the L1 neck is shown as well. The picture in this frame is a close up of the picture above. Here we can see right through the L1-L2 `neck' in the energy surface. L2 is seen as a second black star on the far side of the neck. The stable manifold is wrapped around the earth, obscuring it from view. Its location is however roughly half way between L1 and L2. |
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Description: Another view. |
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Description: The picture shows a Poincare section for the Earth/Moon CRTBP in an integral manifold. The energy (Jacobi constant) of the manifold is between the energy of L1 and L2, so that the L1 ``neck'' is open while the L2 ``neck'' is closed. The coordinates of the section are x and xdot. The section is at y=0 (ydot is then determined by the fixed value of the energy). In this section the dominant features are the secondary KAM tori due to a resonance near x=0.5, and the L1 neck near x=0.8. The ``static'' around the KAM tori is the Birkhoff instability zone. Trajectories in this zone exhibit chaotic motion. The section was constructed by taking a grid of 100 orbits with initial conditions on the x axis (so xdot_0 = 0). For each such orbit, 1000 crossings of the Poincare section are computed. This is repeated three more times with varying initial xdot. The the resulting image (a large part of which is shown here) shows 400,000 crossings. This computation was done in MatLab, and took roughly 100hrs (in FORTRAN or C++ this would run much faster). The data is available in .mat form on my MatLab page. |
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Description: This section illustrates the rich structure of the problem. The picture shows x between about 0.1 and 0.28 and xdot between about 0.0 and 1.5. Then we are much closer to the earth than to L1, and the phase space is packed quite tightly with secondary tori (secondary when we think of the CRTBP as a perturbation of the Kepler problem). Even though the region is rich with tori there are Birkhoff zones between them, and it is possible for orbits to escape or diffuse from the region via ``resonance hopping''. They can wind out from between the tori chaoticly, and move into regions where the influence of the moon is stronger. |
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Description: The black dots in this picture are taken from Poincare section data near the moon, similar to the pictures described above. The red and blue dots (or curves) are data obtained by computing the intersection of the unstable, and stable manifolds of a Lyapunov orbit with the Poincare section. Then a blue dot approaches the Lyapunov orbit in forward time, and a red dot approaches it in backward time. It can be seen from the picture that the stable and unstable manifolds intersect transversely. This implies much. In the first place such an intersection point would approach the Lyapunov orbit in both forward and backward time, so orbits homoclinic to the Lyapunov orbit exist. On the other hand, the transverse crossing of the stable and unstable manifolds also implies the existence of a topological horse shoe near the Lyapunov orbit, so the dynamics there must be very rich. In particular, there exist infinitely many periodic orbits, of arbitrarily long period which ``shadow'' the homoclinic orbit. In other words, they begin near the Lyapunov orbit, move out to near an intersection of the stable and unstable manifold, and return to the vicinity of the Lyapunov orbit. Since the stable and unstable manifolds intersect on the far side of the moon, very far from the Lyapunov orbit, this is global behavior. |
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Description: Knowing that there must be periodic orbits near the Lyapunov orbit and having the intersections of the stable and unstable manifolds allows us to actually compute them (here the low dimension of the problem is essential as we are basically doing graphical analysis). This picture shows a periodic orbit (in configuration space) that begins near the Lyapunov orbit, flies by the moon several times, passes near the intersection, returns, and closes itself off. The outward trip is blue and the return is green. A homoclinic orbit (which can be read off the above Poincare section) is used as the initial guess for a differential corrections procedure which then finds the initial conditions of a nearby periodic orbit. (The material from the last several slides will eventually be added to Note set five). |
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