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Advanced Celestial Mechanics. Questions. Question 1 Explain in main outline how statistical distributions of binary energy and eccentricity are derived in the three-body problem. Escape cone. Density of escape states. Question 2. Calculate the potential above an infinite plane. . . .
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Advanced Celestial Mechanics. Questions Question 1 Explain in main outline how statistical distributions of binary energy and eccentricity are derived in the three-body problem.
Question 2 Calculate the potential above an infinite plane.
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Question 3 • Write the acceleration between bodies 1 and 2 in the three-body problem using only the relative coordinates i.e. in the Lagrangian formulation. Use the symmetric term W.
Question 4 • Show that in the two-body problem the motion takes place in a plane. Derive the constant e-vector, and derive its relation to the k-vector. Draw an illustration of these two vectors in relation to the orbit.
Question 5 • Define true anomaly and eccentic anomaly in the two-body problem. Derive the transformation formula between these two anomalies. Define also the mean anomaly M.
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Question 6 • Define the scattering angle in the hyperbolic two-body problem, and derive its value using the eccentricity. Derive the expression of the impact parameter b as a function of the scattering angle.
Question 7 • Derive the potential at point P, arising from a source at point Q, a distance r’ from the origin. Define Legendre polynomials and write the first three polynomials.
Question 8 • Show that the shortest distance between two points is a straight line using the Euler-Lagrange equation.
Question 9 • Write the Lagrangian for the planar two-body problem in polar coordinates, and write the Lagrangian equations of motion. Solve the equations to obtain Kepler’s second law.
Question 10 • If the potential does not depend on generalized velocities, show that the Hamiltonian equals the total energy. Use Euler’s theorem with n=2.
Question 11 • Write the Hamiltonian in the planar two-body problem in polar coordinates. Show that the is a constant.
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Question 12 The canonical coordinates in the two-body problem are Use the generating function To derive Delaunay’s elements.