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Advanced Celestial Mechanics. Questions

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

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  1. 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.

  2. Escape cone

  3. Density of escape states

  4. Question 2 Calculate the potential above an infinite plane.

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  7. 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.

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  9. 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.

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  11. 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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  13. 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.

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  15. 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.

  16. Question 8 • Show that the shortest distance between two points is a straight line using the Euler-Lagrange equation.

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  19. 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.

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  22. 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.

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  24. Question 11 • Write the Hamiltonian in the planar two-body problem in polar coordinates. Show that the is a constant.

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  27. Question 12 The canonical coordinates in the two-body problem are Use the generating function To derive Delaunay’s elements.

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