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# Advanced Celestial Mechanics. Questions - PowerPoint PPT Presentation

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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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.

Question 13
• Show that the Hamiltonian in the three-body problem is
• Write the Hamiltonian equations of motion for the three-body problem
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Question 14

Show that in the hierarchical three-body problem

Make use of the canonical coordinates and the Hamiltonian