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Advanced Celestial Mechanics. Questions
Question 1Explain 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 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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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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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.
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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.
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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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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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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.
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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
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