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 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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