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.

.

• .

.

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.

.

• .

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.

.

• .

• .

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

.

• .

.

Question 14

Show that in the hierarchical three-body problem

Make use of the canonical coordinates and the Hamiltonian

.

.

.