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Advanced Celestial Mechanics. QuestionsPowerPoint Presentation

Advanced Celestial Mechanics. Questions

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Advanced Celestial Mechanics. Questions

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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.

Calculate the potential above an infinite plane.

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- 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.

- 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.

- 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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- 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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- 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.

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

- 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.

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

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The canonical coordinates in the two-body problem are

Use the generating function

To derive Delaunay’s elements.

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- 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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Show that in the hierarchical three-body problem

Make use of the canonical coordinates and the Hamiltonian