Lectures

NAST005 – Celestial mechanics I (4/0, Ex)

David Vokrouhlický

Motion in gravitational field, recapitulation of elements of analytical mechanics. Two-body problem. Three-body problem in two approximations: (i) restricted problem, and (ii) Hill's problem. For students of the first year of master studies in astronomy.

• A brief historical overview.
• A brief overview of analytical mechanics: Lagrange and Hamiltonian approach; Lagrange equations of the second kind; Hamilton equations; canonical transformations; Poisson and Lagrange brackets; symplectic matrix; Hamilton-Jacobi equation; particle in one-diemnsional potential.
• Two-body problem: Basic formulation; transformation of barycenter; relative coordinate; momentum and angular momentum integrals; Binet equation; Kepler equation and variants for parabolic and hyperobolic motions; orbital and non-singular orbital elements; solution of the two-body problem using Hamilton-Jacobi equation; Delaunay variables; elliptic expansions (Bessel functions; Hansen functions).
• Circular restricted problem of three bodies: Equations of motion in the inertial and synodic reference systems; Jacobi integral; Tisserand criterion; Hill's planes of zero velocity; stationary solutions (Lagrange points); stability of stationary solutions.
• Elliptic restricted problem of three bodies: Nechvile's transformation to rotating and pulsating coordinate system; non-integrability; stationary solutions and their stability.
• Hill's problem: Jacobi coordinates; equations of motion in synodic reference system; Hill's surfaces of zero velocity; lunar origin; theory of lunar motion; variational solution.