The Analytical Foundations of Celestial Mechanics

3-vector according analytic function angular arbitrary assumption Astr barycentric barycentric coordinate belonging canonical transformation central configuration characteristic exponents clear coefficients collinear collision condition conservative considered const constant h corresponding defined degrees of freedom denote determined differential equations eccentric anomaly elliptic energy constant energy h energy integral equilibrium solution equivalent exists fact finite follows Fourier series G. D. Birkhoff given solution Hamiltonian function Hence homographic homographic solution identical implies independent integration constants invariable plane invariant Jacobi Jacobi equations Jacobian Lagrangian Lagrangian function linear m-matrix m₁ Math matrix motion P₁ pair periodic periodic function Pfaffian phase space planar plane positive numbers problem of three proof prove respectively restricted problem rotation satisfied scalar scalar function shows singularities solution path theorem theory three bodies tion vanishes variable vector velocity Wintner X₁ y)-plane