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 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 fixed value 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 pair periodic periodic function Pfaffian phase space planar plane positive numbers power series problem of three proof prove respectively rotation satisfied scalar scalar function shows singularities solution path symmetric theorem three bodies tion vanishes variable vector velocity y)-plane