The second time through, it describes the positions and variables of the two bodies as a single point in a 12-dimensional phase space, describes the behavior of the bodies as a Hamiltonian system, and uses symplectic reductions to shrink the phase space to two dimensions before solving it to produce Kepler's laws of planetary motion in a more direct and principled way.
Topics covered in this part include manifolds, vector fields and differential forms, pushforwards and pullbacks, symplectic manifolds, Hamiltonian energy functions, the representation of finite and infinitesimal physical symmetries using Lie groups and Lie algebras, and the use of the moment map to relate symmetries to conserved quantities.
[1][2][4] Reviewer William Satzer writes that this book "makes serious efforts to address real students and their potential difficulties" and shifts comfortably between mathematical and physical views of its problem.
"[7] Reviewer Ivailo Mladenov notes with approval the book's attention to example-first exposition, and despite pointing to a minor inaccuracy regarding the nationality of Sophus Lie, recommends it to both undergraduate and graduate students.
[6] Reviewer Richard Montgomory writes that the book does "an excellent job of leading the reader from the Kepler problem to a view of the growing field of symplectic geometry".