Brouwer’s satellite solution redux∗

The paper’s Eq. (11) gives the second-order averaged Hamiltonian with a single g-dependent term proportional to cos(2g), and Eq. (12) shows how a suitable choice of C1 ensures the contribution 〈H̃∗_{0,2}〉 cancels that g-dependent term. The paper then chooses C1 exactly as in the candidate solution and concludes with the g–independent second-order Hamiltonian in Eq. (14), completing Brouwer’s reduction at O(ε^2) with a single canonical transformation. The candidate’s steps (differentiating C1, using G=Lη, inserting into Eq. (12), simplifying, and verifying cancellation against Eq. (11)) reproduce the paper’s logic and constants term-for-term. Hence both are correct and essentially the same proof.