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Action-angle coordinates are a tool commonly used in celestial mechanics tosystematically parametrize and store general solutions of the equations ofmotion of astrophysical bodies. I perturbatively construct action-anglecoordinates for bound test particle motion in static, spherically symmetricspace-times using a post-circular expansion. Then I specialise the expressionsto the motion in the gravitational fields of Schwarzschild black holes and giveexplicit formulas for the Hamiltonian to the 10th power in the radial action(20th power in eccentricity), and the transformation to angle coordinates up tothe 8th harmonic with respect to a relativistic orbital anomaly. The resultsprovide a closed-form perturbative solution for the orbital motion parametrizedby coordinate time that will find applications in the modelling of compactbinary inspirals and other fields of astrophysics.

Action-angle coordinates for black-hole geodesics I: Spherically symmetric and Schwarzschild