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I was looking at the problem of a particle in a Coulomb field in three dimensions. Time-translation invariance and spherical symmetry ensures that the energy and angular momentum are conserved, allowing us to write the action as

A = -εt + L ∅ +f(r)

The form of f(r) in terms of ε and L and r can be obtained by substituting in Hamlton-Jacobi equations.

After having obtained the expression for the action, the author differentiates it with respect to ε and equating it to a constant t0 to obtain the trajectory r(θ).

I do not understand this last step. Can anyone explain or give me a reference?