## Online material

### Appendix A: Computing the angles

The angles result from the derivatives of the generating function *W*(** x**,

**) w.r.t. the actions (A.1)(Binney & Tremaine 2008, Eq. (3.204)). The complete generating function for a spherical system is given by (modified from Eq. (3.220), Binney & Tremaine 2008) (A.2)where we integrate over the particles trajectory in phase space, which means for example that during a whole radial period, the particle transverses twice the branch from pericentre to apocentre, but once in reversed direction. The latitudinal momentum**

*J**p*

_{ϑ}is (A.3)where cos

*i*=

*L*

_{z}/

*L*. The radial momentum

*p*

_{r}is (A.4)where

*L*=

*J*

_{θ}+ |

*J*

_{φ}| and

*J*

_{φ}≥ 0 is assumed

^{8}.

We find the radial angle as (A.5)where we note that the angles are always defined modulo 2*π* because there is no information on how many loops a particle made around the Galactic centre. The derivative of the generating function is given by (A.6)where the conditions on *p*_{r} are necessary to take the right branch of *f*_{1}. After one full period, we find *θ*_{r} = 2*π* as expected.

For the azimuthal angle we find (A.7)where we find *φ* using the (quadrant-aware) arctangent (A.8)The derivative *W*_{r,H} has already been worked out for *θ*_{r}, and the other derivatives are

The function *W*_{ϑ,Jφ} is oscillatory in nature (w.r.t. *ϑ*), while the combination *W*_{r,Jφ} + Ω_{φ}*W*_{r,H} is also oscillatory (w.r.t. *r*): after one radial period it evaluates to −2*L**f*_{2}(*r*_{apo}) + 2Ω_{φ}*f*_{1}(*r*_{apo}) = 0.

The latitudinal angle is given by (A.11)Because we assume *J*_{φ} ≥ 0, we find *W*_{r,Jϑ} = *W*_{r,Jφ} and Ω_{ϑ} = Ω_{φ}. The remaining derivative of the generating function is (A.12)where the term *W*_{r,Jϑ} + Ω_{ϑ}*W*_{r,H} vanishes after one radial period, while the term *W*_{ϑ,Jϑ} contains the dependence on *ϑ* and increases by 2*π* after one period in *ϑ*.

### Appendix B: Transformation equations

The linearised transformation between action-angle coordinates and Cartesian coordinates is (B.1)For simplicity, we provide here the matrix for the 2D case (i.e. when the orbit is in the plane). In that case, *J*_{θ} = 0 and *J*_{φ} = *L* = *L*_{z}, so that we find (B.2)where the above terms are given by (B.7)The functions *κ* and *η* are and the *W*_{Ji,Jj} are found by differentiating the generating function

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