# The Lissajous–Kustaanheimo–Stiefel transformation

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## Abstract

The Kustaanheimo–Stiefel transformation of the Kepler problem with a time-dependent perturbation converts it into a perturbed isotropic oscillator of four-and-a-half degrees of freedom with additional constraint known as bilinear invariant. Appropriate action–angle variables for the constrained oscillator are required to apply canonical perturbation techniques in the perturbed problem. The Lissajous–Kustaanheimo–Stiefel (LKS) transformation is proposed, leading to the action–angle set which is free from singularities of the LCF variables earlier proposed by Zhao. One of the actions is the bilinear invariant, which allows the reduction back to the three-and-a-half degrees of freedom. The transformation avoids any reference to the notion of the orbital plane, which allowed to obtain the angles properly defined not only for most of the circular or equatorial orbits, but also for the degenerate, rectilinear ellipses. The Lidov–Kozai problem is analysed in terms of the LKS variables, which allow a direct study of stability for all equilibria except the circular equatorial and the polar radial orbits.

## Keywords

Perturbed Kepler problem Regularization KS variables Lissajous transformation Lidov–Kozai problem## 1 Introduction

The Kustaanheimo–Stiefel (KS) transformation is probably the most renowned regularization technique for the three-dimensional Kepler problem. In the planar case, the conversion of the Kepler problem into a harmonic oscillator has been known since Goursat (1889) and Levi-Civita (1906), but its extension to the three-dimensional problem took many decades of futile efforts. Finally, Kustaanheimo (1964) discovered that the way to the third dimension is not direct, but requires a detour through a constrained problem with four degrees of freedom. The KS transformation gained popularity in the matrix–vector formulation of Kustaanheimo and Stiefel (1965), but it is much easier to interpret and generalize in the language of quaternion algebra, very closely related to the original spinor formulation of Kustaanheimo (1964).

The most common use of the KS transformation is the numerical integration of perturbed elliptic motion, where many intricacies introduced by the additional degree of freedom can be ignored, although—as recently demonstrated by Roa et al. (2016)—they can be quite useful in the assessment of a global integration error. Analytical perturbation methods for KS-transformed problems often follow the way indicated by Kustaanheimo and Stiefel (1965) and developed by Stiefel and Scheifele (1971): variation of arbitrary constants is applied to constant vector amplitudes of the KS coordinates and velocities. But those who want to benefit from the wealth of canonical formalism require a set of action–angle variables of the regularized Kepler problem.

The first step in this direction can be found in the monograph by Stiefel and Scheifele (1971), where the symplectic polar coordinates are introduced for each separate degree of freedom. However, this approach does not account for degeneracy of the problem and thus is unfit for the averaging-based perturbation techniques. Moreover, no attempt was made to relate this set to the constraint known as the ‘bilinear invariant’, effectively reducing the system to three degrees of freedom. Both problems have been resolved by Zhao (2015), who proposed the ‘LCF’ variables [presumably named after Levi-Civita (1906) and Féjoz (2001)]. In his approach, the motion in the KS variables is considered in an osculating ‘Levi-Civita plane’ (Deprit et al. 1994) as a two-degree-of-freedom problem. The third degree of freedom is added by the pair of action–angle variables orienting the plane. The redundant fourth degree is hidden in the definition of the Levi-Civita plane. The transformed Keplerian Hamiltonian depends on a single action variable, the other two actions being closely related to the angular momentum and its projection on the polar axis. Interestingly, the result is identical to the ‘isoenergetic variables’ found by Levi-Civita (1913) without regularization.

The LCF variables respect the degeneracy and bring the oscillations back to three degrees of freedom. Yet they possess a significant weakness: they are founded on the orientation of a plane determined by the angular momentum. Whenever the angular momentum vanishes (even temporarily), the angles become undetermined and equations of motion are singular. It turns out that seeking the proximity to the Delaunay variables, Zhao (2015) reintroduced the singularities of unregularized Kepler problem. Of course, some singularities are inevitable when the problem having spherical topology is mapped onto a torus of action–angle variables. But there is always some freedom in the choice of the singularities. Recalling that the main purpose of regularization is to allow the study of highly elliptic and rectilinear orbits, we find it worth an effort to construct the action–angle set that—unlike the LCF variables—is regular for this class of motions.

The main goal of the present work is to derive an alternative set of the action–angle variables which is not based upon the notion of an orbital plane (thus avoiding singularities when the orbit degenerates into a straight segment) and to test it on some well-known astronomical problem. Section 2 introduces some preliminary concepts related to the KS coordinate transformation in the language of quaternions. We use its generalized form with an arbitrary ‘defining vector’ (Breiter and Langner 2017), which helps to realize how the choice of the KS1 or KS3 convention allows or inhibits the use of the Levi-Civita plane in the construction of the action–angle sets. We have also benefited from the opportunity to polish and extend the geometrical interpretation given to the KS transformation by Saha (2009). In Sect. 3, we complement the KS coordinates with their conjugate momenta and provide the Hamiltonian function in the extended phase space as the departure point for further transformations. Section 4 builds the new action–angle set—the Lissajous–Kustaanheimo–Stiefel (LKS) variables. Two independent Lissajous transformations are followed by a linear Mathieu transformation. In Sect. 5, we show how to interpret the new variables not only in terms of the Lissajous ellipses, but also by the reference to the angular momentum and Laplace vectors of the Kepler problem. As an application, we discuss the classical Lidov–Kozai problem (Sect. 6), showing that stability of rectilinear orbits can be discussed directly in terms of the LKS variables, which has not been possible using the Delaunay or the LCF framework. Conclusions and future prospects are presented in the closing Sect. 7.

## 2 KS transformation in quaternion form

### 2.1 Quaternion algebra

### 2.2 KS coordinates transformation

#### 2.2.1 Generalized definition

#### 2.2.2 Fibres

A noninjective nature of the KS map had been known since its origins, although only recently it has been considered more an advantage than a nuisance (Roa et al. 2016).

### 2.3 KS quaternions *more geometrico*

But if \({\hat{\mathbf {x}}} = -\mathbf {c}\), the situation is different. Observing that then the ellipse from Fig. 1 turns into a circle, we conclude that the fibre consists exclusively of the pure quaternions \(\mathsf {v} = (0, \sqrt{\alpha r} \,\hat{\mathbf {f}})\), where \(\hat{\mathbf {f}}\) is any vector orthogonal to \(\mathbf {c}\).

### 2.4 Bilinear form \({\mathscr {J}}\) and LC planes

#### 2.4.1 Definitions

*P*of quaternions being the linear combinations of \(\mathsf {u}\) and \(\mathsf {w}\), hence such that the form \({\mathscr {J}}\) on any two of them equals 0, was dubbed the ‘Levi-Civita plane’ by Stiefel and Scheifele (1971). We will use the name ‘LC plane’, although, strictly speaking, a (hyper-)plane in a space of dimension 4 should be spun by three basis quaternions.

*P*, the second basis quaternion should be

#### 2.4.2 KS map of an LC plane

Once the LC plane has been defined, a question arises about the possibility of restricting the motion in KS variables to this subspace. But such restriction implies that the motion in ‘physical’ configuration space \({\mathbb {R}}^3\) is planar.

*P*onto a plane \(\varPi \) in ‘physical’ \({\mathbb {R}}^3\) space. Using the basis of two orthonormal quaternions \(\mathsf {u}\) and \(\mathsf {w} = \mathsf {u} (0, \mathbf {f})\), we consider their linear combination

*P*, is, by the definition (8),

#### 2.4.3 KS1 and KS3 setup

Some particular choices of the first basis quaternion \(\mathsf {u}\) deserve a special comment. Inspecting Eq. (34), we notice three obvious cases leading to \(\mathbf {c}\) positioned in the plane of motion: a pure scalar \(\mathsf {u} = \pm \,(1,\mathbf {0})\), or pure quaternions: \(\mathsf {u} = (0, \pm \,\mathbf {f})\), and \(\mathsf {u} = (0, \pm \, \mathbf {c})\). The basis vectors \({\hat{\mathbf {x}}}_1\), resulting from Eq. (31), are \(\mathbf {c}\), \(-\,\mathbf {c}\), and \(\mathbf {c}\), respectively. The last case, i.e. \(\mathsf {u} = (0, \mathbf {c})\), has been the most common choice in Celestial Mechanics since the first paper of Kustaanheimo (1964). It allows the most direct identification of the LC plane with the plane of motion, both spanned by the same vectors (or pure quaternions) \(\mathsf {u}^\natural = {\hat{\mathbf {x}}}_1 = \mathbf {c}\), and \(\mathsf {w}^\natural = {\hat{\mathbf {x}}}_2 = \mathbf {c} \times \mathbf {f}\). The freedom of choice for \(\mathbf {f}\) (any vector perpendicular to \(\mathbf {c}\)) permits to identify \(\mathbf {c}\) and \(-\mathbf {f}\) with the basis vector \(\mathbf {e}_1\) and \(\mathbf {e}_3\) of the particular reference frame used to describe the planar (\(x_3=0\)) motion. For this reason, let us call the KS transformation based upon the paradigmatic choice \(\mathbf {c}=\mathbf {e}_1\), the KS1 transformation.

## 3 Canonical KS variables in the extended phase space

*t*. The fact that \(x^*(t)=t\) is a direct consequence of the way its conjugate momentum \(X^*\) appears in Eq. (36), because

*t*on \(x^*\). The momentum \(X^*\) itself evolves according to

^{1}Then, the transformation

*Q*(Arnold et al. 1997)

It is worth noting that with an elementary choice of \(\alpha = k_1 (X^*)^{k_2}\), the expression in the square bracket evaluates to a single number \(k_2\), and the multiplier \(k_1\) has no influence on canonicity; hence, it can be selected at will—for example to conserve (or to modify) the units of time and length.

*t*to the Sundmann time \(\tau \), related by

*a*.

## 4 Action–angle variables

### 4.1 LLC and LCF variables

When the motion is planar, with \(x_3=0\), an appropriate action–angle set *l*, *g*, *L*, *G* can be created using a combination of the Levi-Civita (Levi-Civita 1906) and Lissajous transformations (Deprit and Williams 1991). This approach has been recently revisited and discussed by Breiter and Langner (2018). Viewed as the special case of the KS framework, the Lissajous–Levi-Civita (LLC) variables are inherently attached to the KS1 setup, requiring the identification of the LC plane \(P \subset {\mathbb {H}}'\) of pure quaternions and the plane of motion \(\varPi \subset {\mathbb {R}}^3\). A generalization of this approach was proposed by Zhao (2015). Roughly speaking, he attached the LC plane to an osculating plane of motion \(\varPi \) and added the third action–angle pair *h*, *H* orienting \(\varPi \) in \({\mathbb {R}}^3\) by direct analogy with the third Delaunay pair: longitude of the ascending node, and projection of the angular momentum on the axis \({\hat{\mathbf {x}}}_3\). As noted by the author, this approach has the same drawbacks as in the Dalunay set—in particular, the singularity when the orbit in physical space is rectilinear, thus having no unique orbital plane.

### 4.2 Lissajous–Kustaanheimo–Stiefel (LKS) variables

#### 4.2.1 Intermediate set

*P*, but merely project \(\mathsf {v}\) on two orthogonal subspaces. The orthogonality is readily checked by

*s*to be different from \(v^*\), while retaining its conjugate \(S=V^*\). Only then, the 1-forms are conserved up to the total differential

This transformation is merely an intermediate step, but before the final move let us inspect the meaning and properties of the variables in the Kepler problem defined by \({\mathscr {K}}'_0=0\). As a generic example, we take a heliocentric orbit in physical phase space with the following Keplerian elements: major semi-axis \(a=10\,\,\mathrm {au}\), eccentricity \(e=0.5\), inclination \(I=10^\circ \), argument of perihelion \(\omega _\mathrm {o} = 60^\circ \), longitude of the ascending node \(\varOmega =10^\circ \), and the initial true anomaly \(f=60^\circ \). From these elements, we compute first the position \(\mathbf {x}(0)\) and momentum \(\mathbf {X}(0)\), and then the representative KS3 quaternions \(\mathsf {v}(0)\) and \(\mathsf {V}(0)\)—an SKS vector given by Eq. (19), and its conjugate momentum defined by Eq. (40), both with \(\mathsf {c}=\mathsf {e}_3\). These initial conditions are labelled with black dots in Fig. 2a. The ellipses described in the \((v_0,v_3)\) and \((v_1,v_2)\) planes have different semi-axes and different eccentricities; however, both are traversed in the same direction—retrograde (clockwise) in the discussed example. The retrograde motion follows from the fact that \(G_{03}=G_{12} < 0\) (the momenta are equal due to the postulate (41), where \(\left( \bar{\mathsf {v}} \wedge {\bar{\mathsf {V}}} \right) \cdot \mathsf {e}_3 = (G_{03}-G_{12})/2\)). The constant angles \(g_{03}\) and \(g_{12}\), measured counterclockwise, position the ellipses in the coordinate planes. The initial angles \(l_{03}\) and \(l_{12}\) are marked according to the geometrical construction similar to that of the eccentric anomaly. Comparing our Fig. 2 with Fig. 1 of Deprit (1991), the readers may note the reverse direction of the \(l_{ij}\) angle. The difference comes from the fact that Deprit (1991) assumed \(G > 0\), i.e. the prograde (counterclockwise) motion along the Lissajous ellipse. Yet, regardless of the sign of \(G_{ij}\), equations of motion imply \(\mathrm {d}l_{03}/\mathrm {d}\tau = \mathrm {d}l_{12}/\mathrm {d}\tau = \omega > 0\).

^{2}In other words, one of the two minor semi-axes in each of the ellipses in Fig. 2 can be chosen at will as the reference one.

Recalling the fibration property of the KS variables, we have plotted the ellipses obtained from the same Cartesian \(\mathbf {x}(0)\) and \(\mathbf {X}(0)\), but with the KS3 initial conditions \(\mathsf {v}(0)\) and \(\mathsf {V}(0)\) right-multiplied by \(\mathsf {q}(\pi /2)=\mathsf {c}=\mathsf {e}_3\), according to Eq. (11) in the KS3 case. The results are displayed in Fig. 2b. Not only the initial conditions, but the entire ellipses are rotated by \(90^\circ \) in the \((v_0,v_3)\) plane and by \(-\,90^\circ \) in the \((v_1,v_2)\) plane. The momenta \(L_{ij}, G_{ij}\), and the angles \(l_{ij}\) remain intact, compared to Fig. 2a. The new angles positioning the ellipses are \(g'_{03} = g_{03}+\pi /2\), and \(g'_{12} = g_{12}-\pi /2\), but their sum has not changed: \(g'_{03}+g'_{12} = g_{03}+g_{12}.\)

#### 4.2.2 Final transformation

*s*and

*S*retained unaffected. One may easily verify that (64) amounts to an elementary Mathieu transformation; thus, the complete composition

*s*through

Two features of the above expressions for \(\mathsf {x}\), \(\mathsf {X}\), and \(x^*\) deserve special attention. First, none of them depends on \(\gamma \), which means that any dynamical system primarily defined in terms of \(\mathbf {x}\), \(\mathbf {X}\), and time, conserves the value of \(\varGamma \). Secondly, the expressions for the Cartesian coordinates and momenta in the extended phase space do not depend on the particular choice of \(\alpha (S)\) and \(\omega (S)\); the choice affects only the form of the Hamiltonian \({\mathscr {M}}\).

## 5 LKS variables and orbital elements

Let us interpret the variables forming the LKS set—first the momenta and then their conjugate angles—by showing their relation to the Keplerian elements or the Delaunay variables.

### 5.1 LKS momenta

*G*becomes clear once we find the pullback of the orbital angular momentum \(\mathbf {G}_\mathrm {o}\) by \(\zeta \), obtaining

*G*to be twice the projection of the orbital angular momentum on the third axis (i.e. twice the Delaunay action \(H_\mathrm {o}\)). Whenever the Hamiltonian admits the rotational symmetry around \(\mathbf {e}_3\), the momentum

*G*will be the first integral of the system.

*L*, we have to distinguish the pure Kepler problem and the perturbed one. In the former case, we can set \({\mathscr {M}}_0=0\) in Eq. (66), finding

*S*can be expressed in terms of the major semi-axis

*a*as \(S=\mu /(2a)\), which justifies the direct link between the values of

*L*and of the Delaunay action \(L_\mathrm {o}\)

*L*and \(2 L_\mathrm {o}\) generally differ in a perturbed problem due to the fact that \(L_\mathrm {o}\) is always defined by \({\mathscr {H}}_0\) alone, whereas the definition of LKS momentum

*L*depends on the complete Hamiltonian \({\mathscr {H}}_0+{\mathscr {R}}\) through the value of \(S=X^*\) (the latter fixed by the restriction to the manifold \({\mathscr {H}}=0\)).

*G*may be either positive or negative, but the above inequality guarantees that all coefficients in Eq. (67) are real.

### 5.2 LKS angles

As already mentioned, the angle \(\gamma \) is a cyclic variable, absent in the pullback of any Hamiltonian \({\mathscr {H}}\) by \(\zeta \). Actually, \(\gamma \) is the ‘KS angle’ parameterizing the fibre of KS variables \((\mathsf {v},\mathsf {V})\) mapped into the same point in the \((\mathbf {x},\mathbf {X})\) phase space. Thus, unless we are interested in some topological stability issues (Roa et al. 2016), the angle can be ignored.

*l*. As expected, its values in the pure Kepler problem are equal to a half of the orbital eccentric anomaly

*E*. Indeed,

*g*and \(\lambda \) in \(\mathbf {G}_\mathrm {o}\) and \(\mathbf {J}\) is similar. But if the norms of the vectors are evaluated, one finds

*g*proves it to be some rotation angle; the presence of \(\lambda \) means that this angle plays a different role (and is somehow related to the eccentricity

*e*).

*e*,

*I*, and \(\omega _\mathrm {o}\).

Interestingly, whenever the argument of pericentre \(\omega _\mathrm {o}\) exists, the statement \(\sin {4\lambda }=0\) means \(\cos {\omega _\mathrm {o}}=0\). Thus, any \(\lambda = k\pi /4\) refers to \(\omega _\mathrm {o}=\pi /2\) or \(\omega _\mathrm {o}=3\pi /2\).

*g*comes out of Eqs. (84) and (85): let us create the sum of normalized vectors \(\mathbf {M}'/\Vert \mathbf {M}'\Vert +\mathbf {N}'/\Vert \mathbf {N}'\Vert \) and let us rotate the resulting vector by \(\pi /2\) counterclockwise, obtaining

*g*is a half of the longitude of \(\mathbf {M}_\mathrm {m}\), or of \(-\mathbf {M}_\mathrm {m}\), depending on the sign of \(\cos {2\lambda }\). Whichever the case, changing the value of

*g*we perform a simultaneous rotation of both \(\mathbf {N}'\) and \(\mathbf {M}'\) by the same angle. Indirectly, it means the rotation of the orbital plane (if it exists) around the third axis, which makes

*g*a relative of the ascending node longitude.

### 5.3 Special orbit types

Let us inspect how some specific orbit types are mapped onto the LKS variables. The discussion is restricted to the elliptic orbits (\(0 \leqslant e \leqslant 1\)) in the pure Kepler problem (Table 1).

#### 5.3.1 Circular orbits

Particular orbits and their relation to the LKS variables

Orbit type | LKS variables | Undetermined angles |
---|---|---|

Generic circular | \(\varLambda =0, 0<|G|<L, \lambda =(2k+1)\frac{\pi }{4}\) | None |

Circular, polar | \(\varLambda =0, G=0, \,\lambda =(2k+1)\frac{\pi }{4}\) | None |

Circular, equatorial | \(\varLambda =0, |G|=L\) | \(l,g,\lambda \) |

Generic radial | \(G=0, 0< |\varLambda |<L, \lambda =k \frac{\pi }{2}\) | None |

Radial, equatorial | \(G=0, \varLambda =0, \lambda =k \frac{\pi }{2}\) | None |

Radial, polar | \(G=0, |\varLambda |=L \) | \(l,g,\lambda \) |

Generic equatorial | \(\varLambda =0, 0< |G|<L, \lambda =k \frac{\pi }{2}\) | None |

The values of \(\lambda \) mentioned above well coincide with the interpretation from Sect. 5.2. In circular orbits, the Cartan vectors \(\mathbf {N}\) and \(\mathbf {M}\) are collinear and opposite; thus, the angle \(\theta =\pi \), and its projection \(\theta '\) remains \(\pm \pi \) as long as the orbit is not equatorial. Thus \(\lambda = \theta '/4 = \pm \pi /4\), plus any multiple of \((2\pi )/4\).

Another explanation of the LKS variables for \(e=0\) can be given by inspecting the Lissajous ellipses in Fig. 2. The orbital distance *r* is the sum of \(\rho ^2_{03} =v_0^2+v_3^2\) and \(\rho ^2_{12} = v_1^2+v_2^2\), both divided by \(\alpha \). In order to secure a constant \(r= (\rho ^2_{12}+\rho ^2_{03})/\alpha \), it is not necessary that both \(\rho _{ij}\) are constant; enough if they oscillate with the same amplitude and a phase shift of \(\pm \pi /2\). Equal amplitudes result from \(L_{12}=L_{03}\) (because \(G_{12}=G_{03}\) by \(\varGamma =0\)), hence \(\varLambda = L_{12}-L_{03}=0\). The phase shift condition is given by \(l_{12}-l_{03} = 2 \lambda = (2k+1) \pi /2\), which means the values of \(\lambda \) as above.

The case of constant \(\rho _{ij}\), mentioned above, should be related to some special kind of a circular orbit. Indeed, since it needs \(L_{12}=|G|/2=L_{03}\), i.e. two circles of equal radii in Fig. 2, we obtain the circular equatorial orbits with \(\varLambda =0\) and \(|G|=L\) (prograde or retrograde, depending on the sign of *G*). Observe that due to the lack of distinct semi-axes in the two circles, the angles \(l_{ij}\) and \(g_{ij}\) are undefined, and so are *l*, *g*, \(\gamma \), and \(\lambda \). But still one can use properly defined ‘longitudes’ \(l+g\) or \(l-g\) in the prograde, and retrograde cases, respectively—at least until some ‘virtual singularities’ appear (Henrard 1974). In terms of the Cartan vectors, \(\mathbf {M}= -\mathbf {N} = \mathbf {G}_\mathrm {o}\), so \(\mathbf {M}'=\mathbf {N}'=\mathbf {0}\), making the angles *g* and \(\lambda \) undetermined.

#### 5.3.2 Radial orbits

In terms of the Lissajous ellipses in \((v_1,v_2)\) and \((v_0, v_3)\) planes from Fig. 2, \(G=0\) means that both degenerate into straight segments. The motion along the segments must obey \(l_{12}=l_{03}+k\pi \), to guarantee that \(v_0=v_1=v_2=v_3=0\) at the same epoch. The direction of \(\mathbf {x}(\mathsf {v})\) is determined by the difference of lengths of the two segments: equatorial orbits result if the segments have the same length, whereas polar orbits require that one of the segments collapses into a point. In the latter case, *l* and \(\lambda \) are undetermined, but \(l+\lambda =l_{02}\) or \(l-\lambda =l_{03}\) retain a well-defined meaning for an appropriate sign of \(\varLambda \). Problems with the definition of \(g_{ij}\) due to the vanishing minor axes are only apparent, because they can be solved by an alternative definition: instead of ‘position angle of the minor semi-axis’, one can equally well say ‘position angle of the major semi-axis minus \(\pi /2\)’.

#### 5.3.3 Equatorial orbits

*G*require \(\lambda = k \pi /2\), where \(k \in {\mathbb {Z}}\). These are the same values as in the case of radial orbits, which makes sense, because \(G=0\) should bring us to the radial equatorial orbit.

For an elliptic (\(e\ne 0\)) equatorial orbit, the Cartan vectors \(\mathbf {N}\) and \(\mathbf {M}\) may form different angles \(\theta \), but since they lie in a polar plane, the projection of these angle is always \(\theta '=0\), exactly as in the radial orbit case—thus the same values of \(\lambda \).

The two Lissajous ellipses in Fig. 2 must have the same semi-axes, and \(l_{12}=l_{03}+k\pi \). This is necessary to obtain \(v_1^2+v_2^2=v_0^2+v_3^2\), which guarantees \(x_3=0\) for all epochs, according to Eq. (9) in the KS3 setup.

#### 5.3.4 Polar orbits

Polar orbits are generically indicated by the simple condition \(G=0\). It is only in the special cases where the angle \(\lambda \) comes into play: circular polar orbits (\(\varLambda =0\)) need \(\lambda =(2k+1)\pi /4\), whereas radial polar orbits \((|\varLambda |=L)\) are the ones where \(\lambda \) is undetermined. Since \(G=0\), both Lissajous ellipses degenerate into segments, but their lengths may be different, and the phase shift arbitrary.

#### 5.3.5 Singularities

*g*become undetermined: circular equatorial orbits with \(|G|=L, \varLambda =0\) and rectilinear polar orbits with \(|\varLambda |=L, G=0\). These four points are the vertices of the square on the \((G,\varLambda )\) plane defined by the constraint \(|G|+|\varLambda | \leqslant L\). However, all four edges of the square leave the angles undetermined. This is related to the fact that:

- (a)
\(L=G+\varLambda \) (upper right edge in Fig. 4) means \(L_{03}=G_{03}\), i.e. prograde circular motion on \((v_0,v_3)\) plane with undetermined \(l_{03}\) and \(g_{03}\) (but \(l_{03}+g_{03}\) is well defined),

- (b)
\(L=-G+\varLambda \) (upper left edge in Fig. 4) means \(L_{03}=-G_{03}\), i.e. retrograde circular motion on \((v_0,v_3)\) plane with undetermined \(l_{03}\) and \(g_{03}\) (but \(l_{03}-g_{03}\) is well defined),

- (c)
\(L=G-\varLambda \) (lower right edge in Fig. 4) means \(L_{12}=G_{12}\), i.e. prograde circular motion on \((v_1,v_2)\) plane with undetermined \(l_{12}\) and \(g_{12}\) (but \(l_{12}+g_{12}\) is well defined),

- (d)
\(L=-G-\varLambda \) (lower left edge in Fig. 4) means \(L_{12}=G_{12}\), i.e. retrograde circular motion on \((v_1,v_2)\) plane with undetermined \(l_{12}\) and \(g_{12}\) (but \(l_{12}-g_{12}\) is well defined).

*g*become undefined.

## 6 Application to the Lidov–Kozai problem

### 6.1 Derivation of the secular model

*t*by its formal twin \(x^*\), obtaining

*r*, which leads to a relatively concise form

*S*is set to give \({\mathscr {M}}=0\), but ignoring the contribution of \({\mathscr {Q}}\) we may estimate that \(s \approx \tau /n\), where

*n*is the Keplerian mean motion.

According to the standard Lie transform method (e.g. Ferraz-Mello 2007), the mean variables can be introduced by a nearly canonical transformation that converts \({\mathscr {M}}\) into \({\mathscr {N}}={\mathscr {N}}_0+{\mathscr {Q}}'\), with \({\mathscr {N}}_0 = {\mathscr {M}}_0\) and \({\mathscr {Q}}'\) being constant along the phase trajectory generated by \({\mathscr {N}}_0\). Up to the first order, the new perturbation \({\mathscr {Q}}'\) is simply the average of \({\mathscr {Q}}\) with respect to \(\tau \), assuming \(l=\tau +l_0\) and \(s=\tau /n\).

*n*and both frequencies can be treated as irrational; even if they are not, the resonance will occur in high degree harmonics with practically negligible amplitudes. In these circumstances, any product of sine or cosine of \(2n_\mathrm {p} s = 2 (n_\mathrm {p}/n) \tau \) with a function which is either constant or \(2\pi \)-periodic in \(\tau \) has the zero average.

^{3}Then \({\mathscr {Q}}'\) simplifies to

Both \({\mathscr {N}}\) and the classical secular Hamiltonian of the Lidov–Kozai problem share the same property: they are reduced to one degree of freedom. In our case, it is the canonically conjugate pair \((\lambda ,\varLambda )\) instead of the usual Delaunay pair of the argument of pericentre and the angular momentum norm. All other momenta are constant and will be treated as parameters. However, there is a fundamental difference between our formulation and the classical approach: the equations of motion for \(\lambda \) and \(\varLambda \) are not singular for most of the radial orbits.

### 6.2 Secular motion and equilibria

*G*: 0.9

*L*, 0.75

*L*and 0. The phase plane has been clipped to \(-\pi \leqslant \lambda \leqslant \pi \), because the reaming range of \(\lambda \) is a simple duplication of the plotted phase portrait.

^{4}equilibria. The bottom panel of Fig. 3 confirms this observation: the points (0, 0), \((90^\circ ,0)\), and \((-\,90^\circ ,0)\) are surrounded by closed, oval-shaped contours. Intersection of any of the integral curves plotted in the bottom panel with the vertical lines \(\lambda = 0\) or \(\lambda =\pm \,90^\circ \) marks a temporary passage through the radial orbit degeneracy.

As far as the polar radial orbits (with \(|\varLambda |=L\)) are concerned, Eq. (107) becomes singular, but this singularity is purely geometrical. Such orbits should be located at the upper and lower edges of the bottom panel in Fig. 3, where \(\lambda \) is undetermined. But since the integral curves approaching the edges become parallel to them, one should expect that polar radial orbits are stable equilibria (which is actually the case, if the analysis is performed in terms of vectors \(\mathbf {G}_\mathrm {o}\) and \(\mathbf {J}\), or simply observing that for \(G=0\) the Hamiltonian \({\mathscr {N}}\) has the local maxima at \(\varLambda = \pm L\) regardless of the value of \(\lambda \)).

Actually, the presence of \(\varLambda \) as a factor of the first of Eq. (107) means that for any value of \(|G| \ne L\), the equilibria exist at \((\lambda = j\,\pi /4,\varLambda =0)\), as shown in Fig. 3. For even \(j=2k\), the equilibria refer to equatorial orbits with the eccentricity depending on *G* through \(e=\sqrt{1-(G/L)^2}\). It is easy to check the they are the local minima of the Hamiltonian \({\mathscr {N}}\); hence, the equatorial orbits are stable. The circular equatorial case with \(|G|=L\) is problematic, because then the upper and lower limits of \(\varLambda \) merge, and in order to prove that these are actually the stable equilibria, one has to resort to the analysis of \(\mathbf {G}_\mathrm {o}\) and \(\mathbf {J}\) vectors.

*G*(equatorial if \(G=0\), prograde for \(G>0\) and retrograde when \(G<0\)). Their stability depends on the ratio

*G*/

*L*. Unlike in the Delaunay chart, variational equations can be formulated directly in the phase plane of \((\lambda ,\varLambda )\), leading to the eigenvalues that are pure imaginary for \((G/L)^2 > 3/5\). Thus circular orbits are stable for inclinations below \(I_1=\arccos {\sqrt{3/5}} \approx 39^\circ \!.23\) and above \(I_2=\arccos {-\sqrt{3/5}} \approx 140^\circ \!.77\). At these critical values, a bifurcation occurs: when \((G/L)^2 < 3/5\) circular orbits become unstable and two stable equilibria are created at \((\lambda = (2k+1)\,\pi /4, \varLambda = \pm \varLambda _\mathrm {c})\) (see the middle panel of Fig. 3). Recall that, in general case of inclined, elliptic orbits, this value of \(\lambda \) means the argument of pericentre equal \(\pi /2\) or \(3\pi /2\). The value of \(\varLambda _\mathrm {c}\) is the root of the first of Eq. (107) with \(\varLambda \ne 0\) and \(\cos {4\lambda }= -1\), i.e.

Figure 4 shows all the equilibria and their stability, with the dashed lines marking the unstable equilibrium. The edges of the \((G,\varLambda )\) square (upper and lower boundaries of the plots in Fig. 3) may not be attached to any of the values of \(\lambda \), but we added the black dots at the corners to show the stable equilibria of the special type as the natural limits of the stable branches (solid lines).

It is not unusual that all action–angle-like variables with bounded momentum suffer from indeterminate angle at the boundary of its conjugate. The LKS variables cannot be different, even if many cases, problematic in the Delaunay chart, have been located inside the boundaries of \(\varLambda \). For each value of \(G \ne 0\) (and \(|G| \ne L\)), there exist integral curves passing through both the extremes: \(\varLambda = L - |G|\) and \(\varLambda = -L+ |G|\). In the top or the middle panel of Fig. 3, they are seen as four disjoint fragments; for example, the two open curves approaching the edges at \(\lambda \pm 22^\circ \!.5\) are the fragments of such an integral curve. There is no singularity in these orbits (see Sect. 5.3.5) other than the indeterminacy of longitude at the poles of a sphere (Deprit 1994).

## 7 Conclusions

While commenting a transformation due to Fukushima, Deprit (1994) observed that it amounts to swapping singularities, and immediately added ‘This remark is not meant to diminish its practical merit, quite the contrary’. The LKS variables we have presented also ‘trade in singularities’, but the rule of trade we propose is to spare the radial, rectilinear orbits (except the polar ones) at the expense of some other types. The exceptions include mostly a family of expendable, rank-and-file orbits with \(e=\sin {I}\) and the lines of apsides perpendicular to the lines of nodes—the cases easily tractable without the KS regularization and unlikely to focus attention by becoming equilibria in typical problems of Celestial Mechanics. More we regret the problems caused by the polar radial, and equatorial circular orbits. Nevertheless, we believe that more has been gained than lost. Enough to enumerate the orbits that remain regular points in our chart: circular inclined, equatorial elliptic, and all radial (except the polar ones). Thanks to refraining from the use of orbital plane in their construction, the LKS variables are better fitted to study highly elliptic orbits than any other action–angle set known to the authors.

The analysis of the quadrupole Lidov–Kozai problem in Sect. 6 suggests that the LKS variables may be a handy tool in the analysis of the more problematic cases, like the eccentric, octupolar Lidov–Kozai problem. In the latter, the ‘orbital flip’ phenomenon occurs: changing the direction of motion with the passage through an equatorial rectilinear orbit phase (Lithwick and Naoz 2011). Previous attempts to discuss this phenomenon in terms of the action–angle variables (e.g Sidorenko 2018) faced the problems which may possibly be resolved with the newly presented parameterization.

Some of the readers might be sceptical about the unnecessary duplication of the phase space resulting from the LKS transformation \(\zeta \). Indeed, Fig. 3 covers the whole phase space of in terms of the argument of pericentre \(\omega _\mathrm {o}\), although it has been clipped to the half range of \(\lambda \). This feature can be trivially removed by means of a symplectic transformation \((\lambda ,\varLambda ) \rightarrow (2 \lambda , \varLambda /2)\), and similarly for other conjugate pairs. We have not made this move in the present work for the sake of retaining the fundamental, angle-halving property of both the Levi-Civita and the Kustaanheimo–Stiefel transformations. Avoiding factor 2 in the arguments of sines and cosines in Eqs. (69) and (72), we would introduce the factor \(\frac{1}{2}\) in the expressions for \(\mathsf {v}\) and \(\mathsf {V}\). Let us mention that the restriction of the LKS transformation to \((\mathsf {v},\mathsf {V}) \rightarrow (l,g,h,\gamma ,L,G,H,\varGamma )\) can be useful also in the studies of perturbed, four-degree-of-freedom oscillators, not necessarily resulting from the KS transformation (e.g. Crespo et al. 2015; van der Meer et al. 2016). In that case, unwanted spurious singularities may arise in course of the Birkhoff normalization, when the multiple of angle does not properly match the power of action.

Having based the LKS variables upon the KS3 variant of the KS transformation, we do not exclude a possibility of performing a similar construction within the KS1 framework. But then the *G* and \(\varLambda \) variables will be the projections of the angular momentum and the Laplace–Runge–Lenz vectors on the \(x_1\) axis. With such a choice, the Lidov–Kozai Hamiltonian (104) would depend on both *g* and \(\lambda \), with the rotation symmetry hidden deeply in some complicated function of all variables, instead of the obvious \(G=\text{ const }\).

## Footnotes

- 1.
- 2.
The statements about ‘the Lissajous variables [...] determined unambiguously from the Cartesian variables’ made by Deprit (1991) should not be taken too literally.

- 3.
The general definition of the average for a function \(f(\tau )\) is \(\lim _{\tau \rightarrow \infty } \tau ^{-1} \int _0^\tau f(\tau ')\mathrm {d}\tau '\), so its value for a quasi-periodic function is null. When \(f(\tau )\) is

*T*-periodic, this definition simplifies to the standard \(T^{-1} \int _0^T f(\tau )\mathrm {d}\tau \). - 4.
The word ‘stable’ is a bit paradoxical in this context, because it means that the motion starting in such an orbit will inevitably end up in collision with the central body.

## Notes

### Compliance with ethical standards

### Conflict of interest

The authors S. Breiter and K. Langner declare that they have no conflict of interest.

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