The rotation of a triaxial rigid body in the absence of external torques is known to be integrable [1, 2]. In particular, the canonical transformation to Andoyer variables [3] reduces the free rigid body rotation to an integrable, one degree of freedom Hamiltonian, which immediately shows the preservation of the total angular momentum and allows for the representation of the possible solutions by contour plots of the reduced Hamiltonian [4]. However, because the solution to the torque-free motion depends on elliptical integrals and elliptic functions, the closed form solution is replaced in practical applications by corresponding expansions in trigonometric functions truncated to some order —as it is customary for integrable problems in which the solution depends on special functions and may further require series inversion [5, 6]. In particular, useful expansions by Kinoshita apply for the case in which the triaxiality of the rigid body is small [7] and for the case in which the rigid body rotation is close to the axis of either maximum or minimum momentum of inertia [8, 9] —the order of these expansions was later extended by other authors [10]. Alternatively, the expansions in trigonometric functions of the free rigid body solution can be directly constructed using perturbation theory [11, 12], an approach that systematizes the computation of higher orders of the expansions and eases the construction of perturbation solutions in the presence of external torques [13], as opossed to the typical first order approach [14, 15, 16].
Perturbation approaches to the torque-free motion of a rigid body start from the decomposition of the free rigid body Hamiltonian in Andoyer variables into a “main problem” and a perturbation term, the solution of the main problem being achievable in elementary functions, contrary to the special functions required in the solution of the full problem. When the triaxiality is small, the selection of either the axisymmetric case or the spherical rotor as the main problem results in a completely reduced zeroth order Hamiltonian that only depends on momenta of the canonical set of Andoyer variables, a fact that simplifies application of the perturbation method because, then, Andoyer variables result to be the action-angle variables of the main problem [11]. Practical application of the perturbative scheme based on the uniaxial model to the case of a non-rigid Earth (rigid mantle and liquid core) have been made in [17].
On the contrary, while the perturbative character of short-axis-mode (SAM) rotation can be shown directly in Andoyer variables, the consequent complete reduction of the main problem by solving the Hamilton-Jacobi equation is required in order to set up an efficient perturbative integration scheme [12]. Indeed, the main problem of SAM rotation is now a simplification of the free rigid body Hamiltonian, contrary to a reduction [18, 19]. That is, it involves the same Andoyer variables as the free rigid body problem, which include coordinates as well as their conjugate momenta, and, in consequence, Andoyer variables are not action-angle variables of the zeroth order Hamiltonian.
As far as the complete reduction of a Hamiltonian is unique [20], the canonical transformation used in finding the completely reduced Hamiltonian is irrelevant to some extent, and the standard approach is to find the action-angle variables of the integrable problem by solving the classical Hamilton-Jacobi equation or variations of it [21]. However, action-angle variables commonly involve singularities for particular configurations in which some angles may be not defined, and become ill-defined for values close to those causing the singularities. The classical instance comes from the orbital motion of a particle, where the action-angle reduction of the Kepler problem by solving the Hamilton-Jacobi equation is achieved in Delaunay variables, in which the argument of the periapsis is not defined for circular orbits, and the argument of the node is not defined in the case of equatorial orbits. Nevertheless, these singularities are of “virtual” nature [22] and can be avoided using non-singular variables. Poincaré canonical coordinates and conjugate momenta avoid the singularities in the orbital motion [23], yet non-canonical sets of variables are commonly preferred in different applications [24, 25, 26].
In the case of rotational motion, Andoyer canonical variables, which are the action angle variables of the uniaxial rotation, show singularities in different configurations. Alternative variables [27, 28] remove the singularities of Andoyer variables but at the cost of creating singularities at different locations [29].
Then, in spite of the standard approach used in [12] of finding the action-angle variables that completely reduce the main problem of SAM rotation is perfectly correct, it happens that the variables used share the same deficiencies of Andoyer variables. In particular, they may become ill-defined when the axis of instantaneous rotation evolves very close to the axis of maximum (or minimum) inertia. Therefore, in practice one must to take some care when applying the solution to this kind of motion, which is precisely the case in which the separation of the free rigid body rotation into the main problem of SAM rotation and a perturbation applies.
On the other hand, it will be shown in Section 2.3 that the perturbative arrangement of the free rigid body Hamiltonian in the case of SAM rotation is immediately disclosed when using non-singular variables of Poincaré type, cf. [30]. In these variables, the free rigid body Hamiltonian takes the form of the Hamiltonian of the simple harmonic oscillator disturbed by additional quartic polynomial terms. Hence, since it is well known that the use of complex variables renders very efficient the construction of higher order analytical solutions to perturbed harmonic oscillators (see, for instance, [31, 32, 33]), an additional transformation to complex variables is carried out in Section 3 that converts the integration of the free rigid body Hamiltonian by perturbations into a simple exercise of polynomial algebra.
The polynomial structure of the perturbation is not preserved, in general, when perturbation torques are taken into account. However, it is shown in Section 4 that, when the rotation is close enough to the axis of maximum inertia, the gravity-gradient perturbation can be easily tackled within the same perturbative scheme.
Finally, it is worth mentioning that how much fruitful expanding the special functions in which the solution of the torque-free motion inherently depends into trigonometric ones can be, there are specific cases in which approximate analytical solutions to perturbed attitude motion can be achieved directly in closed form [34, 35]. These kinds of solutions widen applicability by avoiding the constraint to particular physical characteristics or dynamical configuration, and have been recently proposed as an appealing alternative for practical application to actual problems of astrodynamics [36].
The free rigid body Hamiltonian is plainly stated in terms of Andoyer [3] canonical variables in the form, cf. Eq. (4) of [37],
where the constants
It belongs to Sadov [38] the merit of finding the transformation to action-angle variables that completely reduces Hamiltonian (1). His approach resorts to a complete solution of the Hamilton-Jacobi equation and involves the elliptic integrals of the first and third kinds, both of them in their incomplete and complete versions. Because the transformation from Andoyer to action-angle variables cannot be obtained explicitly, the completely reduced Hamiltonian must remain also as an implicit function of the action variables. A modern, abridged re-derivation of Sadov’s solution, which does not need to deal explicitly with the time when solving the Hamilton-Jacobi equation, can be found in [35]. This solution is summarized in the Appendix for completeness.
Working independently of Sadov, Kinoshita [7] succeeded also in finding the transformation of the free rigid body Hamiltonian to action-angle variables, yet he expressed his solution in terms of the Heuman’s Lambda function rather than the elliptic integral of the third kind. However, it must be reminded that the
To make the use of the free rigid body solution practical in the study of the rotation under external torques, the closed form solution is customarily expanded in terms of trigonometric functions. Both authors, Sadov and Kinoshita, provided the necessary expansions in powers of the Jacobi’s nome, which is a function of the elliptic modulus that improves convergence over the direct use of the elliptic modulus in the expansions. Alternatively, the expansion of the closed form solution can be avoided in the particular cases in which some small quantity is identified in the original Hamiltonian. Then, the free rigid body Hamiltonian can be rearranged as a perturbation Hamiltonian and the solution of the integrable problem is directly attacked by perturbation methods [6].
Regrettably, because the modulus of the elliptic functions that solve the torque free motion depends on the energy of each particular solution, which is in fact the Hamiltonian, neither the Jacobi’s nome nor the elliptic modulus are useful parameters to be identified in the original Hamiltonian in order to apply a perturbation approach. Still, as shown in the Appendix, it happens that this elliptic modulus can be decomposed into a product that splits the physical and dynamical characteristics of the motion, and, luckily, each kind of feature is easily recognized in Hamiltonian (1) after simple rearrangement.
Indeed, Andoyer’s [3] original arrangement of the rigid body Hamiltonian
where the relations
define the physical parameters
Thus
in which the zeroth order term
corresponds to an oblate axisymmetric body (
which is due to the lack of axial symmetry of the rigid body, depends on the angle
An alternative perturbative arrangement has been recently proposed for rigid bodies rotating close to its axis of maximum inertia, irrespective of their triaxiality [12]. In that case
where
is taken as the integrable part,
is a perturbation ∣𝒫∣ ≪ ℳ.
Now, the zeroth order Hamiltonian (5), which has been dubbed the main problem of SAM rotation in [12], is not a completely reduced Hamiltonian, and, on the contrary, it involves the same variables as the free rigid body Hamiltonian (2). Therefore, to approach the solution of the torque-free motion by perturbations in the case of SAM rotation, it is convenient first to find the action-angle variables that completely reduce Eq. (5). This transformation has been achieved in [12] by solving the Hamilton-Jacobi equation. However, it will be shown in the next Section that the complete reduction of Eq. (5) can be immediately obtained without need of solving the Hamilton-Jacobi equation when using non-singular variables of the Poincaré type.
Andoyer variables are singular for
Straightforward computations show that the differential form
The inverse transformation to Eqs. (7)–(10) is
and, by direct replacement in Eq. (5), it is easy to see that it converts the Hamiltonian of the main problem of the SAM rotation into
in which
is a function of the rigid body’s moments of inertia. Alternatively
One easily recognizes in Eq. (15) the Hamiltonian of a harmonic oscillator of (non-dimensional) frequency
completely reduces the Hamiltonian of the harmonic oscillator to a function of only the momentum
which, in view of the definition of
In this way —formulation of the main problem Hamiltonian in the nonsingular variables in Eqs. (7)–(10) followed by the Poincaré transformation in Eq. (17)— the computation of the action-angle variables of the main problem of the SAM rotation carried out in [12] is dramatically abridged to the composition of the canonical transformations
On the other hand, the use of action-angle variables, while customary, is not a requirement in perturbation theory. Indeed, in view of Eq. (6) takes the form of a quartic polynomial after applying the transformation to nonsingular variables in Eq. (7)–(10), viz.
the perturbation solution can be directly constructed in Cartesian variables.
For perturbed harmonic oscillators, it is well known that the use of complex variables makes the whole perturbation approach very efficient, cf. [31, 32, 33]. Therefore, the perturbation Hamiltonian given by Eqs. (15) and (19) is next reformulated in complex variables.
As before, the differential form
If, besides, the choices Ω =
are made, then Eq. (15) is rewritten in the real (
whereas Eq. (19) takes the form
Then, an analytical approximation to the flow stemming from Eqs. (25) and (26) can be computed with the Lie transforms method [41]. Since this method is standard these days, details are not provided here and interested readers are referred to textbooks in the literature, as, for instance [42, 43].
The Lie derivative ℒℳ associated to the Eq. (25), is given by the Poisson bracket operator ℒℳ = { ;ℳ}, viz.
and the partial differential equation ℒℳ(
However, because Eq. (26) does not depend on
Then, for any integers
and, therefore, ℒℳ(
In consequence, the solution of the homological equation becomes a trivial operation in complex variables. Indeed, any monomial
to the generating function. In this way the need of solving partial differential equations is completely avoided.
The perturbation procedure starts writing the free rigid body Hamiltonian in the form of the Taylor series expansion, viz.
where
where
which, as expected, are the same as those in Table 2 of [12] after adjusting subindices and scaling by
The transformation from prime to original variables
is obtained by successive evaluations of Deprit’s triangle
using the generating function 𝒮 = ∑
are computed using Eq. (29). The first few
The construction of the perturbation solution by Lie transforms results extremely efficient in this way. As a demonstration of the performance, it has been extended to the order 35 in the small parameter
This fact is illustrated in Fig. 1 where the label
To further emphasize the efficiency of the selection procedure used with the complex variables approach, as opposite to the usual, time consuming, averaging procedures, these times are shown in Fig. 2 relative to the time required in choosing the different terms of the transformed Hamiltonian in prime variables in Eq. (30). In this figure it is noted that the time spent in the computation of the generating function remains between 5 and 10 times higher than the time spent in the computation of the new Hamiltonian term except for the lower orders, where the ratio can be lower. On the other hand, the time spent in the computation of the known terms of the homological equation is similar to the time spent in filling Deprit’s triangle at each step, and it grows high with the order of the perturbation theory. In particular, we checked that the computation of
The perturbation approach based on the main problem of SAM rotation is also feasible for motion under external torques. In the particular case of gravity-gradient perturbations due to a distant body, one can make the simplifying assumption that the disturbing body moves with Keplerian motion. If, besides, the orbital and inertial planes are assumed to match, the gravity-gradient effect is derived from the potential
where
where
After carrying out the rotations involved in Eq. (34), it is found that the majority of perturbation terms that comprise Eq. (33) are factored by sin
Notably, to this order of approximation, Eq. (35) does not depend either on
Now, the full Lie derivative in Eq. (27) is involved in the solution of the homological equation because
and assume that there is not coupling with the previous terms of the perturbation theory, a particular solution of the homological equation for the order
Short-axis mode rotation of a free rigid body is naturally decomposed into a main problem and a perturbation, a fact that leads to the straightforward integration of the rotation by perturbation series. When using non-singular variables of the Poincaré type, the main problem has the form of a harmonic oscillator, whose frequency is related to the triaxiality of the rigid body, whereas the perturbation is a quartic polynomial. Then, the use of complex variables makes the construction of the perturbation solution trivial. The polynomial character of the perturbation does not persist, in general, when the motion is affected by external torques. However, when the rotation is close to the axis of maximum inertia, the gravity-gradient perturbation can also be approached in complex variables.