Astronomy Astrophysics

A&A 415, 1187–1199 (2004) Astronomy DOI: 10.1051/0004-6361:20034058 & c ESO 2004 Astrophysics

Gauge freedom in the N-body problem of celestial mechanics

M. Efroimsky1 and P. Goldreich2

1 US Naval Observatory, Washington, DC 20392, USA e-mail: [email protected] 2 IAS, Princeton NJ 08540 & CalTech, Pasadena, CA 91125, USA e-mail: [email protected]

Received 8 July 2003 / Accepted 22 October 2003

Abstract. The goal of this paper is to demonstrate how the internal symmetry of the N-body celestial-mechanics problem can be exploited in orbit calculation. We start with summarising research reported in (Efroimsky 2002, 2003; Newman & Efroimsky 2003; Efroimsky & Goldreich 2003) and develop its application to planetary equations in non-inertial frames. This class of problems is treated by the variation- of-constants method. As explained in the previous publications, whenever a standard system of six planetary equations (in the Lagrange, Delaunay, or other form) is employed for N objects, the trajectory resides on a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. The freedom in choosing this submanifold reveals an internal symmetry inherent in the description of the trajectory by orbital elements. This freedom is analogous to the gauge invariance of electrodynamics. In traditional derivations of the planetary equations this freedom is removed by hand through the introduction of the Lagrange constraint, either explicitly (in the variation-of-constants method) or implicitly (in the Hamilton-Jacobi approach). This constraint imposes the condition (called “osculation condition”) that both the instantaneous position and velocity be fit by a Keplerian ellipse (or hyperbola), i.e., that the instantaneous Keplerian ellipse (or hyperbola) be tangential to the trajectory. Imposition of any supplementary constraint different from that of Lagrange (but compatible with the equations of motion) would alter the mathematical form of the planetary equations without affecting the physical trajectory. However, for coordinate-dependent perturbations, any gauge different from that of Lagrange makes the Delaunay system non- canonical. Still, it turns out that in a more general case of disturbances dependent also upon velocities, there exists a “generalised Lagrange gauge”, i.e., a constraint under which the Delaunay system is canonical (and the orbital elements are osculating in the phase space). This gauge reduces to the regular Lagrange gauge for perturbations that are velocity-independent. Finally, we provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.

Key words. celestial mechanics – reference systems – solar system: general – methods: N-body simulations – methods: analytical – methods: numerical

1. Introduction introduced by Newton and Euler and furthered by Lagrange for his studies of planet-perturbed cometary orbits and only 1.1. Prefatory notes later applied to planetary motion (Lagrange 1778, 1783, 1808, On the 6th of November 1766 young geometer Giuseppe 1809, 1810). The method was based on an elegant mathemati- Lodovico Lagrangia, invited from Turin at d’Alembert’s rec- cal device, the variation of constants emerging in solutions of ommendation by King Friedrich the Second, succeeded Euler differential equations. This approach was pioneered by Newton as the Director of Mathematics at the Berlin Academy. in his unpublished Portsmouth papers and described very suc- Lagrange held the position for 20 years, and this fruitful period cintly in Cor. 3 ans 4 of Prop. 17 in Book I of his “Principia”. of his life was marked by an avalanche of excellent results, and The first attempts of practical implementation of this tool were by three honourable prizes received by him from the Académie presented in a paper on Jupiter’s and Saturn’s mutual distur- des Sciences of Paris. All three prizes (one of which he shared bances, submitted by Euler to a competition held by the French with Euler) were awarded to Lagrange for his contributions to Academy of Sciences (Euler 1748), and in the treatise on the lu- celestial mechanics. Among these contributions was a method nar motion, published by Euler in 1753 in St. Petersburg (Euler 1753). However, it was Lagrange who revealed the full power Send offprint requests to: M. Efroimsky, of this approach. e-mail: [email protected]

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20034058 1188 M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics

number of ways of dividing the actual planet’s movement into motion along its orbit and the simultaneous evolution of the orbit. Although the physical trajectory is unique, its description (i.e., its parametrisation in terms of Kepler’s elements) is not. A map between two different (though physically equivalent) sets of orbital elements is a symmetry transformation (a gauge transformation, in physicists’ jargon). Lagrange never dwelled upon that point. However, in his treatment (based on direct application of the method of variation of constants) he passingly introduced a convenient mathematical condition which removed the said ambiguity. This condition and possible alternatives to it will be the topics of Sects. 1–3 of this paper. In 1834–1835 Hamilton put forward his theory of canonical transformations. Several years later this approach was furthered by Jacobi who brought Hamilton’s technique into astronomy and, thereby, worked out a new method of deriving the planetary equations (Jacobi 1866), a method that was soon accepted as standard. Though the mathematical content of the Hamilton-Jacobi theory is impeccably correct, its application Fig. 1. Two coplanar ellipses, having one common focus, are assumed to astronomy contains a long overlooked aspect that needs at- to rotate about this focus, always remaining within their plane. Let tention. This aspect is: where is the Lagrange constraint tacitly a planet be located at one of the points of the ellipses’ intersection, imposed, and what happens if we impose a different constraint? P, and assume that the rotation of the ellipses is such that the planet This issue will be addressed in Sect. 4 of our paper. is always at the instantaneous point of their intersection. At some in- The main line of reasoning and the principal results pre- stant of time, the rotation of one ellipse will be faster than that of the sented in this paper are the following. In a concise introduction, other. On these grounds one may state that the planet is rapidly mov- to be presented in the next subsection, we derive Eq. (16) which ing along the slower rotating ellipse. On the other hand, though, it is is the most general form of the gauge-invariant perturbation also true that the planet is slowly moving along the faster rotating ellipse. Both viewpoints are equally valid, because one can divide, in an equation of celestial mechanics, written in terms of a disturbing infinite number of ways, the actual motion of the planet into a motion force. Then we transform it into (25), which is the most gen- along some ellipse and a simultaneous evolution of this ellipse. The eral gauge-invariant perturbation equation expressed through a Lagrange constraint (7) singles out, of all the sequences of evolving disturbance of the Lagrangian. The next step is to show that, in ellipses, that unique ellipse sequence which is always tangential to the the case of velocity-dependent perturbations, the equations for physical velocity of the planet. the Delaunay elements are, generally, not canonical. However, they become canonical in a special gauge (which is, generally, Below we shall demonstrate that the equations for the in- different from the customary Lagrange constraint). stantaneous orbital elements possess a hidden symmetry sim- Our other goal is to explore how the freedom of gauge ilar to the gauge symmetry of electrodynamics. Derivation of choice is supplemented by the freedom of choice of a coor- the Lagrange system involves a mathematical operation equiv- dinate frame in which to implement the method of variation of alent to the choice of a specific gauge. As a result, trajec- constants. This investigation, carried out in Sect. 3, provides a tories get constrained to some 9-dimensional submanifold in physical example of a nontrivial gauge being instrumental in the 12-dimensional space constituted by the orbital elements simplifying orbit calculations. In this way we demonstrate that and their time derivatives, as demonstrated in (Efroimsky 2002, gauge freedom in celestial dynamics is not only a cultural ac- 2003; Newman & Efroimsky 2003; Efroimsky & Goldreich quisition, but can be exploited to improve computations. 2003). However, the choice of this submanifold is essentially ambiguous, and this ambiguity gives birth to an internal sym- 1.2. Osculating elements vs. orbital elements metry. The symmetry is absent in the two-body case, but comes into being in the N-body setting (N 3) where each body fol- We start in the spirit of Lagrange. Before addressing the lows an ellipse of varying shape whose≥ time evolution contains N-particle case, Lagrange referred to the reduced two-body an inherent ambiguity. problem, For a simple illustration of this point imagine two coplanar µ r ¨r + = 0, ellipses sharing one focus (Fig. 1). Suppose they rotate at dif- r2 r ferent rates in their common plane. Let a planet be located at r rplanet rsun,µG(mplanet + msun), (1) one of the intersection points of these ellipses. The values of the ≡ − ≡ elliptic elements needed to describe its trajectory would depend whose generic solution, a Keplerian ellipse or a hyperbola, can upon which ellipse was chosen to parameterise the orbit. Either be expressed, in some fixed Cartesian frame, as set would be equally legitimate and would faithfully describe the physical trajectory. Thus we see that there exists an infinite r = f (C1, ..., C6, t) , r˙ = g (C1, ..., C6, t) , (2) M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics 1189 where be interpreted as parameters of an instantaneous ellipse (in a ∂ f bound-orbit case) or an instantaneous hyperbola (in a flyby sit- g (3) uation). Lagrange found it natural to make the instantaneous ≡ ∂t !C=const · orbital elements Ci such that, at each moment of time, this el- Since the problem (1) constitutes by three second-order dif- lipse (hyperbola) coincides with the unperturbed (two-body) ferential equations, its general solution naturally contains six orbit that the body would follow if the disturbances were to adjustable constants Ci. At this point it is irrelevant which par- cease instantaneously. In other words, these ellipses (hyperbo- ticular set of the adjustable parameters is employed. (It may lae) are tangential to the physical trajectory. This is the well be, for example, a set of Lagrange or Delaunay orbital ele- known condition of osculation, which is mathematically im- ments or, alternatively, a set of initial values of the coordinates plemented by (7). The appropriate orbital elements are called and velocities.) osculating elements. This way, Lagrange restricted his use of Following Lagrange (1808–1810), we take f as an ansatz the orbital elements to elements that osculate in the reference for a solution to the N-particle problem, the disturbing force frame wherein ansatz (5) is employed. Lagrange never explored 1 acting at a particle being denoted by ∆F : alternative options; he simply imposed (7) and used it to derive µ r his famous system of equations for orbital elements. r¨ + =∆F. (4) r2 r Such a restriction, though physically motivated, is com- Now the “constants” become time dependent: pletely arbitrary from the mathematical point of view. While the imposition of (7) considerably simplifies the subsequent r = f (C1(t), ..., C6(t), t) , (5) calculations it in no way influences the shape of the physical whence the velocity trajectory and the rate of motion along it. As emphasised in Efroimsky (2002, 2003) and Newman & Efroimsky (2003), a dr ∂ f ∂ f dC ∂ f dC = + i = g + i , (6) choice of any other supplementary constraint dt ∂t ∂C dt ∂C dt i i i i X X ∂ f dCi = Φ(C1, ..., C6, t), (8) acquires an additional term, (∂ f/∂Ci)(dCi/dt). ∂C dt Xi i Substitution of (5) and (6)P into the perturbed equation of Φ being an arbitrary function of time and parameters Ci, motion (4) leads to three independent scalar second-order dif- 3 ferential equations which contain one independent parameter, would lead to the same physical orbit and the same velocities . Substitution of the Lagrange constraint (7) by its generalisa- time, and six functions Ci(t). These are to be found from the three scalar equations comprised by the vector Eq. (4), and tion (8) would not influence the motion of the body, but would this makes the essence of the variation-of-constants method. alter its mathematical description (i.e., would entail different However, the latter task cannot be accomplished in a unique solutions for the orbital elements). Such invariance of a phys- way because the number of variables exceeds, by three, the ical picture under a change of parametrisation is called gauge number of equations. Hence, though the physical trajectory freedom or gauge symmetry. It parallels a similar phenomenon (comprised by the locus of points in the Cartesian frame and well known in electrodynamics and, therefore, has similar con- by the values of velocity at each of these points) is unique, sequences. On the one hand, a “good” choice of gauge often its parametrisation through the orbital elements is ambiguous. simplifies solution of the equations of motion. (In Sect. 3 we This circumstance was appreciated by Lagrange, who amended provide a specific application to motion in an accelerated co- the system, by hand, with three independent conditions, ordinate system.) On the other hand, one should expect a dis- placement in the gauge function Φ, owing to the accumulated ∂ f dC i = 0, (7) error in the constants. ∂C dt Xi i A simple case of the same motion being described by 2 two different families of instantaneous ellipses is presented in and thus made it solvable . His choice of constraints was mo- Fig. 1. Two coplanar ellipses have a common focus and are ro- tivated by both physical considerations and mathematical ex- tating about it, in the same plane but at different rates. Assume pediency. Physically, the set of functions (C1(t), ..., C6(t)) can that one ellipse is rotating rapidly and another slowly. Then it 1 The treatment offered by Lagrange, as well as its gauge-invariant will be legitimate to state that the point of their intersection, P, generalisation presented in (Efroimsky 2002, 2003; Newman & is moving rapidly along a slowly rotating ellipse. At the same Efroimsky 2003), addressed only the case of a coordinate-dependent time, it will be right to say that it is moving slowly along a perturbation ∆F(r). The treatment presented in this paper covers dis- swiftly rotating ellipse. turbing forces ∆F(r, ˙r, t) that are arbitrary vector-valued functions of Derivation of the conventional Lagrange and Delaunay position, velocity, and time. planetary equations by the variation-of-constantsmethod incor- 2 Insertion of (5) into (4) makes Eq. (4) a vector equation of the porates the Lagrange constraint (Brouwer & Clemence 1961). second order, with respect to functions Ci(t). This is equivalent to three scalar equations of the second order. However, these may be 3 In principle, one may also impart Φ with dependence upon the interpreted as three scalar equations of the first order, written for six parameters’ time derivatives of all orders. This would yield higher- variables Ci(t), C˙i(t), provided we amend them with six trivial equa- than-first-order time derivatives of the Ci in subsequent developments tions C˙i(t) = dCi(t)/dt. This gives, for each particle, nine scalar first- requiring additional initial conditions, beyond those on r and ˙r,tobe order equations for twelve quantities Ci(t), C˙i(t). Three constraints specified in order to close the system. We avoid this unnecessary com- will then make this system fully defined. plication by restricting Φ to be a function of time and the Ci. 1190 M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics

Both systems of equations get altered under a different gauge This is the most general form of the gauge-invariant perturba- choice, as we now show. The essence of a derivation suitable tion equations of celestial mechanics. In the Lagrange gauge, for a general choice of gauge starts with (6) from which the when the Φ terms are absent, we can obtain an immediate so- formula for the acceleration follows: lution for the individual dCi/dt by exploiting the well known d2 r ∂g ∂g dC dΦ expression for the Poisson-bracket matrix which is inverse to = + i + = the Lagrange-bracket one and is offered in the literature for the dt2 ∂t ∂C dt dt Xi i two-body problem. (Be mindful that our brackets (15) are de- ∂2 f ∂g dC dΦ fined in the same manner as in the two-body case; they contain + i + (9) ∂2t ∂C dt dt · only functions f and g that are defined in the unperturbed, two- Xi i body, setting.) In an arbitrary gauge the presence of the term Together with the equation of motion (4), it yields: proportional to ∂Φ/∂C j on the left-hand side of (16) compli- cates the solution for dC /dt, but only to the extent of requir- ∂2 f µ f ∂g dC dΦ i + + i + =∆F, (10) ing the resolution of a set of six simultaneous linear algebraic ∂t2 r2 r ∂C dt dt Xi i equations. r r = f . ≡| | | | To draw to a close, we again emphasise that, for fixed in- The vector function f was from the beginning introduced as teractions and initial conditions, all possible (i.e., compatible a Keplerian solution to the two-body problem; hence it must and sufficient) choices of gauge conditions expressed by the obey the unperturbed Eq. (1). On these grounds the above for- vector function Φ lead to a physically equivalent picture. In mula simplifies to: other words, the real trajectory is invariant under reparametrisations permitted by the ambiguity of the choice of gauge. This ∂g dC dΦ i =∆F (11) invariance has the following meaning. Suppose the equations ∂C dt − dt · Xi i of motion for C1, ..., C6, with some gauge condition Φ imposed, render the solution C1(t), ..., C6(t). The same equations This equation describes the perturbed motion in terms of of motion, with another gauge Φ˜ enforced, furnish the solu- the orbital elements. Together with constraint (8) it consti- tion C˜i(t)thathasadifferent functional form. Despite this dif- tutes a well-defined system which may be solved with respect ference, both solutions, Ci(t)andC˜i(t), when substituted back to dCi/dt. An easy way to do this is to use the elegant trick in (5), yield the same curve r(t) with the same velocitiesr ˙(t). suggested by Lagrange: to multiply the equation of motion In mathematics this situation is called a fiber bundle, and it by ∂ f/∂Cn and to multiply the constraint by ∂g/∂Cn.The ˜ − gives birth to a 1-to-1 map of Ci(t) onto Ci(t), which is merely former operation results in a reparametrisation. In physics this map is called a gauge transformation. The entire set of these reparametrisations constitute ∂ f ∂g dC j ∂ f ∂ f dΦ = ∆F , (12) a group of symmetry known as a gauge group. ∂C  ∂C dt  ∂C − ∂C dt n Xj j  n n   Just as in electrodynamics, where the fields E and B stay in-   µ while the latter gives variant under gradient transformations of the four-potential A , so the invariance of the orbit implements itself through the ∂g ∂ f dC j ∂g form-invariance of expression (5) under the aforementioned = Φ. (13) −∂C  ∂C dt  −∂C map. The vector r and its full time derivativer ˙, play the role n Xj j  n   of the physical fields E and B, while the Keplerian coordi-   µ Having summed these two equalities, we arrive at: nates C1, ..., C6 play the role of the four-potential A .

dC j ∂ f ∂ f dΦ ∂g No matter whether the role of constants Ci is played by [Cn C j] = ∆F Φ, (14) the six Kepler variables (e, a, M ,ω,Ω, i ) or by some six dt ∂Cn − ∂Cn dt − ∂Cn 0 Xj combinations thereof (like, say, the Delaunay set (27) or the

[Cn C j] standing for the unperturbed (i.e., deﬁned as in the Jacobi set or the Poincare set), these constants are always called two-body case) Lagrange brackets: “orbital elements”. Only in the case when the condition (7) of Lagrange is imposed will these orbital elements be called ∂ f ∂g ∂ f ∂g [Cn C j] (15) “osculating”. This is called an osculating solution, because the ≡ ∂Cn ∂C j − ∂C j ∂Cn · particles’ positions and velocities are instantaneously tangent It was selected above that Φ is a function of time and of the to, i.e., touch, the Keplerian ellipses (or hyperbolae). It is easy to see that, by virtue of the diﬀerentiation chain rule, the oscu- “constants” Ci, but not of their time derivatives. Under this con- vention, the above equation may be shaped into a more conve- lation condition (7) remains form-invariant under a transition nient form by splitting the full derivative of Φ into a partial and from the Kepler set of elements to any other set. This is most a convective one, and by moving the latter to the left-hand side: natural, because the osculation condition is purely geometric one and does not depend upon our preferences in the choice of ∂ f ∂Φ dC j convenient parameters. [Cn C j] + = ∂C ∂C ! dt Xj n j A comprehensive discussion of all the above-raised issues ∂ f ∂ f ∂Φ ∂g can be found in Efroimsky (2003). The interconnection be- ∆F Φ (16) tween the internal symmetry and multiple time scales, both ∂Cn − ∂Cn ∂t − ∂Cn · M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics 1191 in celestial mechanics and in a more general ODE context, is 2.2. Gauge-invariant planetary equations addressed in Newman & Efroimsky (2003). When the expression (21) for the most general force emerging within the Lagrangian formalism is substituted into the gauge- 2. Planetary equations in an arbitrary gauge invariant perturbation Eq. (14), it yields:

2.1. Lagrangian and Hamiltonian perturbations dC j [C C ] = n j dt We can proceed further by restricting the class of perturbations Xj we consider to those in which ∆F is derivable from a perturbed ∂ f ∂ ∆ ∂ f d ∂ ∆ ∂g Lagrangian. Such a restricted class of disturbances is still broad L Φ + L Φ, (22) ∂C ∂r − ∂C dt ∂˙r ! − ∂C enough to encompass most applications of celestial-mechanics n n n perturbation theory. This happens largely because we shall con- [Cn C j] being the Lagrange brackets defined through (15). sider Lagrangian perturbations dependent not only upon coor- Since, for a velocity-dependent disturbance, the chain rule dinates but also upon velocities. It will enable us to describe ∂ ∆ ∂ ∆ ∂ f ∂ ∆ ∂˙r in a perturbative manner not only physical but also velocity- L = L + L = ∂C ∂r ∂C ∂˙r ∂C dependent, inertial forces emerging in non-inertial frames of n n n ∂ ∆ ∂ f ∂∆ ∂(g + Φ) reference. This, in its turn, will provide us with an opportunity L + L , (23) to work in the coordinate system precessing with the planet of ∂r ∂Cn ∂˙r ∂Cn interest. takes place, formula (22) may be reshaped into: Let the unperturbed dynamics be described by the unit- mass Lagrangian (r, ˙r, t) = ˙r 2/2 U(r, t), canonical mo- dC j 0 [Cn C j] = L − 2 dt mentum p = ˙r and Hamiltonian 0(r, p, t) = p /2 + U(r, t). Xj H The disturbed motion will be described by the new, perturbed, ∂ ∆ ∂ ∆ ∂Φ ∂ f d ∂g ∂ ∆ functions: L L Φ + L (24) ∂Cn − ∂˙r ∂Cn − ∂Cn dt − ∂Cn ! ∂˙r ! · ˙r 2 (r, r˙, t) = 0 +∆ = U(r, t) +∆ (r, ˙r, t), (17) Φ L L L 2 − L Next we group terms so that the gauge function everywhere appears added to ∂(∆ )/∂˙r, and bring the term proportional ∂ ∆ to dC /dt to the left handL side of the equation: p = ˙r + L, (18) j ∂˙r ∂ f ∂ ∂ ∆ dC j and [Cn C j] + L + Φ = ( ∂Cn ∂C j ∂˙r !) dt p 2 Xj (r, p, t) = p ˙r = + U +∆ , 2 H −L 2 H ∂ 1 ∂ ∆ 2 ∆ + L 1 ∂ ∆ ∂Cn  L 2 ∂˙r !  ∆ (r, p, t) ∆ L , (19)   H ≡− L−2 ∂˙r ! ∂g ∂ f ∂ ∂ ∆ ∂ ∂ ∆ + + L Φ + L (25) ∆ being introduced as a variation of the functional form, i.e., − ∂Cn ∂Cn ∂t ∂˙r ∂Cn ! ∂˙r ! · asH∆ (r, p, t) (r, p, t). The Euler-Lagrange equa- H≡H −H0 These modifications help us to recognise the special nature of tions written for the perturbed Lagrangian (17) will give: the gauge Φ = ∂(∆ )/∂˙r, which will be the subject of dis- ∂U cussion in the next− subsection.L r¨ = +∆F, (20) − ∂r Contrast (25) with (16): while (16) expresses the variation- where the new term of-constants method in the most general form it can have in terms of the disturbing force ∆F(r, ˙r, t), Eq. (25) renders the ∂ ∆ d ∂ ∆ ∆F L L (21) most general form in the language of a Lagrangian perturba- ∂r dt ∂˙r ≡ − ! tion ∆ (r, ˙r, t). is the disturbing force. Expression (21) reveals that whenever TheL applicability of so generalised planetary equations in the Lagrangian perturbation is velocity-independent, it plays analytical calculations is complicated by the nontrivial nature the role of the disturbing function. Generally, though, the dis- of the left-hand sides of (16) and (25). Nevertheless, the struc- turbing force is not equal to the gradient of ∆ , but has an extra ture of these left-hand sides leaves room for analytical simpli- L term generated by the velocity dependence. fication in particular situations. One such situation is when the Examples in which a velocity dependent ∆ has been used gauge is chosen to be in a celestial mechanics setting include: the treatmentL of inertial ∂ ∆ forces in a coordinate system tied to the spin axis of a precess- Φ = L + η(r, t), (26) ing planet (Goldreich 1965) and the velocity-dependent cor- − ∂˙r rections to Newton’s law of gravity in the relativistic two-body η(r, t) being an arbitrary vector function linear in r. (It may problem (Brumberg 1992). be, for example, proportional to r or may be equal, say, to Our next goal will be to employ the above formula (21) to a cross product of r by some time-dependent vector.) Under translate the gauge-invariant perturbation Eq. (16) from the lan- these circumstances the left-hand side in (25) reduces to the guage of disturbing forces into that of Lagrangian disturbances. Lagrange brackets. The situation becomes especially simple 1192 M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics when ∂∆ /∂˙r happens to be linear in r, in which case we may form in one special gauge, one that coincides with the Lagrange put η(r, tL) = ∂∆ /∂˙r and, thus, employ the trivial Lagrange gauge when the perturbation bears no velocity dependence. The gauge Φ = 0 insteadL of the generalised Lagrange gauge. We issue is explained at length in our previous paper (Efroimsky & shall encounter one such example in Sect. 3.4. Goldreich 2003). Here we offer a brief synopsis of this study. As already stressed above, the Lagrange brackets are gauge-invariant because functions f and g are defined within 2.3. Generalised Lagrange gauge wherein the unperturbed, two-body, problem (1–3) that lacks gauge freedom. For this reason, one may exploit, to solve (25), the the Delaunay-type system is canonical well-known expression for the inverse of matrix [Ci C j]. Its Equation (22) was cast in the form of (25) not only to demon- elements are simplest (and are either zero or unity) when one strate the special nature of the gauge chooses as the “constants” the Delaunay set of orbital variables ∂ ∆ (Plummer 1918): Φ = L, (30) − ∂˙r C = L, G, H, M ,ω,Ω i { 0 } but also to single out the terms in the square brackets on the L √µa, G µa 1 e2 , H µa 1 e2 cos i, (27) right-hand side of (25): together, these terms give exactly the ≡ ≡ q − ≡ q − Hamiltonian perturbation. Thus, we come to the conclusion where µ G(msun + mplanet)and(e, a, M0,ω,Ω, i) are the that in the special gauge (30) our Eq. (25) simplifies to Keplerian≡ elements: e and a are the eccentricity and semima- dC j ∂ ∆ jor axis, M0 is the mean anomaly at epoch, and the Euler an- [C C ] = H (31) n j dt − ∂C · gles ω, Ω, i are the the argument of pericentre, the longitude Xj n of the ascending node, and the inclination, respectively. As emphasised in the preceding subsection, the gauge in- The simple forms of the Lagrange and Poisson brackets in variance of definition (15) enables us to use the standard Delaunay elements is the proof of these elements’ canonicity (Lagrange-gauge) expressions for the matrix inverse to [C C ], in the unperturbed, two-body, problem: the Delaunay elements n j to get the planetary equations from (31). give birth to three canonical pairs (Q , P ) corresponding to a i i Comparing (31) with (28), we see that in the general case vanishing Hamiltonian: (L, M ), (G, ω), (H, Ω). In a 0 of an arbitrary ∆ (r, ˙r, t) one arrives from (31) to the same perturbed setting, when only− a position-dependent− disturbing− equations as fromL (28), except that now they contain ∆ in- function R(r, t) is “turned on”, it can be expressed through the stead of ∆ . When the orbit is parametrised by the Delaunay− H Lagrangian and Hamiltonian perturbations in a simple manner, variables, thoseL equations take the form: R(r, t) =∆(r, t) = ∆ (r, t), as can be seen from the formulae presentedL in the− previousH subsection. Under these cir- dL ∂ ∆ d( M ) ∂ ∆ = H , − 0 = H , (32) cumstances, the Delaunay elements remain canonical, provided dt ∂( M0) dt − ∂L the Lagrange gauge is imposed (Brouwer & Clemence 1961). − This long known fact can also be derived from our Eq. (25): dG ∂ ∆ d( ω) ∂ ∆ = H , − = H , (33) if we put Φ = 0 and assume ∆ velocity-independent, we dt ∂( ω) dt − ∂G arrive to L −

dC j ∂ ∆ dH ∂ ∆ d( Ω) ∂ ∆ [Cn C j] = L, (28) = H , − = H (34) dt ∂Cn dt ∂( Ω) dt − ∂H · Xj − where which is a symplectic system. For this reason we name this special gauge the “generalised Lagrange gauge”. In any diﬀerent ∆ =∆ ( f(C, t), t)=R ( f(C, t), t) = gauge Φ the equations for the Delaunay variables would con- L L ∆ ( f(C, t), t) . (29) tain Φ-dependent terms and would not be symplectic. (Those − H gauge-invariant equations, for both Lagrange and Delaunay This, in its turn, results in the well known Lagrange system of elements are presented in Efroimsky 2003 and Efroimsky & planetary equations, provided the parameters Ci are chosen as Goldreich 2003.) This analysis proves the following. the Kepler elements. In case they are chosen as the Delaunay Theorem: though the gauge-invariant equations for elements, then (28) leads to the standard Delaunay equations, Delaunay elements are, generally, not canonical, they be- i.e., to a symplectic system wherein the pairs (L, M ), 0 come canonical in the “generalised Lagrange gauge”. (G, ω), (H, Ω) again play the role of canonical variables,− but the− Hamiltonian,− in distinction to the unperturbed case, no That this Theorem is not merely a mathematical coincidence, longer vanishes, instead being equal to ∆ = ∆ . but has deep reasons beneath it, will be shown in Sect. 4 In our more general case, the perturbationH − dependsL also where the subject is to be approached from the Hamilton-Jacobi upon velocities (and, therefore, ∆ is no longer equal viewpoint. to ∆ ). Beside this, the gauge Φ isL set arbitrary. As demon- The above Theorem gives one example of the gauge for- strated− H in Efroimsky (2003), under these circumstances the malism being of use: an appropriate choice of gauge can con- gauge-invariant Delaunay-type system is no longer canonical. siderably simplify the planetary equations (in this particular However, it turns out that this system regains the canonical case, it makes them canonical). M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics 1193

According to (18), the momentum can be written as: As evident from (41), our choice of a particular gauge is equiv- ∂∆ ∂∆ alent to decomposition of the physical motion into a movement p = ˙r + L = g + Φ + L, (35) g ∂˙r ∂˙r with velocity along the instantaneous ellipse (or hyperbola, in the flyby case), and a movement associated with the ellipse’s which, in the generalised Lagrange gauge (30), simply (or hyperbola’s) deformation that goes at the rate Φ.Itisthen reduces to: tempting to state that a choice of gauge is equivalent to a choice p = g. (36) of an instantaneous comoving reference frame whereby we de- Vector g was introduced back in (2–3) to denote the functional scribe the motion. Such an interpretation is, however, incom- dependence of the unperturbed velocity upon the time and the plete. Beside the fact that we decouple the physical velocity in a certain proportion between g and Φ, it also matters which parameters Ci. In the unperturbed, two-body, setting this velocity is equal to the momentum canonically conjugate to the physical velocity (i.e., velocity relative to what frame) is decou- position r (this is obvious from (18), for zero ∆ ). This way, pled in this proportion. In other words, our choice of the gauge in the unperturbed case equality (36) is fulfilledL trivially. The does not yet exhaust all freedom: we can still choose in which fact that it remains valid also under perturbation means that, frame to write ansatz (40). We may write it in inertial axes or in in the said gauge, the canonical momentum in the disturbed some accelerated system. For example, in the case of a satellite setting is the same function of time and “constants” as in the orbiting an accelerated and precessing planet it is convenient to unperturbed, two-body, case. Thus, we have shown, follow- write the ansatz for the planet-related position vector. ing Goldreich (1965), Brumberg et al. (1971), and Ashby & The above kinematic formulae (40)–(42) do not yet contain Allison (1993)4 that the instantaneous Keplerian ellipses (hy- information about our choice of the reference system in which perbolae) defined in gauge (30) osculate the trajectory in phase we implement the variation-of-constants method. This infor- space. mation shows up at the next stage, when expression (42) is in- serted into the dynamical equation of motionr ¨ = (µr/r3) + Not surprisingly, the generalised Lagrange gauge (30) re- − duces to Φ = 0 in the simple case of velocity-independent ∆F to yield: disturbances. ∂g dC dΦ ∂ ∆ d ∂ ∆ i + =∆F = L L (43) ∂Ci dt dt ∂r − dt ∂˙r ! · 3. Gauge freedom and freedom of frame choice Complete information about the reference system in which we 3.1. Osculating ellipses described in different frames put the method to work (and, therefore, in which we define of reference the orbital elements Ci) is contained in the expression for the perturbation force ∆F. For example, if the operation is car- The essence of the variation-of-constants method in celestial ried out in an inertial coordinate system, ∆F contains physical mechanics is the following. A generic two-body-problem solu- forces solely. However, if we wish to implement the variation- tion expressed by of-constants approach in a frame moving with a linear acceler- r = f (C, t) , (37) ation a,then∆F also contains the inertial force a. In case this coordinate system rotates relative to inertial ones− at a rate µ, ∂ f = g (C, t) , (38) then ∆F also includes the inertial terms 2 µ r˙ µ˙ r µ − × − × − × ∂t !C (µ r). In considering the motion of a satellite orbiting an × ∂g µ f oblate precessing planet it is most reasonable, though not oblig- = , (39) ∂t ! − f 2 f atory, to apply the method (i.e., to define the time derivative) C in axes that precess with the planet. However, this reasonable is employed as an ansatz to solve the disturbed problem: choice of coordinate system still leaves us with the freedom of r = f(C(t), t), (40) gauge nomination. This will become clear in the example con- sidered below in Sects. 3.2–3.5. ∂ f ∂ f dC r˙ = + i = g + Φ, (41) ∂t ∂C dt i 3.2. Relevant example ∂g ∂g dC dΦ r¨ = + i + Gauge freedom of the perturbation equations of celestial me- ∂t ∂Ci dt dt chanics finds an immediate practical implementation in the de- µ f ∂g dCi dΦ scription of test particle motion around an precessing oblate = 2 + + (42) − f f ∂Ci dt dt · planet (Goldreich 1965). It is trivial to extend this to account 4 Treatment presented in these publications was equivalent to for acceleration of the planet’s centre of mass. choosing the generalised Lagrange gauge. Below, in Sect. 3.3, we Our starting point is the equation of motion in the inertial shall consider that development from the viewpoint of gauge-invariant frame: theory. The papers Goldreich (1965), Brumberg et al. (1971), and ∂U Ashby & Allison (1993) are unique examples of non-osculating el- r 00 = , (44) ∂r ements being employed. One more such example appeared in 1987 when Borderies & Longaretti (1987) put forward their theory of geo- where U is the total gravitational potential and time deriva- metric elements, to be used in the the planetary ring dynamics. tives in the inertial axes are denoted by primes. Suppose that 1194 M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics the planet’s spin axis precesses at angular rate µ(t) and that the variation-of-constants Eq. (43), ∆ is given by formula (46) acceleration of its centre of mass is given by a(t). Physically, and ∆ by (50). L H this acceleration originates both due to the circumsolar motion We now choose to describe the motion in the generalised and due to the gravitational pull from the other planets (Krivov Lagrange gauge (30), so the expression (Φ + µ r) on the 1993). It is a very natural (and very common in celestial and right-hand side of (50) vanishes (as follows from (47)),× and the planetary dynamics) technique to switch to a comoving or/and expression for ∆ in terms of f and g has the form: corotating reference frame, in order to better visualise the prop- H ∆ = R( f, t) + µ ( f g) a f . (51) erties of the physical system. For example, in oceanography H − · × − · and atmospheric science almost all work is carried out in a At the same time, the generic expression for the frame corotating with the Earth. In our case, our preference will variation-of-constants method given by (25) simplifies be to use not a corotating but rather a coprecessing frame, i.e., to (31). Insertion of (51) therein leads us to a coordinate system attached to the planet’s centre of mass and precessing (but not spinning) with the planet. In the new coor- dCi ∂ dinate frame the inertial forces modify the equation of motion [Cr Ci] = R( f, t) + µ ( f g) a f . (52) dt ∂Cr · × − · so that it assumes the form: Interestingly, this equation does not containµ ˙ even though it is ∂U r¨ = 2µ r˙ µ˙ r µ (µ r) a, (45) valid for nonuniform precession. ∂r − × − × − × × − As explained in Sect. 2.3, in the generalised Lagrange time derivatives in the accelerated frame being denoted by dots. gauge the vector g is equal to the canonical momentum p = To implement the variation-of-constants approach in terms ˙r + ∂ ∆ /∂˙r. In the case when the velocity dependence of ∆ L L of the orbital elements defined in the accelerated frame, we note is called into being by inertial forces, the momentum is, ac- that the disturbing force on the right-hand side of (45) is gen- cording to (47), erated according to (21) by: ∂ ∆ p = r˙ + L = r˙ + µ r, (53) 1 ∂˙r ∆ (r, r˙, t) = R + ˙r (µ r) + (µ r) (µ r) a r, (46) × L · × 2 × · × − · which is the particle’s velocity relative to the inertial frame co- wherewedenotebyR(r, t) the gravitational-potential pertur- moving with the accelerated, rotating frame. In this sense we bation (which is the perturbation of the overall gravitational may say that our elements are defined in the accelerated, rotat- potential U). Since ing frame, but osculate in the comoving inertial one. In the Appendix we provide explicit expression for each ∂ ∆ L = µ r, (47) of the partial derivatives of µ J that appears in the planetary ∂r˙ × Eq. (52). · the corresponding Hamiltonian perturbation reads:

1 ∂ ∆ 2 3.4. Elements defined in the accelerated, rotating ∆ = ∆ + L = H −  L 2 ∂r˙ !  frame, that osculate in this frame   R +p (µ r) a r = R + (r p) µ a r , (48) Here we not only define the elements in the accelerated, rotat- − · × − · − × · − · ing frame, but we also make them osculate in this system, i.e., with vector J = r p being the satellite’s orbital angular mo- × we make them satisfy Φ = 0. In this gauge, expression (50) mentum in the inertial frame. takes the following form: According to (35) and (47), the momentum can be written as: ∆ = R( f, t) + µ ( f g) + (µ f) (µ f) a f , (54) H − · × × · × − · p = g + Φ + µ f, (49) while Eq. (25), after some algebra5, looks like this: × whence the Hamiltonian perturbation becomes dC ∂ ∆ [C C ] i = H n i dt − ∂C ∆ = R + ( f g) µ + (Φ + µ f) (µ f) a f . (50) n H − × · × · × − · ∂ f ∂g +µ g f · ∂Cn × − × ∂Cn ! 3.3. Elements defined in an accelerated, rotating ∂ f ∂ frame, that osculate in the comoving inertial frame µ˙ f (µ f) (µ f) . (55) − · × ∂Cn ! − × ∂Cn × In this subsection we recall a calculation carried out by When substituting (54) into (55), it is convenient to rent the Goldreich (1965), Brumberg et al. (1971), and Ashby & expression for ∆ apart and to group the term (µ f) (µ f) Allison (1993) and demonstrate that it may be interpreted as H × · × an example of nontrivial gauge fixing. 5 Due to (47), the second term on the left-hand side in (25) is pro- Let us implement the variation-of-constants method in a portional to [∂(µ f)/∂C ] C˙ = µ Φ and, therefore, vanishes. The × j j × frame that is accelerating at rate a and rotating at angular second term on the right-hand side simplifies in accordance with the rate µ relative to some inertial system S. This means that, in the simple rule A (B C) = (C A) B. · × × · M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics 1195 with the last term on the right-hand side of (55): problem is perturbed, the role of perturbation being played by the relativistic correction to the Newton law of gravity. dCi ∂ [Cn Ci] = R( f, t) + µ ( f g) a f Interestingly, this correction depends not only upon the posi- dt ∂Cn · × − · tions, but also upon the velocities of the binary components ∂ f ∂g +µ g f (Brumberg 1992). For this reason, to simplify the orbit integra- · ∂Cn × − × ∂Cn ! tion of a binary, the generalised Lagrange gauge (not the cus- ∂ f ∂ tomary Lagrange constraint) should be imposed. In this gauge µ˙ f + (µ f) (µ f) . − · × ∂Cn ! × ∂Cn × the calculations will be very considerably simplified (and, for example, it is in the generalised Lagrange gauge that the equa- In so writing (56) we have deliberately cast it into a form that tions for the Delaunay elements will retain their canonicity). eases comparison with (52). Another simple example is a non-relativistic reduced two- In the Appendix we set up an apparatus from which the par- body problem with a variable mass. In this case, too, the tial derivatives of the inertial terms with respect to the orbital Lagrangian acquires a velocity-dependent correction. Hence, elements may be obtained. We also show that some of these in this case, the orbital elements will be convenient to intro- derivatives vanish. However, a complete evaluation of the iner- duce in the generalised Lagrange gauge, not in the customary tial input to the planetary equations in the ordinary Lagrange Lagrange gauge. gauge involves a long and tedious calculation which we do not carry out. 4. Planetary equations and gauges 3.5. Comparison of the two gauges in the hamilton-Jacobi approach One of the powers of gauge freedom lies in the availability of In this section we demonstrate that the derivation of planetary equations in the N-particle (N 3) case, performed through gauge choices that simplify the planetary equations, as we can ≥ see from contrasting (52) with (56). While the latter equation the medium of Hamilton-Jacobi method, implicitly contains a is written under the customary Lagrange constraint (i.e., for gauge-fixing condition not visible to the naked eye. We present elements osculating in the frame where they are defined), the a compact account of our study; a comprehensive descrip- former equation is written under a nontrivial constraint called tion containing technical details may be found in Efroimsky the “generalised Lagrange gauge”. The simplicity of (52), in & Goldreich (2003). contrast with (56), is evident. The Hamilton-Jacobi analysis rests on the availability of different canonical descriptions of the same physi- By identifying the parameters Ci with the Delaunay variables, one arrives from (52) and (56) to the appropriate cal process. Any two such descriptions, (q, p, (q, p)) and (Q, P, (Q, P)), correspond to different parametrisationsH of Delaunay-type equations (see Appendix I to Efroimsky & H ∗ Goldreich 2003). The Lagrange equations corresponding to the same phase flow, and both obey Hamilton’s equations. Due (52) and to (56) may be easily derived from each of these two to the latter circumstance the infinitesimally small variations equations by choosing Ci as the Kepler elements and using the dθ = p dq dt (56) appropriate Lagrange brackets. −H Although the planetary equations are much simpler in the and generalised Lagrange gauge than in the ordinary Lagrange dθ˜ = P dQ ∗ dt (57) gauge, some of these differences are less important than oth- −H µ2 ers. In many physical situations, though not always, the and are perfect differentials, and so is their difference µ˙ terms in (56) are of a higher order of smallness compared to those linear in µ, and therefore may be neglected, at least for dW dθ˜ dθ = P dQ p dq ( ∗ ) dt. (58) sufficiently short times6. − ≡ − − − H −H Here, vectors q, p, Q, and P each contain N components. Given a phase flow parametrised by a set (q, p, (q, p, t)), 3.6. Further applications it is always useful to simplify the description by aH canonical transformation to a new set (Q, P, (Q, P, t)), with the new In the above example of a satellite orbiting a wobbling planet, H ∗ an evident simplification of the planetary equations (both in the Hamiltonian ∗ being constant in time. Most advantageous are transformationsH that nullify the new Hamiltonian , because Lagrange and Delaunay forms) was achieved through imposi- H ∗ tion of the generalised Lagrange gauge. This optimal gauge then the new canonical equations render the variables (Q, P) differed from the standard Lagrange constraint, because in constant. A powerful method of generating such transforma- the said example the Lagrangian perturbation depended upon tions stems from (58) being a perfect differential. It is sufficient velocities. to consider W to be a function of the time and only two other A similar situation emerges in the relativistic two-body canonical variables, for example q and Q. Then (58) may be problem. In the relativistic dynamics, even the two-body written as: ∂W ∂W ∂W 6 As an example of an exception to this rule, we mention Venus dt dQ dq = whose wobble is considerable. This means that, for example, theµ ˙ − ∂t − ∂Q − ∂q term cannot be neglected in computations of circumvenusian orbits. P dQ p dq + ( ∗) dt (59) − H−H 1196 M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics from which it follows that and ∂W ∂W ∂ ∆ ∂ ∆ ∂q ∂ ∆ ∂p P = , p = , P˙ = H = H H (65) − ∂Q ∂q − ∂Q − ∂q ∂Q − ∂p ∂P ∂W (q, p, t) + = ∗(Q, P, t). (60) H ∂t H into the expression for the velocity: Inserting the second equation into the third and assuming that ∂q ∂q ∂q q˙ = + Q˙ + P˙. (66) ∗(Q, P, t) is simply a constant, we get the famous Jacobi ∂t ∂Q ∂P Hequation: This leads to: ∂W ∂W q, , t + = ∗ (61) ∂q ∂q ∂q ∂q ∂q ∂ ∆ H ∂q ! ∂t H q˙ = + H ∂t ∂Q ∂P − ∂P ∂Q! ∂q whose solution furnishes the transformation-generating func- ∂q ∂p ∂q ∂p ∂ ∆ tion W. The elegant power of the method becomes most vis- + H ∂Q ∂P ∂P ∂Q ∂p ible if the constant ∗ is set to zero. Under this assumption − ! the reduced two-bodyH problem is easily resolved. Starting with ∂ ∆ ∂q = g + H ,g, (67) the three spherical coordinates and their canonical momenta as ∂p !q, t ≡ ∂t (q, p), one arrives to canonically conjugate constants (Q, P) that coincide with the Delaunay elements (27): where we have taken into account that the Jacobian of the (Q1, P1) = (L, M0); (Q2, P2) = (G, ω); canonical transformation is unity: (Q , P ) = (H,− Ω). − 3 3 − ∂q ∂p ∂q ∂p Extension of this approach to the N-particle problem begins = 1. (68) with consideration of a disturbed two-body setting. The num- ∂Q ∂P − ∂P ∂Q ber of degrees of freedom is still the same (three coordinates q To establish the link between the regular variation-of-constsnts and three conjugate momenta p), but the initial Hamiltonian is method and the canonical treatment, compare (67) with (41). perturbed: We see that the symplectic description necessarily imposes a ∂( +∆ ) ∂( +∆ ) particular gauge Φ=∂ ∆ /∂p. q˙ = H H , p˙ = H H (62) H ∂p − ∂q · It can be easily demonstrated that this special gauge coincides with the generalised Lagrange gauge (30) discussed in While in (59)–(61) one begins with the initial Hamiltonian H Sect. 2.2. To that end one has to compare the Hamilton equa- and ends up with ∗ = 0, the method may be extended tion for the perturbed Hamiltonian (19), to the perturbed settingH by accepting that now we start with a disturbed initial Hamiltonian +∆ and arrive, through ∂ ( +∆ ) ∂ ∆ H H q˙ = H H = p + H , (69) the same canonical transformation, to an equally disturbed ∂p ∂p eventual Hamiltonian ∗ +∆ =∆ . Plugging these new Hamiltonians into (59)H leadsH to cancellationH of the distur- with the definition of momentum from the Lagrangian (17), bance ∆ on the right-hand side, whereafter one arrives to the ∂ ( (q, q˙, t) +∆ (q, q˙, t)) ∂ ∆ same equationH for W(q, Q, t) as in the unperturbed case. Now, p L L = q˙ + L (70) ≡ ∂q˙ ∂q˙ · however, the new canonical variables are no longer conserved but obey the dynamical equations: Equating the above two expressions immediately yields: ∂ ∆ ∂ ∆ Q˙ = H , P˙ = H (63) ∂ ∆ ∂ ∆ ∂P − ∂Q · Φ H = L (71) ≡ ∂p !q, t − ∂q˙ !q, t Because the same generating function is used in the perturbed and unperturbed cases, the new, perturbed, solution (q, p)isex- which coincides with (30). Thus, the transformation generated pressed through the perturbed “constants” Q(t)andP(t)inthe by W(q, Q, t) is canonical only if the physical velocityq ˙ is split same manner as the old, undisturbed, q and p were expressed in a fashion prescribed by (67), i.e., if (71) is fulfilled. This is through the old constants Q and P. This form-invariance pro- exactly what our Theorem from Sect. 2.2 says. vides the key to the N-particle problem: one should choose To summarise, the generalised Lagrange constraint, Φ= the transformation-generating function W to be additive over ∂ ∆ /∂q˙, is tacitly instilled into the Hamilton-Jacobi method. − L the particles and repeat this procedure for each of the bodies, Simply by employing this method (at least, in its straightfor- separately. ward form), we automatically fix the gauge7. By sticking to the Armed with this preparation, we can proceed to uncover Hamiltonian description we sacrifice gauge freedom. the implicit gauge choice made in using the Hamilton-Jacobi 7 method to derive evolution equations for the orbital elements. An explanation of this phenomenon from a different viewpoint is offered in Sect. 6 of Efroimsky (2003), where the Delaunay equations To do this we substitute the equalities: are derived also through a direct change of variables. It turns out that ∂ ∆ ∂ ∆ ∂q ∂ ∆ ∂p the outcome retains the symplectic form only if an extra constraint is Q˙ = H = H + H (64) ∂P ∂q ∂P ∂p ∂P imposed by hand. M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics 1197

Above, in Sect. 2.3, we established that in the generalised the disturbance depends not only upon positions but also upon Lagrange gauge the momentum coincides with g. We now can velocities, another constraint (which we call the “generalised get to the same conclusion from (67), (70) and (71): Lagrange constraint”) turns out to be stiffly embedded in the Hamilton-Jacobi development of the problem. ∂ ( (q, q˙, t) +∆ (q, q˙, t)) p L L = q˙ Φ=g. (72) Unless a specific constraint (gauge) is imposed by hand, ∂ ≡ q˙ − the planetary equations assume their general, gauge-invariant, Thus, implementation of the Hamilton-Jacobi theory in ce- form. In the case of a velocity-independent disturbance, any lestial mechanics demands the orbital elements to osculate in gauge different from that of Lagrange drives the Delaunay sys- phase space. Naturally, this demand reduces to that of regular tem away from its symplectic form. If we permit the disturbing osculation in the simple case of velocity-independent ∆ . force to depend also upon velocities, the Delaunay equations L retain their canonicity only in the generalised Lagrange gauge. Interestingly, in this special gauge the instantaneous ellipses 5. Conclusions (hyperbolae) osculate in phase space. In the article thus far we have studied the topic recently Briefly speaking, N-body celestial mechanics, expressed in raised in the literature: the planetary equations’ internal sym- terms of orbital elements, is a gauge theory, but it is not strictly metry that stems from the freedom of supplementary condi- canonical. It becomes canonical in the generalised Lagrange tion’s choice. The necessity of making such a choice con- gauge. strains the trajectory to a 9-dimensional submanifold of the 12-dimensional space spanned by the orbital elements and Acknowledgements. The authors are grateful to William Newman and their time derivatives. Similarly to the field theory, the choice Victor Slabinski for their help in improving the manuscript, to Sergei of the constraint (=the choice of gauge) is vastly ambiguous Klioner for drawing the authors’ attention to the paper by Brumberg and reveals a hidden symmetry instilled in the description of et al., and to Michael Nauenberg for the information on Newton’s pri- the N-body problem in the language of orbital elements. ority in developing the variation-of-constants method. Research by We addressed the issue of internal freedom in a sufficiently ME was supported by NASA grant W-19948. Research by PG was general setting where a perturbation to the two-body problem partially supported by NSF grant AST 00-98301. depends not only upon positions but also upon velocities. Such situations emerge when relativistic corrections to Newton’s Appendix law are taken into account or when the variation-of-constants method is employed in rotating systems of reference. In this Appendix we set up an apparatus from which one may We derived the most general form of the gauge-invariant evaluate the partial derivatives with respect to the orbital ele- perturbation equation of celestial mechanics, written in terms ments of inertial terms that appear in the planetary equations of a disturbing force. Then we transformed it into the most gen- derived in Sects. 3.3 and 3.4. We then show that some of these eral gauge-invariant perturbation equation expressed through derivatives vanish. Following that, we derive explicit expressions for each derivative of µ ( f g), which provides a com- the Lagrangian disturbance. · × Just as a choice of an appropriate gauge simplifies solution plete analytic evaluation of the rotational input in the gener- of the equations of motion in electrodynamics, an alternative alised Lagrange gauge. The topic is further developed (and the (to that of Lagrange) choice of gauge in the celestial mechan- appropriate generalised Lagrange system of equations is pre- ics can simplify orbit calculations. We provided one such ex- sented) in Efroimsky & Goldreich (2003). ample, a satellite orbiting a precessing planet. In this exam- To find the explicit form of the dependence f = f(Ci, t), it ple, the choice of the generalised Lagrange gauge considerably is conventional to introduce an auxiliary set of Cartesian coor- simplifies matters. To achieve this simplification, we not only dinates q, with an origin at the gravitating centre, and with the exercised our right to choose a convenient gauge, but we also first two axes located in the plane of orbit. The q coordinates chose a preferred coordinate system in which to implement the are easy to express through the semimajor axis a, the eccentric- variation-of-constants method. This interplay of the two types ity e and the eccentric anomaly E: 2 of freedom enabled us to eliminate some of the mathematical q1 a (cos E e) , q2 a √1 e sin E, q3 = 0, (73) complications associated with the inertial forces. Not surpris- ≡ − ≡ − where E itself is a function of the semimajor axis a, the eccen- ingly, it has turned out to be convenient to define the orbital tricity e, the mean anomaly at epoch, M , and the time, t.The elements in the precessing frame of the planet; however, for 0 time dependence is realised through the Kepler equation the sake of mathematical simplification, it also turned out to be beneficial to make these elements osculate in a different, iner- E e sin E = M, (74) − tial frame of reference. where We have explained where the Lagrange constraint tacitly t 1/2 3/2 enters the Hamilton-Jacobi derivation of the Delaunay equa- M M + µ a− dt. (75) ≡ 0 tions. This constraint turns out to be an inseparable (though not Zt0 easily visible) part of the method: in the case of momentum- The inertial-frame-related position of the body reads: independent disturbances, the N-body generalisation of the r = f (Ω, i,ω,a, e, M0; t) = two-body Hamilton-Jacobi technique is legitimate only if we use orbital elements that are osculating. In the situation where Rˆ (Ω, i,ω) q (a, e, E(a, e, M0, t)) , (76) 1198 M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics

Rˆ (Ω, i,ω) being the matrix of rotation from the orbital-plane- To continue, we note that in the two-body setting the ratio J/µ, related coordinate system q to the ﬁducial frame (x,y,z)in is known to be equal to a(1 e2) wˆ where wˆ is a unit vec- − which the vector r is deﬁned. This rotation is parametrised by tor perpendicular to the unperturbedp orbit’s plane. Moreover, in the three Euler angles: inclination, i; the longitude of the node, planet-associated noninertial axes (x,y,z) with corresponding Ω; and the argument of the pericentre, ω. unit vectors (xˆ, yˆ, zˆ), the normal to the orbit is expressed by: In the unperturbed two-body setting the velocity is wˆ = xˆ sin i sin Ω yˆ sin i cos Ω+zˆ cos i. (85) expressed by − ∂ Hence, g = f (Ω, i,ω,a, e, M ; t) = ∂t 0 ∂ ( f g) ∂ a(1 e2) 1 1 e2 µ = µ wˆ − = µ (86) ∂E ∂q × p r − Rˆ (Ω, i,ω) (77) · ∂a · ∂a 2 a ⊥ ∂t ! ∂E ! · a, e, M0 a, e and One can similarly calculate partial derivatives of f with respect ∂ ( f g) ∂ a(1 e2) ae2 to M0: µ × = µ wˆ − = µ (87) · ∂e · p ∂e − r1 e2 ⊥ ∂ − f (Ω, i,ω,a, e, M0; t) = where µ = µx sin i sin Ω µy sin i cos Ω+µz cos i is ∂M0 the orthogonal-to-orbit⊥ component− of the precession rate. The ∂E ∂q remaining two derivatives look: Rˆ (Ω, i,ω) , (78) ∂M0 ! ∂E ! ∂ ( f g) ∂wˆ a, e, t a, e µ × = a 1 e2 µ = · ∂Ω q − · ∂Ω whence it becomes evident that ∂ f/∂M0 is parallel to g and, 2 hence, a 1 e µx sin i cos Ω+µy sin i sin Ω (88) q − ∂ f n o g = 0. (79) and × ∂M ! 0 Ω, i,ω,a, e, t ∂ ( f g) ∂wˆ µ × = a 1 e2 µ = By a similar trick it is possible to demonstrate that ∂( f g)/∂M0 · ∂i q − · ∂i × is proportional to ∂( f g)/∂E and, therefore, to ∂( f g)/∂t. 2 × × a 1 e µx cos i sin Ω µy cos i cos Ω µz sin i . (89) Hence, this derivative vanishes (because in the two-particle q − − − n o case the cross product f g is an integral of motion). This As for the derivatives of a f, they may be calculated di- × · vanishing of ∂( f g)/∂M0, along with (79), implies: rectly from the expression for f(Ω,ω,i, a, e, M ; t)presented × 0 ∂g above. However, the resulting expressions are cumbersome so f = 0. (80) we do not present them here. × ∂M0 !Ω, i,ω,a, e, t

In the situation when the parameters Ci are implemented by the References Delaunay elements, a similar sequence of calculations leads to Ashby, N., & Allison, T. 1993, Cel. Mech. Dyn. Astron., 57, 537 ∂ f g = 0 (81) Borderies, N., & Longaretti, P. Y. 1987, Icarus, 72, 593 Brouwer, D., & Clemence, G. M. 1961, Methods of Celestial × ∂M0 !Ω,ω,L, G, H, t Mechanics (NY & London: Academic Press), Chapter XI and, appropriately, to: Brumberg, V. A., Evdokimova, L. S., & Kochina, N. G. 1971, Cel. Mech., 3, 197 ∂g f = 0. (82) Brumberg, V. A. 1992, Essential Relativistic Celestial Mechanics × ∂M0 !Ω,ω,L, G, H, t (Bristol: Adam Hilger) Efroimsky, M. 2002, Equations for the orbital elements. Hidden sym- We can proceed much farther in the generalised Lagrange metry. Preprint No. 1844 of the Institute of Mathematics and its gauge, at least in so far as derivatives of the rotational input Applications, University of Minnesota. J/µ = f g are concerned. (We remind that here and ev- × http://www.ima.umn.edu/preprints/feb02/1844.pdf erywhere µ stands for the reduced mass, while µ denotes the Efroimsky, M. 2003, [astro-ph/0212245] precession rate.) Efroimsky, M., & Goldreich, P. 2003, J. Math. Phys., 44, 5958 As we proved above, this cross product is independent Euler, L. 1748, Recherches sur la question des inégalites du mou- of M0 and, hence, vement de Saturne et de Jupiter, sujet proposé pour le prix de l’année, Berlin. For modern edition see: Euler L. Opera mech. ∂( f g) µ × = 0. (83) et astron. (Birkhauser-Verlag, Switzerland, 1999) · ∂M0 Euler, L. 1753, Theoria motus Lunae exhibens omnes ejus in- aequalitates etc. Impensis Academiae Imperialis Scientarum Since J is orthogonal to the orbit plane, it is invariant under Petropolitanae (St. Petersburg, Russia). For modern edition see: rotations of the orbit within its plane, whence Euler L. Opera Mech. Astron. (Switzerland: Birkhauser-Verlag, ∂( f g) 1999) µ × = 0. (84) · ∂ω Goldreich, P. 1965, AJ, 70, 5 M. Efroimsky and P. Goldreich: Gauge freedom in the N-body problem of celestial mechanics 1199

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