On action-angle coordinates and the Poincaré coordinates

This article is a review of two related classical topics of Hamiltonian systems and celestial mechanics. The first section deals with the existence and construction of action-angle coordinates, which we describe emphasizing the role of the natural adiabatic invariants “∮γp dq”. The second section is the construction and properties of the Poincaré coordinates in the Kepler problem, adapting the principles of the former section, in an attempt to use known first integrals more directly than Poincaré did.


ACTION-ANGLE COORDINATES
1.1. Main Statement and Comments Let (M, ω) be a symplectic manifold of dimension 2n and F : M → B be a fibration whose fibers M b , b ∈ B, are Lagrangian, 1) compact and connected submanifolds of M . Roughly speaking, the theorem of the action-angle coordinates says that locally in the neighborhood of a base point, the universal model for F is the canonical projection where B o is a domain of R n , T n = R n /Z n is the n-torus and M o is endowed with the standard symplectic form ω o = j dI j ∧ dθ j . All maps considered here are smooth.
The coordinates I and θ are called action and angle coordinates, respectively. * E-mail: jacques.fejoz@dauphine.fr 1) Sometimes, such a fibration is called a (reducible) real polarization of M [49]. {F j , F j } = 0 (∀i, j) rk F (x) = n (∀x ∈ M ), such that the levels of F are compact and connected. Any of the components F i 's is then often called an integrable Hamiltonian system. 2) Due to a classical theorem of Ehresman [17,32], F is a fibration. The tangent space of its fibers is generated by the Hamiltonian vector fields of F i , i = 1, . . . , n. Since the F i commute, fibers are Lagrangian and the hypotheses of the theorem are satisfied.
The history of action-angle coordinates has known several stages, which can be sketched as follows.
• Early versions of Liouville [36] or Jacobi [25] focus on the possibility of local integration of a Hamiltonian system (or, more generally, of an ordinary differential equation) by quadratures. They single out the hypothesis of n commuting independent first integrals, but they do not provide a topological description of the quasiperiodic tori foliation of the phase space. • In the course of the 19th century, astronomers fill this gap, realizing the importance and the non-genericity of the foliation of the phase space in "multiply periodic" solutions, 3) in particular in integrable approximations of the planetary problem [43]. • Several versions of the theorem of the action-angle coordinates, in the early 20th century, are related to adiabatic invariants and the Bohr-Sommerfeld quantization (see Remark 3). They are due to Gibbs-Hertz [47, Bd 1, p. 535] (adiabatic invariance of the volume), Burgers [11] (adiabatic invariance of the p i dq i for decoupled systems) and others. Poincaré [30] suggests to replace the Bohr-Sommerfeld rule of quantization by a rule which is invariant, substituting an integral invariant for the p i dq i 's; he also raises the question of uniqueness of the system of adiabatic invariants (see also Einstein's quantization [18]). Epstein discusses how degeneracy induces ambiguity in the choice of adiabatic invariants and thus in quantization [19]. Levi-Civita [34] and Mineur [39] seemingly prove the modern statement on action-angle coordinates. • Landau-Lipschitz treats of adiabatic invariants in a way close to Mineur [29]. Arnold uses a more modern geometric language [3,6]. • Some unnecessary hypotheses, such as the functional independence of the action variables and the exactness of the symplectic form, are removed in [16,26,37], with variants [7,24,35]. Usually these proofs build some angle coordinates by straightening the period lattice of the flow of the first integrals, and then define the action variables as the variables which are symplectically conjugate to the angles. In order to prove integrability by quadratures, one eventually needs to show how these coordinates relate to the natural adiabatic invariants (see, e.g., [16]). • Generalizations in several directions: refined integrability properties [4,27], geometric quantization [49], globality and monodromy of the action [15,16], singular fibrations [33], or, non-commuting integrals (interestingly, the question then relates to weak KAM theory) [12].
Here we will review the proof of Theorem 1 along the lines of Duistermaat and Guillemin-Sternberg [16,24], with only minor differences aiming at practical computations, in relation to Section 2 of this article.
2) In addition to the integrability of differential equations by quadrature, integrability may also refer to Pfaff systems satisfying the hypotheses of the Frobenius theorem or, more generally, to geometric structures satisfying some flatness condition [45].
Proof. By the definition of an action, if x ∈ X and v ∈ V , we have ρ Hence, by differentiating with respect to h, So, if the action is infinitesimally transitive, by the inverse function theorem, the orbit ρ x (V ) of x is open. Since X is compact, X is covered by a finite union of orbits. Since X is connected, there can only be one orbit. Thus, the action is transitive.
Conversely, if the action is not infinitesimally transitive, due to the remark above, ρ x (v) is invertible for no v ∈ V , so the whole orbit ρ x (V ) consists of critical values of ρ x and, by Sard's theorem, has measure zero. So, the action is not transitive.
Assume again that the action is transitive. Let x and y be any two points of X. Since the action is transitive, the stabilizers of x and y are conjugate. Since V is an Abelian group, the stabilizers agree. Let L ⊂ V be the common stabilizer of points of X. As already mentioned, due to the infinitesimal transitivity of the action, ρ x is a local diffeomorphism in the neighborhood of v → v · x for every x and v. Hence L has only isolated points and is thus discrete. So, L is a lattice [8,46]. Since X is compact, L is a maximal lattice. So, V/L is compact and hence an n-torus. Since X is diffeomorphic to V/L, X itself is an n-torus.
The above stabilizer L is called the period lattice of the action.

Example 2.
Let ξ 1 , . . . , ξ n be n vector fields on X, commuting and everywhere independent: Then the "joint flow of the ξ i 's" where ϕ ξ i v i is the time-v i of the flow of ξ i , is an infinitesimally transitive action.

Lagrangian Fibrations
We now aim at proving Theorem 1. Let F : M → B be as in the statement.

Lemma 2. ([24])
There is a transitive action of T * B on M , and fibers of F are n-tori.
)ω) allows us to associate to a cotangent vector β b ∈ T * b B a vectorβ x tangent to M at every point x in the fiber of b, characterized by the equation Since the 1-form F * (x) · β b vanishes on vertical vectors, it induces an element of the normal bundle N * x (M b ) = T * x M/Ker T x F of the fiber at x. Since the fiber is Lagrangian (in other words, equal FÉJOZ to its own ω-orthogonal), ω identifies N * x (M b ) with the vertical tangent space T x (M b ). So,β x is vertical and the map Since the fibers are compact, the vector fieldβ can be exponentiated into a fibered diffeomorphism, which we will simply denote by β.
For this construction to define an action, we have to require that, if is the restriction of a Hamiltonian vector field and, since the fibers are Lagrangian, We have thus defined an action of T b B on the fiber M b for all b ∈ B. The action is infinitesimally transitive. Hence, according to Lemma 1, the action is transitive and the fibers of F are n-tori.
A first consequence of Lemma 2 (and its proof) is the existence of local Lagrangian submanifolds of M which are transverse to the fibers.
where on the left hand-side β is thought of as a diffeomorphism of M . Consequently, there exists a section of F whose image is Lagrangian.
From Lemma 2, and using the fact that F • σ 0 = id, The next lemma is a key step towards understanding the structure of M . We endow T * B with its canonical symplectic form, which we denote by ω o .

Remark 1.
In the construction of Lemma 2, we may replace F by the canonical projection π : seen as a constant vertical vector field. Thus,β o exponentiates (despite the fiber being not compact) into a diffeomorphism β o of T * b B, which is just the vertical translation: The tangent space of T * B at (b, 0) splits into its horizontal and vertical subspaces, The derivative χ maps these subspaces, respectively, to the tangent spaces at σ(b) of the image of σ and of the fiber M b , both of which are Lagrangian too. So it is enough to check that For such vectors β and u, on the one hand, we have 4) ω On the other hand, we have Let us now consider any point γ ∈ T * b B not necessarily on the zero-section. First notice that Now, if u and v are two vectors tangent to T * B at γ, they are of the form Here we choose between the two possible signs of the canonical symplectic form.

Remark 2. The Poincaré lemma relative to a fiber
Let λ be a primitive of ω and γ 1 (b), . . . , γ n (b) be smooth generators of the fundamental group of M b with basepoint σ(b), varying smoothly with b ∈ B. Define I by Proof.
We want to compute the coordinates I i in terms of such well chosen coordinates (J, ϕ), in order to see that (1) Since the map χ is a local diffeomorphism everywhere, the set L = χ −1 (σ(B)) of elements of T * B acting trivially on M is a submanifold of dimension n (with countably many connected components). Besides, χ being symplectic and σ(B) being Lagrangian, L itself is Lagrangian. Due to Lemma 1, the trace of L on a fiber ) is a basis of the fundamental group of M b , the so-defined (β 1 (b), . . . , β n (b)) is a basis of L b over Z. When now b is varied, the covectors β i (b) extend to uniquely defined 1-forms β 1 , . . . , β n , whose disjoint images are n connected components of L. Since L is Lagrangian, the 1-forms β i are closed: there are functions J 1 , . . . , J n over B such that β i = dJ i , i = 1, . . . , n. That the β i 's form a basis of the lattice subbundle entails that the J i 's are independent and thus form a coordinate system over B. Define the dual coordinates ϕ 1 , . . . , ϕ n (as above for I and θ) by letting ϕ i be the tangent vector field ϕ i = ∂/∂J i over B.
Define the primitive of the symplectic form ω. Since χ is symplectic, χ * λ − λ o is closed in T * B and hence exact: for B is simply connected by assumption. Note that S : M → R is L-periodic, and, for every i = 1, . . . , n, the function is constantly equal to some c i ∈ R.
5) This is assumed in [6] and shown using a first version of the action-angle variables in [16]. If X is Lagrangian, this also follows from the fact that the only symplectic invariant of a neighborhood of X is the diffeomorphism class of X itself [38,48].

Remark 3 (Adiabatic invariants).
The action coordinates I i defined by (1.1) play an important role in classical dynamics because of their adiabatic invariance, i.e., their invariance under infinitesimally slow perturbations [40]. They also play a crucial role in the Sommerfeld quantization, which is explained by Ehrenfest's "adiabatic hypothesis": quantities which are to be quantized must be adiabatically invariant, because, on perturbing the system, these quantities would have to remain integral multiples of the Planck constant [41].
There are numerous examples illustrating Theorem 1, for example, in the book [7]. Examples closely related to the topic of the next section are the Delaunay coordinates (see [13,44], or [22, Appendix] 6) for a closer viewpoint), or action-angle coordinates of the non-Newtonian Kepler problem [23].

THE POINCARÉ COORDINATES
The Poincaré variables are symplectic coordinates in the phase space of the Kepler problem, in the neighborhood of horizontal circular motions. Determining such coordinates departs from the abstract setting of the first section in two respects: • The Kepler problem is super-integrable in the sense that it has more independent (noncommuting) first integrals than degrees of freedom, so the dynamics is degenerate and does not determine in itself a full set of coordinates.
• The action of rotations is degenerate at circular Keplerian ellipses, in the sense that dH ∧ dC = 0 with the notations below.
The Poincaré variables are closer to being action-angle coordinates in this situation. Despite being of prime importance in perturbation theory [2,14,21,43] (see also more complicated, Depritlike coordinates of the N -body problem in [14,42]), they have few complete descriptions in the literature (see [9,10,13,44] for proofs at various levels of precision), all of which are based on Poincaré's computation, through the Delaunay coordinates. 7),8) This computation requires a good deal of intuition -which Poincaré did not lack. 6) This appendix is really on the Delaunay coordinates and does not prove the analyticity of the Poincaré coordinates. 7) In the first edition of [1], the Delaunay and Poincaré coordinates are wrongly found nonsymplectic! 8) In the unpublished note [20] on the plane Poincaré coordinates, it is wrongly claimed that the analyticity of the Poincaré coordinates was not proved by Poincaré. Poincaré did prove that the inverse map is analytic (and, in particular, that the Hamiltonian is an analytic function of the Poincaré coordinates), which, due to the inverse function theorem, is equivalent to proving that the coordinate map itself is analytic.
Here we aim at providing a slightly more direct construction (although symplecticity is always more simple to check by relating the Poincaré coordinates to the Delaunay coordinates), trying to find out the definition of the coordinates at the same time as proving their properties.
In the sequel, we will set T n = R n /2πZ n (as opposed to R n /Z n as in the first section) for the sake of convenience.

Reminder on the Plane Kepler Problem
We start with the plane problem because it is an interesting intermediate step, with simpler computations. In this section, we recall some elementary (nonsymplectic) facts.
Consider the equationq references are so numerous that we give up advising any one of them. The phase space is the set • The angular momentum C = Im (qq) = xẏ − yẋ is preserved, as in any central force problem (Kepler's second law).
• The eccentricity vector too is preserved, this time in contrast to other central forces than Newton's. The equation of an orbit can be obtained by eliminating the velocity from C and E: which is the equation of the conic with a focus at the origin (Kepler's first law), of eccentricity = |E| and with directrix D : Re (Ēq) + C 2 = 0.
• The Hamiltonian H = |q| 2 2 − 1 r (denoted H by Lagrange in reference to Huygens [28]) is a first integral too. The dynamics is that of the Hamiltonian vector field of H with respect to the standard symplectic form, ω = Re (dp ∧ dq).
Because of the symmetry about the origin, it is useful to switch to polar coordinates. The cotangent map of the "polar coordinate" map (r, θ) → q = r e iθ is the diffeomorphism 9) Pol : The (pull-backs by Pol of the) first integrals are The polar representation of E is

4)
9) Another way to compute the conjugate variable of θ is to think of Θ as the momentum i(X) p dq of the rotational vector field X.
where the argument g is easily computed when R = 0: it is the argument of the pericenter of the conic (when the conic is a circle and the pericenter is thus not defined, E = 0 is well defined, of course). From now on, we restrict ourselves to negative energies. If a is the semi-major axis, r varies between a(1 − ) and a(1 + ). These two extremal values are the roots of the quadratic equation H = C 2 2r 2 − 1 r , so their sum and product are Hence, using (2.4), , and E = e ig ; (2.5) we will use that these functions are analytic. Now, according to Kepler's second law, the area swept by q grows with constant speed C/2. Since the area of the ellipse is πab = πa 2 √ 1 − 2 , the following relation holds between the period T and the elliptic elements: and, thanks to the remarkable disappearance of the eccentricity (use (2.5)), so does Kepler's third law: We will also use three classical angles, defined when the ellipse is not circular 10) : • the mean anomaly is the angle which is proportional to the area, counting from the pericenter (Kepler's second law says that increases linearly with time); • the eccentric anomaly u is defined in Fig. 1; • the true anomaly v = θ − g.    (2.6) which shows that q is a transcendant function of (see Newton's proof in [5]), that u does vary around the whole circle, etc. We will need one more fact. The anomalies u, v or are not analytic, nor even defined, for circular motions. In contrast, the eccentric longitude w = u + g is analytic. Indeed, elementary geometry again shows that Solving these two equations for cos u and sin u allows us to express trigonometric functions of w in terms of v = θ − g: for example, − 1 cos 2 g cos θ + cos g sin g sin θ + cos g; the right-hand side is real analytic because Θ , and E = e ig is analytic. The function sin w can similarly be seen to be real analytic. So, w is real analytic.

Plane Poincaré Variables
Consider the plane Kepler problem with negative energy and, say, positive angular momentum. The phase space is diffeomorphic to R 3 × T 1 and has coordinates (R, r, Θ, θ) and symplectic form a. Keplerian action variable Λ First consider the problem reduced by rotations, in the symplectic space K = {(R, r)} R 2 , with the angular momentum Θ as a parameter. The reduced Hamiltonian H = H Θ (R, r) has an elliptic singularity at (R, r) = (0, Θ 2 ), corresponding to circular motions. Locally outside the singularity, the energy H and the time t (counted from some section of the flow of H, which we do not want to specify at this stage) form some symplectic coordinates. We would like to switch to some coordinates (Λ, λ) where the action is a well-chosen function of H, so that the dual coordinate λ is an angle, defined modulo 2π: where Λ = Λ(H) and λ = 2π T t. Hence, or, if we choose the primitive vanishing at H = −∞, Up to now, λ is only determined up to the addition of a first integral.

Remark 4.
The above computation of Λ can be recovered from the expression of the action coordinate given in the previous section (up to a factor 2π due to the fact that we then took circles of length 1). Indeed, denoting by X H the Hamiltonian vector field of H, by (φ H ) the flow of the (well-defined) Hamiltonian vector field X t (with "time" H), and by H 0 = 1 2Θ 2 the value of the Hamiltonian for circular motions, The Keplerian action variable Λ lifts to an analytic variable in the non-reduced phase space (analyticity follows from (2.5)).
b. Eccentric variable F One could check that the symplectic transformation (R, r) → (Λ, λ) in the reduced space lifts to an essentially unique symplectic transformation (R, r, Θ, θ) → (Λ, λ, F 1 , F 2 ) in the full phase space, possibly using a generating function. But this computation is more involved than necessary (a similar computation is made in [23] in the non-Newtonian twobody problem at the second order in the eccentricity). We will make a much shorter computation, completed by geometric arguments.
Consider the space E of Keplerian ellipses of fixed energy H < 0. For an ellipse in the plane, to be Keplerian means that it is oriented and has a focus at the origin. In addition to that, Keplerian ellipses in E have a fixed semi-major axis. Including degenerate ellipses corresponding to collision orbits of eccentricity 1, E is diffeomorphic to S 2 . Outside the poles {N, S} corresponding to circular ellipses, it bears the coordinates (Θ, g), where g is the argument of the pericenter. Since the flow of Θ = C consists of rigid rotations in the plane, the Poisson bracket of Θ and g is {Θ, g} = 1, hence the symplectic form induced from ω on E \ {N, S} is ω E = dΘ ∧ dg.
We will focus on the open hemisphere E + of E consisting of direct, non-degenerate ellipses; this domain is defined in E by the inequality Θ > 0. 11) Over E + , the eccentricity vector E is a (complexvalued) real analytic coordinate, unfortunately not symplectic, since, using the expression (2.4), we get Let us first look at the case Λ = 1. We will look for a real analytic symplectic coordinate F obtained by multiplyingĒ (not E because of the negative sign in (2.7)) by a positive real analytic function f of Θ ∈ ]0, 1]: The requirement that F be symplectic is equivalent to imposing that the expression in brackets be zero. In the unknown ϕ = f 2 , the equation becomes Solutions are of the form 11) One can similarly define dual Poincaré coordinates over the hemisphere of negatively oriented ellipses.
That F is symplectic has already been proved. That it is analytic follows from the formula c. Mean longitude λ The variable F can be lifted to the full phase space of the plane Kepler problem. We need to show that the coordinates (Λ, λ, F ) are analytic and symplectic provided that we make an adequate choice of a real analytic section of the flow of Λ to define λ = + cst; this choice of constant corresponds to choosing an analytic Lagrangian section in Lemma 3. Define λ as the mean longitude λ = + g; this choice is primarily motivated by the first argument given in the proof of the following statement.

Lemma 7. The coordinate system (Λ, λ, F ) is real analytic and symplectic in the neighborhood of direct circular Keplerian motions.
Only λ is not yet known to be analytic. Adding g to Kepler's equation (2.6) yields where both E and w are analytic. So λ is analytic.
We need to show that the coordinate system is symplectic. By continuity, it is enough to check this outside circular motions. Recall that the "symplectic polar map" is symplectic: 1 2i F * (dz ∧ dz) = dρ ∧ dφ. So, the question reduces to checking that (Λ, λ, Λ − Θ, −g) is symplectic, or, equivalently, that the Delaunay coordinates (Λ, , Θ, g) are symplectic.
Since the matrix of the symplectic form is the inverse of the matrix of the Poisson structure, we will check that the Poisson brackets are given by the standard matrix. We know that so it is enough to prove that { , g} = 0 on the codimension-2 submanifold = g = 0 (mod 2π), a section of the L-and Θ-flows. So, without loss of generality we may thus assume that the body is on the major axis and that the major axis itself is the first coordinate axis. Then the partial derivatives of and g with respect of x or p y are zero, and

Reminder on the Spatial Kepler Problem
Now consider the same equation as in (2.1) but with q = (x, y, z) ∈ R 3 \ {0}. We will again restrict to negative energy and non-collision motions (q ∧q = 0). Due to the equivariance of the equation by orthogonal symmetries, a solution q(t) is drawn on the vector plane generated by q(0) andq(0).
Redefine the angular momentum and the eccentricity vector, respectively, as with r = q and i(q)C = (q · q)q −q 2 q.
We will need extensions of e ig and of the eccentric longitude w in space. Let R q (α) ∈ SO 3 be the rotation around a vector q ∈ R 3 of angle α, R x (α) = R (1,0,0) (α), and similarly for rotations around the two other vectors of the canonical basis of R 3 . When the orbital plane is not the "horizontal" plane (xy-plane), define • the inclination ι, or the angle of the orbital plane with respect to the horizontal plane, • Δ, the oriented direction of the ascending node (half-line from the center of attraction to the point of the Keplerian ellipse where z = 0 andż > 0), • the longitude of the node ϕ, or the angle between the x-axis and Δ.
SinceẼ lies in the orbital plane, R Δ (−ι)Ẽ is horizontal. The fortunate fact is that this rotation matrix is an analytic function of q andq. Indeed, using a classical decomposition (see [31]), we see that where we have used the auxiliary notation Identifying R Δ (−ι)Ẽ ∈ R 2 × {0} to a complex number, define the analytic variable E = R Δ (−ι)Ẽ = e i(g+ϕ) ∈ C.
Moreover, the computation analogous to the one made in the plane shows that now the eccentric longitude w = u + g + ϕ is analytic.

d. Horizontal variables
The plane coordinates (Λ, λ, F ) extend to real analytic variables in a neighborhood of circular coplanar Keplerian motions in the spatial phase space {(q, q)} = R 3 × (R 3 \ {0}), in the following manner: where a is the semi-major axis. Due to the invariance of the Hamiltonian by rotations, the relation H = − 1 2a still holds in space, showing that a and thus Λ are real analytic.
Only λ is not obviously analytic. But this follows from Kepler's equation 2.6, to which one can add g + ϕ (instead of g in the plane) to get λ = w − sin(w − ϕ − g) = w + Im (Ee −iw ), and from the fact that E and the eccentric longitude w are analytic.
e. Oblique variable The complex-valued variable can be added to (Λ, λ, F ) to form an analytic coordinate system in space. Unfortunately, (Λ, λ, F, Γ) is not symplectic since, in restriction to the tangent space generated by ∂/∂z and ∂/∂ż, Looking for a symplectic modification of Γ of the form G = f (Θ, Φ)Γ by carrying out an analogous computation of the symplectic form as after (2.8), using spherical coordinates, one can find another way to build spatial coordinates from the plane coordinates is described in [50]. G is analytic because Only the property of being symplectic remains to be proved. Since the map (2.11) is symplectic, by continuity it suffices to show that the coordinate system • The three Poisson brackets between the pairs of angles among , g and ϕ vanish. Indeed, as in the plane, the Jacobi identity shows that it is enough to check those Poisson brackets on the submanifold { = g = ϕ = 0 (mod π)}. But on this submanifold the partial derivatives of any of the angles with respect to x, p y or p z vanish.
This completes the proof of Theorem 2.