CANONICAL PERTURBATION THEORY FOR THE ELLIPTIC RESTRICTED-THREE-BODY PROBLEM

Transcription

1 AAS CANONICAL PERTURBATION THEORY FOR THE ELLIPTIC RESTRICTED-THREE-BODY PROBLEM Brenton Duffy and David F. Chichka The distinguishing characteristic of the elliptic restricted three-body problem is a time-varying potential field resulting in non-autonomous and non-integrable dynamics. The purpose of this study is to normalize the system dynamics about the circular case and about one of the triangular Lagrange points by applying a method of canonical perturbation theory introduced by Hori and Deprit in the 1960s. The classic method derives a near-identity transformation for a Hamiltonian function expanded about a single parameter such that the transformed form possesses ideal properties of integrability. In this study, the method is extended to two-parameter expansions and applied to motion about the triangular elliptic Lagrange points. The transformed system is expressed in Birkhoff normal form for which the stability properties may be analyzed using KAM theory and the motion by local integrals and level sets. INTRODUCTION The elliptic restricted three-body problem ERTBP considers the motion of an object of negligible mass within the gravitational potential field generated by two primary bodies that move in closed Keplerian orbits about their common center of mass. The practical example for such a system is a spacecraft flying within the gravitational field of a planet-moon system. If the primary bodies move within circular orbits, the system reduces to that of the circular or classic restricted three-body problem CRTBP. As such, the ERTBP represents a simplification of the full three-body problem, but a generalization of the CRTBP. It is interesting to note that while both the general case and the circular case are autonomous and at least partially integrable, the ERTBP is non-autonomous and possesses no known integrals of motion. Most studies in this area tend to focus on the CRTBP, as exemplified by the extensive analyses included in the books by Szebehely and Meyer, et al. 1, However, a number of important studies were conducted in the late 1960s specifically for the elliptic case. 3, 4, 5, 6, 7 In addition, a resurgence 8, 9, 10 The of interest has occurred in recent years. goal of the present study is to build on these previous studies for an analytical treatment of motion about one of the triangular Lagrange points. The system dynamics are formulated using Hamiltonian mechanics and normalized using a method of canonical perturbation theory developed independently by Hori and Deprit, herein referred to as the Deprit-Hori method. 11, 1 The normalized system assumes a convenient, locally integrable form, which may then be analyzed using classical methods and KAM theory. 13, 14, 15 PhD Candidate in Mechanical and Aerospace Engineering at The George Washington University, Washington, DC. Research Professor in Mechanical and Aerospace Engineering at The George Washington University, Washington, DC. 1

2 HAMILTONIAN FORMULATION OF THE ERTBP In Theory of Orbits, Szebehely provides a useful introduction to the ERTBP including a presentation of Nechvile s transformation of the equations of motion. 1 The transformation expresses the system dynamics within a rotating and pulsating barycentric reference frame as shown in Figure 1. The spacecraft position is normalized by the instantaneous distance between the primaries, which for the circular case is fixed, but for the elliptic case varies periodically with the true anomaly ν. z r r 1 r ν x ν y Figure 1. Rotating and Pulsating Barycentric Reference Frame The state of the third body is represented in terms of normalized Cartesian coordinates q = {q x, q y, q z } and the generalized momenta p = {p x, p y, p z }. The system dynamics are defined in canonical units through the Hamiltonian function where r 1 = Hq, p, e, ν = 1 q x + qy + qz + p x + p y + p z + qy p x q x p y 1 q x + qy + qz 1 + e cos ν + 1 µ + µ, 1 r 1 r q x µ + qy + qz and r = q x + 1 µ + qy + qz are the normalized distances from each of the primaries to the spacecraft. In addition, µ parameterizes the mass ratio between the two primaries, e is the eccentricity of the primaries orbit, and the true anomaly ν serves as the indendent variable. The Hamiltonian function is highly nonlinear, non-autonomous and does not possess the same Jacobi integral as in the CRTBP. 4 It does however possess five Lagrange equilibrium points, but only when expressed relative to the rotating, pulsating reference frame and with ν serving as the independent variable. Previous studies have shown that motion about any of the collinear elliptic Lagrange points is unstable. 1 Motion about the triangular elliptic Lagrange points is linearly stable, but only within a subset of mass ratios and eccentricities defined by Danby s stability curve. 3 In this case, linear stability implies a set of purely imaginary eigenvalues in the linearized phase space for which Lyapunov s indirect method is inconclusive and one may not infer the stability of the full system based on that of the linearized system. Instead, one must treat the nonlinear and non-circular perturbation terms directly. This requires a normalization scheme based on the Deprit-Hori method of canonical perturbation theory followed by an application of KAM theory wherein a complicated, non-integrable system is treated as a small perturbation to a simplified, integrable system.

3 CANONICAL PERTURBATION THEORY Consider a nominal Hamiltonian system perturbed by higher-order terms. For sufficiently small perturbations i.e. within the vicinity of the nominal system, one may normalize the perturbed system about the unperturbed result. The classic approach is called Von Zeipel s method, which considers the expansion of the Hamiltonian function about a small parameter ɛ and derives a nearidentity transformation H = n=0 ɛ n n! H nq, p, ν K = n=0 ɛ n n! K nˆq, ˆp, ν, for which the form for the zero-order terms are left un-changed, H 0 = K 0. The transformation is defined implicitly through a generating function Wˆq, p, ν satsifying q = W p and ˆp = W ˆq. The main disadvantage to Von Zeipel s method is that the generating function contains both new and old variables leading to a mixed variable solution, which must be inverted in order to derive the state transformation equations. An alternative approach was introduced in the late 1960s by Hori and Deprit. 11, 1 Instead of relying on a generating function of mixed variables, the Deprit-Hori method represents the state transformation in terms of the Lie operator, which in the case of Hamiltonian systems, is equivalent to the Poisson bracket. The approach achieves the same result as Von Zeipel s method, the expanded Hamiltonian function is transformed using an expanded generating function, but by using the Lie operator the resultant transformation equations are obtained directly. Deprit-Hori Method of Lie Transforms The Poisson bracket between two functions, H and W, represents a Lie product wherein L W H [H, W] = N i=1 H W H W q i p i p i q i is defined as the Lie derivative of H generated by W both assumed real and analytic in a bounded domain of the phase space. The basis of the Deprit-Hori method is to apply such Lie operators in the framework of canonical perturbation theory for the construction of a canonical transformation Hq, p, ɛ, ν Kˆq, ˆp, ɛ, ν for which the generating function, original Hamiltonian function, and transformed Hamiltonian function are all expanded about a small parameter. The single parameter case is treated in the original papers of Deprit and Hori and discussed in the framework of orbital mechanics in Boccaletti and Pucacco and Meyer, et al., 11, 1, 16 As an extension to the single parameter method, consider a doubly-expanded Hamiltonian system whose higher-order terms are parameterized by two quantities: ɛ and γ. This two-parameter case was treated for autonomous systems in 1985 by Varadi, whose analysis was subsequently extended to three parameters by Ahmed and to N parameters by Andrade. 17, 18, 19 With the aim of applying the method to motion about the elliptic Lagrange points, the two-parameter method is presented here including an extension to non-autonomous systems. 3

4 Theorem 1. Consider a non-autonomous Hamiltonian function expanded about two small parameters as represented by the series Hq, p, ɛ, γ, ν = n! m! H0,0 q, p, ν. A canonical transformation q = Qˆq, ˆp, ɛ, γ, ν and p = P ˆq, ˆp, ɛ, γ, ν may be generated from a pair of functions Wq, p, ɛ, γ, ν and Vq, p, ɛ, γ, ν such that the transformed Hamiltonian function may be constructed term by term in the series Kˆq, ˆp, ɛ, γ, ν = by the extended Deprit recursive equations H r,s+1 H r+1,s = H r,s +1 + n H 0,0 ˆq, ˆp, ν + R 0,0 ˆq, ˆp, ν, n! m! = H r,s m+1,n + n n m L Wj+1,i+1 H r,s n m L Vj+1,i+1 H r,s, and where R 0,0 = S 0,0 S r,s+1 S r+1,s { 0,n 1 S 0,0 m = 0, n 0 T m 1,0 0,0 m 0, n = 0 S 1 0,0 = T m 1,n 0,0 m, n 0 = ν W m+1,n+1 n = S r,s n m +1 + n = S r,s n m m+1,n + L Wj+1,i+1 S r,s L Vj+1,i+1 S r,s, and T 0,0 T r,s+1 T r+1,s = ν V m+1,n+1 n = T r,s +1 + = T r,s m+1,n + n n m n m L Wj+1,i+1 T r,s L Vj+1,i+1 T r,s. The expansion of the explicit state transformation equations as represented by the series q = n! m! q 0,0 ˆq, ˆp, ν and p = n! m! p 0,0 ˆq, ˆp, ν 4

5 may also be constructed using the recursive equations n q r,s+1 = q r,s n m +1 + L Wj+1,i+1 q r,s n q r+1,s = q r,s n m m+1,n + L Vj+1,i+1 q r,s, and p r,s+1 p r+1,s = p r,s +1 + n = p r,s m+1,n + n n m n m where q 0,0 0,0 = ˆq, p 0,0 0,0 = ˆp, and q 0,0 = p 0,0 = 0 for m + n > 0. L Wj+1,i+1 p r,s L Vj+1,i+1 p r,s, A full proof of the theorem is beyond the scope of this paper, but a sketch of the proof is provided here. In the spirit of Deprit s original proof, one may define an extended phase space by appending the independent variable ν and the Hamiltonian to the original state variables q {q, ν} and p {p, H}. The extended Deprit operators are defined as E W ɛ + L W + W ν E V γ + L V + V ν H D W + W ν H H D V + V ν H, which represent a combination of Deprit s original operators, D W = / ɛ+l W and D V = / γ + L V, and the partial derivatives with respect to ɛ and γ. Note that the original Deprit operators satisfy the following conditions: the zeroth-order Deprit operators act as identity operations, DW 0 H = DV 0 H = H, the operators are commutative D WD V = D V D W, and successive applications of the Deprit operators are given by s r DWD s V r = ɛ + L W γ + L V. For a function F q, p, ɛ, γ, ν with no explicit dependence on the Hamiltonian, the extended Deprit operators reduce to the original Deprit operators. Furthermore, under the transformation q = Qˆq, ˆp, ɛ, γ, ν and p = P ˆq, ˆp, ɛ, γ, ν, one may express the mixed partial derivatives of F with respect to ɛ and γ by Eq., n m ɛ n γ m F q, p, ɛ, γ, ν q = Qˆq, ˆp, ɛ, γ, ν p = P ˆq, ˆp, ɛ, γ, ν = DWD n V m F q = Qˆq, ˆp, ɛ, γ, ν p = P ˆq, ˆp, ɛ, γ, ν Having formulated the Deprit operators within the extended phase space, one may now consider the canonical transformation of a non-autonomous Hamiltonian function. For a canonical transformation mapping coordinates q, p to ˆq, ˆp, the transformed Hamiltonian function takes the form K = H Qˆq, ˆp, ɛ, γ, ν, P ˆq, ˆp, ɛ, γ, ν, ɛ, γ, ν + R ˆq, ˆp, ɛ, γ, ν = Ĥ ˆq, ˆp, ɛ, γ, ν + R ˆq, ˆp, ɛ, γ, ν 3 5

6 where Ĥ represents the original Hamiltonian function written explicitly in terms of the state transformation equations and R represents a remainder function. The expansion of Ĥ in a Taylor series about ɛ = 0 and γ = 0 may be represented in terms of the Deprit operators by Ĥˆq, ˆp, ɛ, γ, ν = = n! m! n! m! n m ɛ n Ĥˆq, ˆp, ɛ, γ, ν γm DWD n V m Hq, p, ɛ, γ, ν By introducing the subscripted and superscripted formulation ɛ,γ=0 q = Qˆq, ˆp, ɛ, γ, ν p = P ˆq, ˆp, ɛ, γ, ν. ɛ,γ=0 D s WD r VH = H = n! m! H0,0 n! m! Hr,s, 4 the expansion of the original Hamiltonian function is represented by the series of functions H 0,0 and the expansion of the first part of the transformed Hamiltonian function is represented by Ĥ = = n! m! DWD n V m Hq, p, ɛ, γ, ν n! m! H 0,0 ˆq, ˆp, ν. Within the subscripted and superscripted formulation, one has L W H r,s = L V H r,s = n n q = Qˆq, ˆp, ɛ, γ, ν p = P ˆq, ˆp, ɛ, γ, ν n m L Wj+1,i+1 H r,s n m L Vj+1,i+1 H r,s, such that the terms included in the expansion series under the Deprit operator satisfy n! m! Hr,s+1 = D W n! m! Hr,s = ɛ = n! m! m! Hr,s n! H r,s +1 + n + L W n i m j ɛ,γ=0 n! m! Hr,s L Wj+1,i+1 H r,s 5, 6

7 and likewise n! m! Hr+1,s = H r,s n! m! m+1,n + n n i m j L Vj+1,i+1 H r,s. 6 Thus, comparing Eqs.5 and 6 to Eq.4, all the unknown functions H r,s may be constructed term by term using recursive equations H r,s+1 H r+1,s = H r,s +1 + n = H r,s m+1,n + n n m n m L Wj+1,i+1 H r,s L Vj+1,i+1 H r,s 7 referred to as the extended Deprit recursive equations. The formulae are recursive in the sense that each successive term is dependent only on terms preceding it, starting with the original Hamiltonian H 0,0. The construction may be visualized in the tradition of Deprit s triangle, but now in the form of a pyramid as shown in Figure. H 0,0 0,0 H 1,0 0,0 H 0,0 1,0 H 0,0 0,1 H 0,1 0,0 H,0 0,0 H 1,0 1,0 H 0,0 H 0,1,0 H 0,0 0, 0,1 H 0, 0,0 H 1,1 0,0 H 1,0 0,1 H 0,0 1,1 H 0,0 1,1 H 0,1 1,0 H 1,1 0,0 H 3,0 H,0 H 1,0 H 0,0 3,0 H 0,0 0,3 H 0,1 0, H 0, 0,0 1,0,0 0,1 H 0,3 0,0 Figure. Deprit s Pyramid One must still account for the remainder function. Applying the extended operators to the Hamiltonian itself yields E W H = D W H + W ν E V H = D V H + V ν, 7

8 and for higher-order and mixed terms, EWE s VH r = DWD s VH r + D s 1 W W Dr V ν + Ds WD r 1 V V ν. 8 The first term appearing in Eq. 8 is equivalent to the explicit subsitution of the state transformation equations into the original Hamiltonian function as constructed from the extended Deprit recursive formulae given in Eqs. 7. The other two terms appearing in Eq. 8 comprise the remainder function and may be constructed term by term in the same manner as the Hamiltonian function by first defining the intermediary terms R 0,0 = and then applying the recursive formulae and S 0,0 S r,s+1 S r+1,s T 0,0 T r,s+1 T r+1,s { 0,n 1 S 0,0 m = 0, n 0 T m 1,0 0,0 m 0, n = 0 S 1 0,0 = T m 1,n 0,0 m, n 0 = ν W m+1,n+1 n = S r,s n m +1 + n = S r,s n m m+1,n + = ν V m+1,n+1 n = T r,s n m +1 + n = T r,s n m m+1,n + L Wj+1,i+1 S r,s L Vj+1,i+1 S r,s, L Wj+1,i+1 T r,s L Vj+1,i+1 T r,s. Having derived the recursive equations for Ĥ and R, consider the explicit state transformation equations expanded about ɛ = 0 and γ = 0 as represented by the Taylor series n m q = Qˆq, ˆp, ɛ, γ, ν n! m! ɛ n γ m ɛ,γ=0 = ɛ n n! γ m D n m! W DV m Qˆq, ˆp, ɛ, γ, ν ɛ,γ=0. Introduce the subscripted and superscripted notation q = n! m! q 0,0 ˆq, ˆp, ν p = n! m! p 0,0 ˆq, ˆp, ν 9 8

9 where the zero superscript terms correspond to the identity transformation, q 0,0 = { ˆq m = n = 0 0 m + n > 0 and p 0,0 = { ˆp m = n = 0 0 m + n > 0. By the same logic used previously, the terms in the state transformation equations may be generated using the recursive equations q r,s+1 q r+1,s = q r,s +1 + n = q r,s m+1,n + n n m L Wj+1,i+1 q r,s n m L Vj+1,i+1 q r,s, 10 and p r,s+1 p r+1,s = p r,s +1 + n = p r,s m+1,n + n n m n m L Wj+1,i+1 p r,s L Vj+1,i+1 p r,s. 11 Since the state transformation equations are not explicitly dependent on the Hamiltonian function, the application of the extended Deprit operators and the application of the original Deprit operators yield the same results. Therefore, after deriving the generating functions, the terms generated by the recursive equations above constitute the full state transformation equations. Carrying out the extended Deprit-Hori method to terms of order yields the differential equations K 0,0 = H 0,0 0,0 K 0,1 = H 0,1 0,0 + S 0,0 0,0 = W 1,1 + L W1,1 H 0,0 0,0 + H 0,0 0,1 ν K 1,0 = H 1,0 0,0 + T 0,0 0,0 = V 1,1 ν + L V 1,1 H 0,0 0,0 + H 0,0 1,0 K 0, = H 0, 0,0 + S 0,1 0,0 = W 1, + L W1, H 0,0 0,0 + H 0,0 0, + L W1,1 H 0,0 0,1 + K 0,1 ν K,0 = H,0 0,0 + T 1,0 0,0 = V,1 ν + L V,1 H 0,0 0,0 + H 0,0,0 + L V1,1 H 0,0 1,0 + K 1,0 which are consistent with the single-parameter equations in the form of so-called homological equations. 1 In addition, one must compute the mixed-variable terms appearing in the transformed Hamiltonian, for example, K 1,1 corresponding to the term with first-order dependency in both ɛ and γ. This is a departure from the single-parameter case, which obviously has no such mixed terms. The term K 1,1 may be computed from one of two formulations, K 1,1 = H 1,1 0,0 + S 1,0 0,0 = W,1 + L W,1 H 0,0 0,0 + H 0,0 1,1 + L W1,1 H 0,0 1,0 + L V1,1 K 0,1 ν = H 1,1 0,0 + T 0,1 0,0 = V 1, ν + L V 1, H 0,0 0,0 + H 0,0 1,1 + L V1,1 H 0,0 0,1 + L W1,1 K 1,

10 For each mixed term appearing in the original Hamiltonian, there are two possible avenues to derive the corresponding term in the new Hamiltonian: each using a distinct, but complementary generating function term. Note that this is a bi-product of the Deprit commutation condition D W D V = D V D W. Whether one is applying the Deprit-Hori method about a single parameter or multiple parameters, the key operation is to solve the so-called homological equation appearing in Eqs.1 and 13 in either of the forms W i+1,j ν L Wi+1,j H 0,0 = Q i,j K i,j or V i,j+1 ν L Vi,j+1 H 0,0 = P i,j K i,j 14 All of the terms included in H 0,0, Q i,j, and P i,j are known a priori, either from the expansion of the original Hamiltonian function or from previously derived terms of lesser order. The goal is then to prescribe the desired formulation for K i,j and solve the homological equation for the corresponding term in the generating function. However, one must take some care in choosing the form of K i,j to insure a realizable transformation. The resultant state transformation equations must be everywhere analytic in the original coordinate system, which means they must be locally equivalent to convergent power series in some neighborhood of every point. This condition holds when applying the Deprit-Hori method as an averaging technique wherein the transformed Hamiltonian function is defined by the periodic average of the original Hamiltonian, that is, K i,j = Q i,j, or P i,j. When expressed in action-angle variables, the averaging normalization eliminates quasi-periodic variations in the independent variable ν and fast variables θ i, leaving only secular variations in the slow variables I i. After defining the desired form for K, one may solve the homological equations shown in Eq.14 for the ordered generating functions, W i and V i, to sufficient order. Once derived, the ordered generating functions then facilitate the derivation of the corresponding state transformation equations using the same recursive methodology. The state transformation equations to second-order are given by q = ˆq + ɛl W1,1 ˆq + γl V1,1 ˆq + ɛ L W1, + L W 1,1 ˆq + γ L V,1 + L V 1,1 ˆq + ɛγ L W,1 + L V1,1 L W1,1 ˆq +... p = ˆp + ɛl W1,1 ˆp + γl V1,1 ˆp + ɛ L W1, + L W 1,1 ˆp + γ L V,1 + L V 1,1 ˆp + ɛγ L W,1 + L V1,1 L W1,1 ˆp +... NORMALIZATION OF THE ERTBP Returning to the ERTBP, the Hamiltonian function is expanded about the circular case and about a triangular Lagrange point in the Taylor series Hq, p, e, ν = m=0 n=0 γ m e n m! n! H q, p, ν 15 where γ = q, p parameterizes the distance from the Lagrange point and H corresponds to terms of order q, p m+ and e n. The simplest approximation of the system is the linearized, 10

11 circular case represented by the term H 0,0. In the spirit of the Deprit-Hori method and KAM theory, this is referred to as the unperturbed system while the non-circular and nonlinear terms comprise the higher-order perturbation terms. Unperturbed System The unperturbed system is defined by the linearized, circular part of Eq. 15, which may be derived from the original Hamiltonian function in Eq.1 as H 0,0 = 1 [ ] p x + q y + p y q x + p z + qz 3 [qx + 3qy + ] 31 µq x q y. 8 wherein the generalized coordinates and momenta are expressed relative to the triangular Lagrange point located at q x = µ 1/, q y = ± 3/, and q z = 0. Within the range 0 < µ < µ c = 1 69/9/, the unperturbed system dynamics are characterized by a set of eigenvalues lying along the imaginary axis such that the linearized system is nuetrally stable. The ensuing motion is then in the form of harmonic oscillation about the Lagrange point. To elucidate the oscillatory behavior of the unperturbed system, Breakwell and Pringle provide a linear transformation of the planar CRTBP that expresses the linearized system directly in the form of coupled harmonic oscillators. 0 Extending their transformation to three-dimensions yields H 0,0 = 1 ω s qs + p 1 s ω l ql + 1 l p + ω z qz + p z where the natural frequencies are constant as determined by the mass ratio µ through the equations µ1 µ 1 1 7µ1 µ ω s = ω l = ω z = 1. In the range, 0 < µ < µ c, the planar frequencies satisfy the inequality 0 < ω l < / < ω s < 1. Introducing action-angle variables through the explicit transformation equations I s ω sq s + p s ω s I l ω l q l + p l ω l I z ω zq z + p z ω z yields a Hamiltonian function in the convenient form tan θ s ω s q s p s tan θ l ω l q l p l tan θ z ω z q z p z, 16 H 0,0 = ω s I s ω l I l + ω z I z. 17 Based on the definitions and conditions set forth in Boccaletti and Pucacco, the normalized H 0,0 function shown in Eq. 17 is autonomous in ν, integrable in the sense of Liouville, degenerate, and isoenergetically non-degenerate. 16 The integrability property insures that the solution can be directly expressed using quadratures. In particular, since the system is degenerate, one may express the state trajectories by the equations θ i ν = θ i,0 ± ω i ν ν 0 mod π I i ν = I i,0 11

12 where θ i,0, I i,0 are the initial conditions at ν = ν 0. Note that the action variables I s, I l, and I z are constant and the angular variables θ s, θ l, and θ z propagate linearly with ν. As such, the angular variables are said to wind an invariant torus defined by the constant action variables. Perturbed System The extended linear transformation is further applied to the full nonlinear and non-circular ERTBP such that the perturbed system is expressed in terms of θ i and I i and the unperturbed part H 0,0 assumes the convenient form H 0,0 = ω s I s ω l I l + ω z I z + m+n>0 γ m e n m! n! H q, p, ν. The higher-order, perturbation terms are still explicitly dependent on ν and all six state variables θ i, I i. As such, the perturbed system is still non-autonomous and non-integrable. The Deprit-Hori method is applied with the goal of normalizing the perturbed Hamiltonian function about the unperturbed case. In doing so, the transformed system takes the form of the unperturbed system in that it becomes integrable up to the order of truncation with a phase space foliated by invariant tori. The state transformation equations are defined implicitly through the generating functions W and V, which are themselves defined by the ordered homological equations shown previously in Eqs. 14. Substituting the unperturbed Hamiltonian function H 0,0 = ω s I s ω l I l + ω z I z into Eqs. 14 yields the first-order differential equation ν + ω s ω l + ω z W i+1,j = Q i,j K i,j, θ s θ l θ z whose solution is given by the inversion W i+1,j = ν + ω 1 s ω l + ω z Q i,j K i,j, 18 θ s θ l θ z and likewise for V. The terms on the right-hand side of Eq. 18, namely Q i,j and K i,j, constitute terms from the original Hamiltonian function and those from lower-orders and terms for the new Hamiltonian function. The latter is defined by the periodic average of the former through the operation K i.j Q i,j = 1 4π π π π π Q i,j dν dθ s dθ l dθ z, 19 such that the true anomaly and angular variables are effectively eliminated from the Hamiltonian function leaving only the action-type variables, that is, K = KI i. The corresponding generating functions that produce this result may be derived through the inversion operation in Eq. 18. Since ν and θ i are all periodic in the original Hamiltonian, that is, they all appear within trigonometric terms, the differential inversion operation is equivalent to the explicit substition defined by cos i 1 ν + i θ s + i 3 θ l + i 4 θ z sin i 1ν + i θ s + i 3 θ l + i 4 θ z i 1 + i ω s i 3 ω l + i 4 ω z sin i 1 ν + i θ s + i 3 θ l + i 4 θ z cos i 1ν + i θ s + i 3 θ l + i 4 θ z i 1 + i ω s i 3 ω l + i 4 ω z. 1

13 The term in the denominator goes to zero under resonant conditions including resonance between the out-of-plane dynamics and true anomaly as in cosν θ z. In the former, one must define a different normalized form for K and in the latter, one may apply the substitution cos i 1 ν + i θ s + i 3 θ l + i 4 θ z ν cos i 1 ν + i θ s + i 3 θ l + i 4 θ z sin i 1 ν + i θ s + i 3 θ l + i 4 θ z ν sin i 1 ν + i θ s + i 3 θ l + i 4 θ z. Since this substitution introduces secular terms, the state transformation will exhibit small secular drift in the out-of-plane dynamics. The scale of the drift is relatively small and the period of oscillation is relatively large such that these secular effects are negligible. Transformed Hamiltonian Function Upon applying the Deprit-Hori method, the ERTBP Hamiltonian is transformed into Birkhoff normal form as shown in Eq. 0 e n K = ω s + n! ω e n s,n I s ω l + n! ω l,n I l + I z e n + α ss + n! α ss,n Is e n + α ll + n! α ll,n Il + + α zz + α sz + e n n! α zz,n e n n! α sz,n Iz e n + α sl + n! α sl,n I s I l e n I s I z + α lz + n! α lz,n I l I z + OIi 3, 0 where the constant coefficients are expressed as infinite series dependent on the mass ratio µ and even-powers of the eccentricity. Due to the form of Eq. 1, all odd-powers of the eccentricity are eliminated under the averaging operation shown in Eq. 19. In addition, the Hamiltonian function is made autonomous up to the order of truncation. With the exception of the linear, out-of-plane term, I z, the coefficients included in Eq. 0 are in the form of circular coefficients under perturbation by even-powers of the eccentricity. In particular, the coefficients for the linearized planar system are given by perturbations to the natural frequencies ω s and ω l. Most of the coefficients are too large and complicated to display symbolically, but some of the lower-ordered terms can be expressed compactly as shown in Eq. 1 ω s,1 = ω s1 ω s7 6ω s 1 ω s1 4ω s ω l,1 = ω l1 ωl 7 6ω l ω 1 ωl 1 4ω l α zz = sωl 31 + ωsω l α ss = ω l ω s + 14ω 4 s 71 ω s 1 5ω s ω s ω l ω α sl = sωl 31 ωs1 ωl 1 5ω s1 5ωl 16ω s ωl α sz = 34 9ωs + ωs 4 α ll = ω s81 696ωl + 14ω4 l 16ω 71 ωl 1 5ωl α lz = sω l 34 9ωl + 1 ω4 l. Note that the coefficients are directly related to the system mass ratio µ through the natural frequencies ω i. Further, the results shown in Eqs. 1 are consistent with those derived in previous studies that focused on either the planar CRTBP or the planar linearized ERTBP. 7, 1, 13

14 SYSTEM DYNAMICS IN THE TRANSFORMED PHASE SPACE Having normalized the system Hamiltonian function through the Deprit-Hori method, one may derive the corresponding state transformation equations from Eq. 9 and Eqs. 10 and 11. These then provide the means to relate the motion in the transformed phase space back to the original coordinate system used in Eq. 15. Thus, within a small neighborhood of the Lagrange point as determined by the order of truncation in the transformation, one may model the dynamics of the non-autonomous and non-integrable system in Eq. 15 by the autonomous and integrable system in Eq. 0. Being integrable, the latter system may be treated using classical methods of analysis including modeling the phase space using curves of zero velocity and determining the stability of the Lagrange points using KAM theory. KAM Stability Analysis Since the unperturbed Hamiltonian system K 0,0 = H 0,0 is completely integrable, its phase space is foliated by invariant tori parameterized by the action variables. One may then consider whether these invariant tori persist for small perturbations, that is, in the perturbed, nearly-integrable system of Eq. 0. In 1954, Kolmogorov provided a theorem addressing this issue whose proof was later derived by Moser for the case of twist maps and then by Arnol d for analytic Hamiltonian systems. 13, 14, 15 In their honor, the theorem is referred to as the KAM theorem and may be stated as follows. Consider the Hamiltonian system K = K 0,0 + K, where K 0,0 represents the unperturbed, integrable system and K represents the perturbations of higher-order. If the unperturbed system is non-degenerate or isoenergetically non-degenerate, then for a sufficiently small Hamiltonian perturbation most non-resonant invariant tori do not vanish but are only slightly deformed, so that in the phase space of the perturbed system there are invariant tori densely filled with [quasi-periodic] phase curves winding around them, with a number of independent frequencies equal to the number of degrees of freedom. 16 Note that in quasi-periodic motion the orbit trajectory will never repeat itself, but a single orbit is dense everywhere on the torus, that is, it will eventually cover the entire torus uniformly. Preservation of invariant tori does not necessarily imply stability in the general case. This is only true in the -dimensional case where the invariant tori separate the phase space into two noncommunicating parts, that is, any trajectory originating in the space between two non-resonant tori is necessarily confined to remain in this space for all time. This property is no longer valid for n 3 dimensions for which trajectories could escape through gaps in the tori, which is known as Arnold diffusion. 16 Thus, application of the KAM theorem as a means to prove stability of an equilibrium point is only valid in the -dimensional, planar case. Meyer, et al. presents KAM theory specifically for systems in Birkhoff normal form and re-states the theorem as follows. Consider a Hamiltonian system in Birkhoff normal form, K = K + K 4 + K N + K hot, 14

15 where K is real analytic in the neighborhood of the origin, K = ω s I s ω l I l, K k is a homogeneous polynomial of degree k in I s and I l, and K hot has a series expansion that starts with terms at least of degree N + 1. It is also assumed that the frequencies are sufficiently non-resonant that is, their ratio is irrational. The origin of this system is stable if for at least one k in the range 1 k N, the system satisfies the inequality D k = K k I s = ω l, I l = ω s 0, which implies K does not divide K k. Often it is sufficient to show that the second-order part is non-zero, that is D 4 0. However, if for some cases this term does go to zero, one must then show that a higher-order part is non-zero, for instance D 6 0. Meyer et al. provides a nice proof of the theorem by applying a canonical transformation of the Hamiltonian for small perturbations and in the neighborhood of the origin and invoking Moser s invariant curve theorem to show that the transformed system is stable., 14 Stability of the ERTBP via KAM Theory Based on the preceding introduction of KAM theory, one can easily analyze the stability of motion about the triangular Lagrange points of the ERTBP. First, one must limit the analysis to the planar case and demonstrate that the linearized non-circular system K 0,i = ω s + e n n! ω s,n I s ω l + e n n! ω l,n I l, still constitutes a linearly stable system of harmonic oscillators. This particular question has been studied in the past starting with Danby who numerically demonstrated that the non-circular problem is linearly stable for a range of mass ratios and eccentricities given by the plot shown in Figure 3. Danby s stability curve was subsequently verfied analytically by Aflriend and Rand in 1969 and Deprit in , 6, 7 Figure 3. Danby s Stability Curve The curve shown in Figure 3 was generated from an analysis of the system Floquet exponents as determined by the normalized coefficients of I s and I l in Eq. 0, that is e n σ = ±i ω s + n! ω e n s,n and λ = i ω l + n! ω l,n. 15

16 Within this linearly stable range of mass ratios and eccentricities, nonlinear stability is achieved under the irrationality condition ω s /ω l / Z and the KAM inequality shown in Eq.. Substituting in the normalized Hamiltonian function yields the condition α ss + e n n! α ss,n ω l + α ll + e n n! α ll,n ωs+ α sl + Local Integrals of Motion and Curves of Zero Velocity e n n! α sl,n ω s ω l 0 The normalized equations of motion are expressed within the transformed phase space by Eqs. 4 dθ s dν = K e n = ω s + I s n! ω e n s,n + α ss + n! α ss,n I s e n + α sl + n! α e n sl,n I l + α sz + n! α sz,n I z + OIi di s dν = K θ s = 0, 4 and likewise for θ l, I l and θ z, I z. As such, up to the order of truncation in the normalization, the action-type variables represent local integrals of motion in the transformed phase space corresponding to invariant tori about the Lagrange point. The angular state variables wind the tori according to the linear trajectories θ i = θ i,0 ± Ω i ν ν 0 mod π, where Ω i are the perturbed natural frequencies, which are constant for a given system and defined by e n Ω s = ω s + n! ω e n s,n + α ss + n! α ss,n I s,0 e n + α sl + n! α e n sl,n I l,0 + α sz + n! α sz,n I z,0 + OIi,0 e n Ω l = ω l + n! ω s,n + α ll + e n + α sl + n! α sl,n I s,0 + α lz + e n Ω z = ω z + α zz + n! α zz,n I z,0 e n + α sz + n! α sz,n I sl,0 + α lz + e n n! α ll,n e n n! α lz,n I l,0 I z,0 + OI i,0 e n n! α lz,n I l,0 + OIi,0. For the CRTBP, Hill introduced a method of foliating the phase space by level sets of the Jacobi integral, that is, of the energy. 3, 1 Since the action-type variables represent local integrals of motion 3 16

17 up to the order of truncation, one may generate local level sets that foliate the phase space in the vicinity of the Lagrange point. To do so, one must first provide an adequate level set relation in the tradition of Hill s curves of zero velocity. From the transformation equations introduced in Eqs.16, one may derive the mixed variable equation p s + p l + p z = ω s I s + ω l I l + ω z I z ω sq s + ω l q l + ω zq z, where the term on the left-hand side is the magnitude of the generalized momentum vector, which must be nonnegative for real solutions. Therefore, one may define the constant C = ω s I s + ω l I l + ω z I z to serve as a Jacobi-type integral whose level sets foliate the phase space. This is exemplified in Figure 4, which shows level sets in the vicinity of the triangular Lagrange point of the planar Earth-Moon system µ = 0.014, e = and transformed back to the original coordinate system. Figure 4. Level Sets Green and Trajectory Blue in the Planar Earth-Moon System While the trajectory in Figure 4 spans multiple orbits, the level sets are shown for a given instant in time. Since the state transformation equations are ν-dependent, the level set curves pulsate as ν varies with time. To exaggerate this effect, the eccentricity of the Earth-Moon system is increased to 0. with the results shown at various intervals of ν in Figures 5 and 6. CONCLUSION This study has presented the main ingredients for an analytical treatment of the spatial ERTBP. After presenting an extension to the Deprit-Hori Lie transform canonical perturbation method, the ERTBP was normalized about the circular case and one of the triangular Lagrange points. The transformed Hamiltonian function is autonomous in Birkhoff normal form such that it is integrable up to the order of truncation. The triangular Lagrange points are linearly stable in a subdomain of the µ e phase space in which nonlinear stability is insured away from resonance under the nonlinear stability condition given in Eq. 3. Being integrable, the transformed system possesses local integrals of motion given by the action variables. Transforming these back into the original 17

18 coordinate system provides a means of generating level sets in the tradition of Hill s analysis of the CRTBP. The motion is then bound within level sets as parameterized by integrals of motion. Based on the methodology and results from this study, further analysis is being conducted in other areas including motion about the collinear Lagrange points, motion about the primaries, the incorporation of stabilizing feedback control, and orbit transfer design. REFERENCES [1] V. G. Szebehely, The Theory of Orbits. New York: Academic Press, [] K. R. Meyer, G. R. Hall, and D. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. New York: Springer: Applied Mathematical Sciences, nd ed., 009. [3] J. M. A. Danby, Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies, The Astronomical Journal, Vol. 69, No., 1964, pp [4] V. G. Szebehely, On the Elliptic Restricted Problem of Three Bodies, The Astronomical Journal, Vol. 69, No. 3, 1964, pp [5] G. Contopoulos, Integrals of Motion in the Elliptic Restricted Three-Body Problem, The Astronomical Journal, Vol. 7, 1967, pp [6] K. T. Alfriend and R. H. Rand, Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies, AIAA Journal, Vol. 7, No. 6, 1969, pp [7] A. Deprit and A. Rom, Characteristic Exponents at L 4 in the Elliptic Restricted Problem, Astronomy & Astrophysics, Vol. 5, 1970, pp [8] G. Gómez, A. Jorba, J. Masdemont, and C. Simó, Dynamics and Mission Design Near Libration Points: Vol. IV Advanced Methods for Triangular Points. Singapore: World Scientific Publishing Co., 001. [9] C. Lhotka, C. Efthymiopoulos, and R. Dvorak, Nekhoroshev Stability at L 4 or L 5 in the Elliptic- Restricted Three-Body Problem Application to Trojan Asteroids, Monthly Notices of the Royal Astronomical Society, Vol. 384, No. 3, 008, pp [10] B. Érdi, E. Forgács-Dajka, I. Nagy, and R. Rajnai, A Parametric Study of Stability and Resonances Around L 4 in the Elliptic Restricted Three-Body Problem, Celestial Mechanics and Dynamical Astronomy, Vol. 104, No. 1, 009, pp [11] G. Hori, Theory of General Perturbations with Unspecified Canonical Variables, Publications of the Astronomical Society of Japan, Vol. 18, No. 4, 1966, pp [1] A. Deprit, Canonical Transformation Depending on a Small Parameter, Celestial Mechanics, Vol. 1, 1969, pp [13] A. Kolmogorov, On Conservation of Conditionally Periodic Motions for a Small Change in Hamilton s Function, Doklady Akademii Nauk SSSR, Vol. 98, 1954, pp [14] J. Moser, On Invariant Curves of Area-Preserving Mappings of an Annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. KI. II., 196, pp [15] V. I. Arnol d, Proof of a Theorem of A. N. Kolmogorov on the Preservation of Conditionally Periodic Motions under a Small Perturbation of the Hamiltonian, Uspehi Mat. Nauk, Vol. 18, 1963, pp [16] D. Boccaletti and G. Pucacco, Theory of Orbits, Volume : Perturbative and Geometrical Methods. Berlin: Springer, [17] F. Varadi, Two-Parameter Lie Transforms, Celestial Mechanics, Vol. 36, 1985, pp [18] M. Ahmed, Multiple-Parameter Lie Transform, Earth, Moon, and Planets, Vol. 61, 1993, pp [19] M.Andrade, N-Parametric Canonical Perturbation Method Based on Lie Transforms, The Astronomical Journal, Vol. 136, 008, pp [0] J. Breakwell and R. Pringle, Resonances Affecting Motion near the Earth-Moon Equilateral Libration Points, Progress in Astronautics and Aeronautics: Methods in Astrodynamics and Celestial Mechanics, Vol. 17, 1966, pp [1] A. Deprit and A. Deprit-Bartholomé, Stability of the Triangular Lagrange Points, The Astronomical Journal, Vol. 7, No., 1967, pp [] K. R. Meyer and D. S. Schmidt, The Stability of the Lagrange Triangular Point and a Theorem of Arnold, Journal of Differential Equations, Vol. 6, 1986, pp. 36. [3] G. W. Hill, Researches in the Lunar Theory, American Journal of Mathematics, Vol. 1, No. 1, 1878, pp

19 a ν = 0 Farfield b ν = 0 Nearfield c ν = π/4 Farfield d ν = π/4 Nearfield e ν = 3π/4 Farfield f ν = 3π/4 Nearfield Figure 5. Level Sets Green and Trajectory Blue in the Planar Earth-Moon System with e = 0. 19

20 a ν = π/ Farfield b ν = π/ Nearfield c ν = 5π/4 Farfield d ν = 5π/4 Nearfield e ν = 3π/ Farfield f ν = 3π/ Nearfield Figure 6. Level Sets Green and Trajectory Blue in the Planar Earth-Moon System with e = 0. 0