Variational Integrators and the Three-Body Problem

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1 Variational Integrators and te Tree-Body Problem Evan Gawlik SURF 2006 Final Report Mentor: Dr. Jerrold E. Marsden, Control and Dynamical Systems Co-Mentor: Pilip Du Toit, Control and Dynamical Systems Abstract Tis study applies variational integrators to te tree-body problem, a classic problem from celestial mecanics tat asks for te motion of tree masses in space under mutual gravitational interaction. Te use of variational integrators, a class of numerical metods used to simulate mecanical systems, is of value because tey are known to exibit accurate beavior wit respect to te preservation of pysical constants of motion wen applied to systems wit conserved quantities like energy and momentum. Tis contrasts wit te performance of oter numerical differential equation-solving algoritms like te Runge- Kutta metod, wic typically experience artificial dissipation of conserved quantities over successive iterations. A comparison of te performance of variational integrators versus te fourt-order Runge- Kutta metod applied to te planar circular restricted tree-body problem composes te core of tis study. In particular, te algoritms are evaluated based on teir ability to accurately model individual trajectories, conserve integrals of motion, predict statistical quantities like transport rates, and preserve te structure of Poincaré section plots. In addition to te contributing to te development of numerical metodology, tese investigations ave applications to solar system simulations and to te design of spacecraft mission trajectories. 1 Introduction Predicting te motion of a set of bodies in space under mutual gravitational interaction is a classic problem in te study of celestial mecanics. Consider n masses m i wose motions r i (t) are determined by gravitational forces. From Newton s laws, te differential equations of motion for tese bodies are m i r i (t) = j i Gm i m j rij 2 ˆr ij, (1) were G is te universal gravitational constant and r ij = r j r i. Te n-body problem entails finding eac body s position as a function of time given te initial positions and initial velocities of te n masses. For te case of two masses, a general solution can be found analytically 1. For n 3, te problem permits no general solution, even in te restricted case of a test mass moving in te orbital plane of two circularly orbiting primaries (te planar circular restricted tree-body problem [PCR3BP]) 2. It comes as no surprise ten tat numerical metods play a vital role in studying celestial mecanical problems. Likewise, te PCR3BP is an ideal coice for a test bed upon wic a comparison of numerical metods can be made, owing to bot its simple formulation and its interesting dynamics. 1.1 Numerical metods Tis study focuses on te application of variational integrators to te PCR3BP. Te use of variational integrators, a class of numerical metods used to simulate mecanical systems, is of value because tey are known to exibit accurate beavior wit respect to te preservation of pysical constants of motion wen applied to systems wit conserved quantities like energy and momentum 3. Tey arise from approacing 1

2 mecanical systems from a variational mecanics standpoint. Particularly, a fundamental principle from classical mecanics states tat a mecanical system will evolve suc tat te so-called action integral T 0 L(q(t), q(t)) dt (2) is extremized subject to fixed endpoints q(0) and q(t ), were q is a generalized coordinate vector and L(q(t), q(t)) is te system s kinetic energy minus potential energy at time t. Calculus of variations ten sows tat tis olds only if q(t) satisfies te Euler-Lagrange equations: L q d L = 0. (3) dt q As an example, consider a particle moving in R 3 in te presence of a potential V (q), were q is te usual cartesian position vector (x, y, z). Ten L = 1 2 qt M q V (q) and equation (3) reduces to Newton s second law, force equals mass times acceleration: M q = V (q). (4) To derive a variational integrator, one discretizes te action integral (2) and uses a discrete version of te Euler-Lagrange equations (3) to define a map (q k 1, q k ) (q k, q k+1 ), were q k q(k) and is a time step. Tis map is ten applied recursively to a set of initial conditions (q 0, q 1 ) to produce a discrete curve of points {q k } T/ k=0 tat approximates q(t) over te time interval [0, T ]. Different variational integrators, possibly wit differing orders of accuracy, can be constructed using different quadrature metods for te discretization of te action integral (2) 4. Two distinguising features of constant time-stepping variational integrators applied to conservative mecanical systems are symplecticity (for systems wit one degree of freedom, tis property can be realized as preservation of area in pase space under te discrete Lagrangian map (q k 1, q k ) (q k, q k+1 )) and momentum conservation 5. As a basis for comparison in tis study, te results of applying variational integrators to te PCR3BP are compared wit tose arrived at troug applying te fourt-order Runge-Kutta metod 6, a popular numerical algoritm used to integrate ordinary differential equations of te form q = f(q, t). If te function f is time independent, te Runge-Kutta metod takes te form were q k+1 = q k a a a a 4, (5) a 1 = f(q k ) a 2 = f(q k a 1) a 3 = f(q k a 2) a 4 = f(q k + a 3 ), wic recursively generates a discrete curve of points {q k } T/ k=0 {q(k)}t/ k=0 from initial conditions q Te planar circular restricted tree-body problem (PCR3BP) Te PCR3BP considers te motion of a test mass m 3 = 0 in te presence of te gravitational field of two primary masses m 1 = 1 µ and m 2 = µ in circular orbits about teir center of mass. Witout loss of generality, all units are normalized and positions are defined relative to a rotating coordinate frame wose x-axis coincides wit te line joining m 1 and m 2 and wose origin coincides wit te center of mass of m 1 and m 2, as sown in Fig. 1. Te equations of motion for te test particle are ten 2 2

3 Figure 1: Rotating coordinate system in te planar circular restricted tree-body problem. All units are nondimensionalized. Te coordinate frame rotates counterclockwise wit unit angular frequency. were ẍ 2ẏ = Ω x ÿ + 2ẋ = Ω y, (7) Ω(x, y) = x2 + y 2 1 µ + 2 (x + µ)2 + y + µ 2 (x 1 + µ)2 + y + 1 µ(1 µ) (8) 2 2 and (x, y) denotes te position of m 3 in te rotating frame. It is straigtforward to ceck troug differentiation tat C(x, y, ẋ, ẏ) = 2Ω(x, y) (ẋ 2 + ẏ 2 ) (9) is a constant of motion for tis system. Tis value C is commonly referred to as te Jacobi integral 7. We sall refer to te constant E = C/2 as te energy of te system, taking care not to confuse E wit te sum of te test particle s kinetic and potential energies. In tis study we coose a mass parameter µ = , wic corresponds to te mass ratio for te Sun and Jupiter 8. Tus, solving te equations of motion (6-7) can be pysically interpreted as simulating te motion of a small mass (e.g. a spacecraft or a comet) in te presence of te gravitational field of te Sun-Jupiter system. 2 Results 2.1 Individual trajectories We begin wit an evaluation of te performance of variational integrators and te Runge-Kutta metod applied to calculating individual trajectories. Rater tan attempting to survey a wide range of orbits corresponding to different initial conditions, we focus ere an class of interesting trajectories called transit orbits. Consider expression (9) for te Jacobi constant C. Since C is constant and (ẋ 2 + ẏ 2 ) is a nonnegative quantity, it immediatelly follows tat m 3 is restricted to regions of te (x, y) plane were Ω(x, y) C/2. (10) An example of te allowed regions of motion for a system wit a given Jacobi constant C and mass parameter µ is sown in Fig. 2(a). Notice tat te example in Fig. 2(a) allows for transit orbits trajectories tat (6) 3

4 (a) (b) Figure 2: (a) Regions of allowed motion (wite areas) in te planar circular restricted tree-body problem wit µ = 0.1, C = (b) Grap of te negative of te effective potential Ω(x, y). transition between orbiting outside m 2 s orbit (te exterior region) and orbiting inside m 2 s orbit (te interior region). Tese types of orbits are mimicked in te solar system by comets suc as Oterma, wose orbit in te past century as experienced a resonance transition between an exterior and interior orbit and back wit respect to te Sun-Jupiter system 9. An understanding of te dynamics of orbits like tese can be applied, for example, to te design of low-fuel spacecraft mission trajectories 10. For a toroug discussion of transit orbits and procedures for constructing initial conditions tat generate transit orbits, see Koon et al. 8. In Appendix Figs. 7 and 8, te results of te integration of two sets of transit orbit initial conditions obtained from Koon et al. 8 are sown. For small time steps, te metods agree on te two trajectories associated wit te initial conditions. For larger time steps, te metods disagree. In te case of Appendix Fig. 7, te variational integrator predicts an orbit tat differs qualitatively from te resolved calculation, wile te Runge-Kutta metod generates an orbit consistent wit te resolved calculation. In Appendix Fig. 8, te opposite occurs: te variational integrator follows te bencmark calculation closely, wile te Runge-Kutta metod deviates significantly. In general, comparing individual trajectories reveals few patterns tat caracterize te performance of variational integrators versus te Runge-Kutta metod applied to te PCR3BP. As we ave just observed, one can find cases in wic a variational integrator appears to outperform te Runge-Kutta metod in calculating a trajectory, and vice-versa. In te following two sections, we examine some properties tat distinguis variational integrators from standard numerical metods. 2.2 Energy conservation Preserving constants of motion in numerical simulations bears importance in a variety of pysical situtations. A frictionless pendulum sould oscillate infinitely wit constant amplitude; an energy-conserving integrator will qualitatively reproduce tis beavior, wereas a dissipative integrator will predict decaying oscillation. Naturally, good energy beavior is favorable wen one seeks qualitative accuracy over long time spans. For mecanical systems wit conserved quantities tat arise from symmetries of te system (e.g. invariance of te Lagrangian under translation or rotation), a discrete version of Noeter s teorem from classical mecanics tat guarantees exact conservation of te corresponding momenta (e.g. linear momentum and angular momentum) accompanies te use of a variational integrator 5. Exact energy conservation, on te oter and, does not accompany te use of a constant time-stepping variational integrator for most systems of interest. Noneteless, variational integrators are known to exibit excellent energy beavior wen applied 4

5 Figure 3: Energy vs. time for a direct (counterclockwise) orbit around m 2. E 1 and E 2 correspond to energy levels tat separate different configurations of te forbidden regions, as sown in Appendix Fig. 6. to conservative mecanical systems 3. Fig. 3 sows te energy-conserving performance of a variational integrator compared to te Runge-Kutta algoritm applied to a direct (counterclockwise) orbit about m 2 wit a time step = For reference, energy levels tat correspond to critical values tat separate different configurations of te forbidden regions are plotted. Notice tat wile te variational integrator energy oscillates about its initial value, te Runge- Kutta energy decreases almost linearly. Moreover, te Runge-Kutta metod dissipates enoug energy to lead to canges in te configuration of te allowed regions of motion. Appendix Fig. 6 displays te forbidden region configurations associated wit tese lower energy levels. 2.3 Poincaré section plots It was mentioned previously tat Koon et al. 8 outline a procedure for numerically constructing initial conditions tat generate orbits wit prescribed itineraries. It turns out tat Poincaré sections play a key role in tis process, allowing one to identify regions of pase space associated wit selected trajectories. Likewise, Dellnitz et al. 11 demonstrate te power of identifing caotic regions and resonant regions in pase space wen computing transport rates. Indeed, te ability to obtain accurate pictures of te geometry of pase space as applications to a variety of celestial mecanical problems. To construct a Poincaré section plot, we select a two-dimensional region R = {(x, y, ẋ, ẏ) : C(x, y, ẋ, ẏ) = constant, q i = constant}, were q i {x, y}. We cover a portion of R wit a rectangular grid of points and ten integrate eac point forward in time, recording its position in pase space every time it intersects R. A projection of te recorded positions onto te (y, ẏ) plane for q i = x or te (x, ẋ) plane for q i = y yields a Poincaré surface-of-section plot tat gives information about te structure of pase space in te selected region. Appendix Figs sow a series of Poincaré section plots generated troug te use of variational and Runge-Kutta algoritms wit a range of time steps. In virtually all cases, te variational integrator outperforms te Runge-Kutta metod in capturing te geometry of pase space. For small time steps, te metods rougly agree on te structure of te Poincaré section. For larger time steps, resonant tori deteriorate and te plotted region appears almost entirely caotic wit te use of te Runge-Kutta algoritm. Wit te use of a variational integrator, resonant tori persist for large time steps and regions of caos and stability are easily distinguisable. Note, owever, tat in some cases (e.g. Appendix Figs. 10(b) and 11(b)) te resonant regions predicted by te variational integrator differ from tose predicted by te resolved 5

6 (a) (b) Figure 4: (a) Transport rate (interior to exterior) for initial positions uniformly distributed trougout te interior region wit C = Initial velocities were selected suc tat velocity vectors were oriented counterclockwise at angles nearly perpendicular to teir position vectors, wit a random deviation angle δ [ π 4, π 4 ] from perpendicular. (b) Transport (interior to exterior) as a function of time for fixed time step = calculation in bot teir appearance and location, even for small time steps. Te success of variational integrators and oter symplectic integrators at producing accurate pictures of te geometry of pase space is not unique to te PCR3BP. Indeed, tis penomenon as been observed for a variety of systems 12,13 and was popularized wit a publication by Cannel & Scovel 14 in For an analytical treatment of tis topic, see Hairer, Lubic & Wanner Transport rates Te study of transport in te tree-body problem involves scientists in a surprisingly wide range of fields. For astronomers, an understanding transport penomena plays a key role in studying asteroid escape rates, collision probabilities, and transport of material trougout te solar system 11,15,16. On a muc smaller scale, transport teory in te tree-body problem is of interest to cemists, as it sares an intimate link wit reaction rate teory in molecular dynamics 16. Wit te growing popularity of implementing teoretical metods to determine transport rates, te need for accurate and computationally inexpensive integration algoritms tat can be used to verify teoretical predictions grows accordingly. In tis study, we consider te transport of particles (tat are in counterclockwise orbit about te Sun and ave a specified energy) from te interior region to te exterior region wit respect to te Sun-Jupiter system. As motivation for tis analysis, recall from Section 2.3 tat for large time steps, te Runge-Kutta algoritm generally gives spurious results wen predicting pase space structures: te metod predicts caotic seas were bencmark calculations do not. Te presence of additional caotic regions makes plausible te ypotesis tat te Runge-Kutta metod is likely to predict excessively ig transport rates. Fig. 4(a) sows te results of Monte-Carlo simulations applied to te above transport problem for various time steps. Te variational integrator and te Runge-Kutta metod give nearly indistinguisable results. Bot metods give unreliable transport rates wit large time steps but appear to asymptotically approac an identical limit. Fixing and plotting te transport of particles as a function of time (Fig. 4(b)) reveals little more; te two metods give matcing results tat differ sligtly from te bencmark calculation. Before discarding te suggestion tat Runge-Kutta s lack of structure preservation migt contribute to erroneous predictions of transport rates, recall tat Section 2.2 reveals a case in wic Runge-Kutta s energy dissipation leads to canges in te configuration of m 3 s allowed regions of motion. It may be tat numerical dissipitation is sealing sut te interior region (as sown in Appendix Fig. 6) for some particles in te Monte-Carlo simulation, counterbalancing te influence of Runge-Kutta s introduction of artificial caos. 6

7 3 Conclusions and Furter Study Variational integrators were constructed and compared wit te fourt-order Runge-Kutta metod via application to te PCR3BP. Computational tests verified te superiority of variational integrators over te Runge-Kutta metod in conserving energy and producing accurate Poincarè section plots, but revealed no significant differences between te two algoritms ability to predict individual trajectories and calculate transport rates. Several components of tis study ave immediate possibility for furter study. From a computational standpoint, introducting adaptive time-stepping scemes to te algoritms would be an appropriate complement to tis study. From a celestial mecanical point of view, an extension to tree dimensions (removing te restriction tat m 3 lie in te orbital plane of te primaries) seems more tan fitting. It would also be of interest to furter pursue te issue of transport rate simulations; te ypoteses posed in Section 2.4 are still open questions. 4 Metods 4.1 Derivation of te PCR3BP equations of motion Consider Newton s equations of motion for te full tree-body problem in tree dimensions: m 1 r 1 (t) = Gm 1m 2 r 2 12 m 2 r 2 (t) = Gm 2m 1 r 2 21 m 3 r 3 (t) = Gm 3m 1 r 2 31 ˆr 12 + Gm 1m 3 r13 2 ˆr 13 (11) ˆr 21 + Gm 2m 3 r23 2 ˆr 23 (12) ˆr 31 + Gm 3m 2 r32 2 ˆr 32. (13) Dividing te above equations by m 1, m 2, and m 3, respectively, and letting m 3 approac zero, te equations of motion (11-13) approac r 1 (t) = Gm 2 r12 2 ˆr 12 (14) r 2 (t) = Gm 1 r21 2 ˆr 21 (15) r 3 (t) = Gm 1 r 2 31 ˆr 31 + Gm 2 r32 2 ˆr 32. (16) Notice tat (14) and (15) are merely te equations of motion for te two-body problem, wic can be solved analytically 1. Te problem ten reduces to finding te motion of m 3 in te presence of two bodies wit prescribed orbits. In particular, if m 1 and m 2 are in circular orbit about teir center of mass and te motion of m 3 is restricted to te plane of teir orbit, te problem reduces to one wit two degrees of freedom, te PCR3BP. To study te PCR3BP, coose units suc tat m 1 + m 2 = 1, G = 1, and r 12 = 1. Let m 2 = µ. Now define a counterclockwise rotating coordinate frame wose origin is at te center of mass of m 1 and m 2 and wose x-axis coincides wit te vector r 12, wit m 2 on te positive x-axis. Under te restriction tat m 1 and m 2 ave circular orbits, it follows tat te angular frequency of tis rotating frame is 1 and tat te positions of m 1 = 1 µ and m 2 = µ in tis coordinate system are ( µ, 0) and (1 µ, 0), respectively. See Fig. 1 for a visual aid. Let (x, y) denote te position of m 3 in te rotating frame, and let ( x, ȳ) denote te position of m 3 relative to an inertial frame wose origin is at te center of mass of m 1 and m 2 and wose axes coincide wit tose of te rotating frame at t = 0. It follows tat ( ) ( ) ( x cos t sin t x =. (17) ȳ) sin t cos t y 7

8 Differentiation wit respect to t yields ( ) x = ȳ (ẋ ) cos t x sin t ẏ sin t y cos t. (18) ẋ sin t + x cos t + ẏ cos t y sin t Te kinetic energy of m 3 is tus equal to T (x, y, ẋ, ẏ) = 1 ( x ȳ ) ( ) ( ) m 3 0 x 2 0 m 3 ȳ = 1 (ẋ ) T ( ) (ẋ ) cos t x sin t ẏ sin t y cos t m3 0 cos t x sin t ẏ sin t y cos t 2 ẋ sin t + x cos t + ẏ cos t y sin t 0 m 3 ẋ sin t + x cos t + ẏ cos t y sin t = 1 2 m 3((ẋ y) 2 + (ẏ + x) 2 ). (19) Te potential energy of m 3 is equal to V (x, y) = Gm 1m 3 r 13 Gm 2m 3 r 23 = m 3(1 µ) (x + µ)2 + y 2 m 3 µ (x 1 + µ)2 + y 2. (20) Substituting L = T V into te Euler-Lagrange equations (3) and simplifying, we arrive at te equations of motion for m 3 : ẍ 2ẏ = x ÿ + 2ẋ = y (1 µ)(x + µ) ((x + µ) 2 + y 2 ) µ(x 1 + µ) 3/2 ((x 1 + µ) 2 + y 2 ) 3/2 (21) (1 µ)y ((x + µ) 2 + y 2 ) µ y. 3/2 ((x 1 + µ) 2 + y 2 ) 3/2 (22) Te rigt-and side of equations (21) and (22) can be written as te gradient of an effective potential Ω(x, y), were 1 Ω = x2 + y 2 1 µ + 2 (x + µ)2 + y + µ 2 (x 1 + µ)2 + y + 1 µ(1 µ). (23) 2 2 Te equations of motion (21-22) for te PCR3BP are ten equivalent to (6-7). 4.2 Construction of variatational integrators To construct a variational integrator tat models a mecanical system, we write down an expression L(q(t), q(t)) tat equals te difference between kinetic and potential energies of te system in terms of a set of coordinates q. Variational mecanics tells us tat q(t) must extremize te integral T L(q(t), q(t)) dt subject to 0 fixed endpoints q(0) and q(t ) if it is a solution trajectory of te system. From a discrete standpoint, tis is equivalent to extremizing te sum N 1 k=0 L d (q k, q k+1, ) (24) subject to fixed endpoints q 0 and q N, were {q k } N k=0 is a discrete curve of points tat approximates te exact solution q(t), is a time step, and L d (q k, q k+1, ) is an approximation of te action integral over te time interval [k, (k + 1)]: L d (q k, q k+1, ) It can ten be sown 3 tat tis sum is extremized if (k+1) k L(q(t), q(t)) dt. (25) D 2 L d (q k 1, q k, ) + D 1 L d (q k, q k+1, ) = 0 (26) 1 By convention, te constant 1 µ(1 µ) is added to te effective potential Ω(x, y) so tat a particle at rest at eiter of te 2 equilateral Lagrange points (equilibrium points in te PCR3BP) as a Jacobi constant C = 3. 8

9 for every k, were D i symbolizes te derivative of L d wit respect to te i t argument of L d. Tese results, analogous to te Euler-Lagrange equations from variational calculus, are appropriately named te discrete Euler-Lagrange equations. Tey define a recursive relation for finding q k from a set of initial conditions (q 0, q 1 ) tat can be used to construct algoritms tat model mecanical systems. To integrate te equations of motion (6-7), tree variational integrators were constructed using te discrete least action principle described above. Before detailing teir structure, note tat te use of a rotating coordinate system generates ambiguity wit respect to initial conditions. We sall assume tat all initial conditions are given in te form ( x(0), ȳ(0), x(0), ȳ(0)), i.e. relative to an inertial frame wose axes coincide wit te rotating frame at t = 0. For algoritms tat require initial conditions in te rotating frame format (x(0), y(0), ẋ(0), ẏ(0)), te following relation is used: ( x(0), ȳ(0), x(0), ȳ(0)) = (x, y, ẋ y, ẏ + x). (27) Tis equation can be derived by evaluating (17) and (18) at t = 0. For algoritms requiring initial conditions wit velocities replaced by momenta, we multiply x(0) and ȳ(0) by m 3. Note, owever, tat in all of te algoritms tat follow, no recursive relations contain m 3 upon simplification and cancellation of common factors. Te first two programs constructed use explicit variational integrator algoritms derived from te use of a rectangular and trapezoidal quadrature rules. In te PCR3BP, te Lagrangian as te form L(x, y, ẋ, ẏ) = 1 2 m 3((ẋ y) 2 + (ẏ + x) 2 ) + m 3(1 µ) (x + µ)2 + y 2 + m 3 µ (x 1 + µ)2 + y 2. (28) To implement a rectangle rule variational inegrator, we coose a discrete Lagrangian tat approximates (ẋ, ẏ) wit ( q k+1 q k ) = ( x k+1 x k, y k+1 y k ) and ten approximate (k+1) L(q, q) dt wit te lengt of te k time interval times L evaluated at te left endpoint of te interval: ( L r d(q k, q k+1, ) = L q k, q ) k+1 q k ( ( ) 2 1 = 2 m xk+1 x k 3 y k + 1 ( ) ) 2 2 m yk+1 y k 3 + x k V (x k, y k ). (29) Here, q is te position vector (x, y). Before proceeding to substitute L r d equations (26), expand (26) into its component form: into te discrete Euler-Lagrange L d (q k 1, q k, ) + L d (q k, q k+1, ) x k x k = 0 (30) L d (q k 1, q k, ) + L d (q k, q k+1, ) y k y k = 0. (31) Ten define te x-component of m 3 s momentum at step k as Using (30), we can rewrite (32) as Canging te indices in (32), we can write p x k = L r x d(q k 1, q k, ) (32) k ( ) xk x k 1 = m 3 y k 1. (33) p x k = x k L r d(q k, q k+1, ). (34) p x k+1 = x k+1 L r d(q k, q k+1, ). (35) 9

10 Equations (34-35) and teir y-component anologues p y k = L r y d(q k, q k+1, ) k (36) p y k+1 = L r y d(q k, q k+1, ) k+1 (37) form a set of recursive relations tat can be used to calculate a trajectory from initial conditions (x(0), y(0), p x (0), p y (0)). Specifically, we solve te simultaneous equations (34) and (36) for x k+1 and y k+1, use equations (35) and (37) to compute p x k+1 and py k+1, and repeat for te desired number of iterations. Te curve of points {q k } t f / k=0 calculated in tis fasion automatically satisfies te discrete Euler-Lagrange equations (30-31) by virtue of te manner in wic p x k and py k were defined. To implement a trapezoidal rule variational inegrator, we coose a discrete Lagrangian wic approximates (ẋ, ẏ) wit q k+1 q k = ( x k+1 x k, y k+1 y k ) and take a weigted sum of L(q k, q k+1 q k ) and L(q k+1, q k+1 q k ): L tr d (q k, q k+1, ) = ( 2 L q k, q k+1 q k ) + 2 ( L q k+1, q k+1 q k ( = ( ) m xk+1 x k 3 y k ( ) ) 2 2 m yk+1 y k 3 + x k+1 V (x k+1, y k+1 ) ( + ( ) m xk+1 x k 3 y k + 1 ( ) ) 2 2 m yk+1 y k 3 + x k V (x k, y k ). (38) We ten use equations (34-37) (replacing L r d wit Ltr d ) to calculate a trajectory in te manner described in te previous paragrap. Te final variational integrator constructed uses a midpoint quadrature rule for a discrete Lagrangian defined as follows: ( L mp d (q qk + q k+1 k, q k+1, ) = L, q ) k+1 q k 2 ( ( 1 = 2 m xk+1 x k 3 y k + y k+1 2 ( xk + x k+1 V, y k + y k ) ) ( 2 m yk+1 y k 3 + x ) ) 2 k + x k+1 2 ). (39) Replacement of L r d wit Lmp d in (34-37) leads to equations tat cannot be solved for x k+1 and y k+1 explicitly, as was possible wit te rectangle and trapezoidal quadrature rules. We terefore invoke Newton s metod at eac step to find x k+1 and y k+1 : we introduce te vector x k L mp d (q k, q k+1, ) p x k g(p k, q k, q k+1 ) = (40) d (q k, q k+1, ) p y k and te Jacobian matrix J = y k L mp g 1 g 1 x k+1 y k+1 g 2 x k+1 g 2 y k+1, (41) were g i is te i t component of g. We coose an initial guess q k+1 wit a roug estimate Ten we apply te recursive formula q k+1 (1) = q k. (42) q k+1 (n + 1) = q k+1 (n) J 1 (p k, q k, q k+1 (n))g(p k, q k, q k+1 (n)) (43) 10

11 until a fixed point is reaced. q k+1 can ten be used in (35) and (37) to find p k+1, and te process begins again at (34). In te comparisons made between variational integrators and te Runge-Kutta metod in Section 2, te trapezoidal variational integrator (38) is compared wit te fourt-order Runge-Kutta metod (5) using equal time steps. In doing so, we avoid te use of a computationally expensive implicit metod (te midpoint variational integrator) witout sacrificing te algoritm s order of accuracy (te rectangle variational integrator is first-order accurate, wile te midpoint and trapezoidal variational integrators are bot second-order accurate). 5 References 1. Goldstein, H., Poole, C. & Safko, J. Classical Mecanics (Addison Wesley, San Fransisco, 2002). 2. Szebeely, V. G. Teory of Orbits: Te Restricted Problem of Tree Bodies (Academic Press, New York, 1967). 3. Marsden, J. E. & West, M. Discrete mecanics and variational integrators. Acta Numerica 10, (2001). 4. Lew, A., Marsden, J. E., Oritz, M. & West, M. Variational time integrators. International Journal for Numerical Metods in Engineering 60, (2004). 5. Lew, A., Marsden, J. E., Oritz, M. & West, M. An overview of variational integrators. Finite Element Metods: 1970 s and Beyond, (2004). 6. Weisstein, E. W. Runge-Kutta Metod. From MatWorld A Wolfram Web Resource. ttp://matworld.wolfram.com/runge-kuttametod.tml 7. Valtonen, M. & Karttunen, H. Te Tree-Body Problem (Cambridge University Press, Cambridge, 2006). 8. Koon, W. S., Lo, M., Marsden, J. E. & Ross, S. Heteroclinic connections between periodic orbits and resonance transitions in celestial mecanics. Caos 10, (2000). 9. Koon, W. S., Lo, M. W., Marsden, J. E. & Ross, S. D. Dynamical systems, te tree-body problem, and space mission design. International Conference on Differential Equations, Berlin, 1999, (Fiedler, B., Gröger, K. & Sprekels, J. eds.), World Scientific, (2000). 10. Marsden, J. E. & Ross, S. D. New metods in celestial mecanics and mission design. Bulletin of te American Matematical Society 43, (2005). 11. Dellnitz, M. et al. Transport of Mars-crossing asteroids from te quasi-hilda region. Pysical Review Letters 94, 1-4 (2005). 12. Hairer, E., Lubic, C. & Wanner, G. Geometric Numerical Integration : Structure-Preserving Algoritms for Ordinary Differential Equations (Springer Series in Computational Matematics, Berlin, 2002). 13. Rowley, C. W. & Marsden, J. E. Variational integrators for degenerate Lagrangians, wit application to point vortices. 41st IEEE Conference on Decision and Control (2002). 14. Cannel, P. J. & Scovel, C. Symplectic integration of Hamiltonian systems. Nonlinearity 3, (1990). 15. Jaffé, C. et al. Statistical teory of asteroid escape rates. Pysical Review Letters 89, (2002). 16. Dellnitz, M. et al. Transport in dynamical astronomy and multibody problems. International Journal of Bifurcation and Caos 15, (2005). 11

12 6 Acknowledgments I wis to tank Dr. Jerrold Marsden for providing me te opportunity to work under is direction tis past summer. I tank Pilip Du Toit for providing me guidance and instruction trougout te summer. Tanks also to Nawaf Bou-Rabee for providing toroug responses to eac of my inquiries via . Finally, I tank te SURF program and te Axline benefactors for teir financial support. (Report continues on following page) 12

13 7 Appendix Figure 5: Forbidden regions for energy values in te range of tose displayed in Fig. 3, wit µ = (a) (b) (c) Figure 6: Close up of boxed region in Fig. 5 for (a) E > E 2, (b) E 2 > E > E 1, and (c) E < E 1. 13

14 (a) (b) (c) Figure 7: Transit orbit wit initial conditions x(0) = 1 µ, y(0) = , ẋ(0) > 0, ẏ(0) = 0.085, C = Bot forward and backward integrations of te initial conditions are displayed as single continuous trajectories. Rigt column sows zoomed views of boxed regions in left column. (a) Bencmark calculation. (b) Variational integrator, = (c) Runge-Kutta, =

15 (a) (b) (c) Figure 8: Transit orbit wit initial conditions identical to tose in Fig. 7, only tis time wit y(0) = Bot forward and backward integrations of te initial conditions are displayed as single continuous trajectories. Rigt column sows zoomed views of boxed regions in left column. (a) Bencmark calculation. (b) Variational integrator, = (c) Runge-Kutta, =

16 (a) (b) (c) (d) Figure 9: Poincaré cut at x < 0, y = 0, wit ẏ > 0, C = Variational integrator outputs are sown in te left column, wile Runge-Kutta outputs are sown in te rigt column. (a) Bencmark calculation. (b) = (c) = (d) =

17 (a) (b) (c) (d) Figure 10: Poincaré cut at x = 1 µ, y < 0, wit ẋ < 0, C = Variational integrator outputs are sown in te left column, wile Runge-Kutta outputs are sown in te rigt column. (a) Bencmark calculation. (b) = (c) = (d) =

18 (a) (b) (c) (d) Figure 11: Poincaré cut at x < 0, y = 0, wit ẏ < 0, C = Variational integrator outputs are sown in te left column, wile Runge-Kutta outputs are sown in te rigt column. (a) Bencmark calculation. (b) = (c) = (d) =

19 (a) (b) (c) (d) Figure 12: Poincaré cut at x = 1 µ, y < 0, wit ẋ > 0, C = Variational integrator outputs are sown in te left column, wile Runge-Kutta outputs are sown in te rigt column. (a) Bencmark calculation. (b) = (c) = (d) =

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