Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics (original) (raw)
Research Article| June 01 2000
Control and Dynamical Systems, Caltech 107-81, Pasadena, California 91125
Search for other works by this author on:
Navigation and Mission Design, Jet Propulsion Laboratory M/S: 301-142, 4800 Oak Grove Drive, Pasadena, California 91109-8099
Search for other works by this author on:
Control and Dynamical Systems, Caltech 107-81, Pasadena, California 91125
Search for other works by this author on:
Control and Dynamical Systems, Caltech 107-81, Pasadena, California 91125
Search for other works by this author on:
In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the “interior” and “exterior” Hill’s regions and other resonant phenomena.
REFERENCES
P.
Holmes
, “
Poincaré, celestial mechanics, dynamical systems theory and chaos
,”
Phys. Rep.
193
,
137
–
163
(
1990
).
C. Simó, “Dynamical systems methods for space missions on a vicinity of collinear libration points,” Hamiltonian Systems with Three or More Degrees of Freedom, edited by C. Simó (Kleuwer Academic, Dordrecht, 1999), pp. 223–241.
E.
Belbruno
and
B.
Marsden
, “
Resonance hopping in comets
,”
Astron. J.
113
,
1433
–
1444
(
1997
).
M. Lo and S. Ross, “SURFing the solar system: Invariant manifolds and the dynamics of the solar system,” JPL IOM 312/97, 1997, pp. 2–4.
J.
Llibre
,
R.
Martinez
, and
C.
Simó
, “
Transversality of the invariant manifolds associated to the Lyapunov family of periodic orbits near L2 in the restricted three-body problem
,”
J. Diff. Eqns.
58
,
104
–
156
(
1985
).
M. Lo et al., Genesis Mission Design, Paper No. AIAA 98-4468, 1998.
C.
Conley
, “
On some new long periodic solutions of the plane restricted three body problem
,”
Commun. Pure Appl. Math.
16
,
449
–
467
(
1963
).
C.
Conley
, “
Low energy transit orbits in the restricted three-body problem
,”
SIAM (Soc. Ind. Appl. Math.) J. Appl. Math.
16
,
732
–
746
(
1968
).
R. P. McGehee, “Some homoclinic orbits for the restricted three-body problem,” Ph.D. thesis, University of Wisconsin, 1969.
J. Moser, Stable and Random Motions in Dynamical Systems with Special Emphasis on Celestial Mechanics (Princeton University Press, Princeton, 1973).
R. Abraham and J. E. Marsden, Foundations of Mechanics, 2nd ed. (Addison-Wesley, New York, 1978).
K. R. Meyer and R. Hall, Hamiltonian Mechanics and the n-Body Problem (Springer-Verlag, Applied Mathematical Sciences, Berlin, 1992).
J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Texts in Applied Mathematics, 2nd ed. (Springer-Verlag, Berlin, 1994), Vol. 17.
V. Szebehely, Theory of Orbits (Academic, New York, 1967).
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Texts in Applied Mathmatics Science (Springer-Verlag, Berlin, 1990), Vol. 2.
G. Gómez, A. Jorba, J. Masdemont, and C. Simó, Study Refinement of Semi-Analytical Halo Orbit Theory, Final Report, ESOC Contract No.: 8625/89/D/MD(SC), Barcelona, April, 1991.
S. Wiggins, “Global dynamics, phase space transport, orbits homoclinic to resonances, and applications,” Fields Institute Monographs, American Mathematical Society, 1993.
R. L. Devaney, “Singularities in classical mechanical systems,” Ergodic Theory and Dynamical Systems, edited by A. Katok (Birkhäuer, Basel, 1981), pp. 211–333.
M. Lo and S. Ross, “Low energy interplanetary transfers using invariant manifolds of L1, L2 and halo orbits,” AAS/AIAA Space Flight Mechanics Meeting, Monterey, California, 9–11 February 1998.
D. F. Lawden, Optimal Trajectories for Space Navigation (Butterworth, London, 1963).
W. S.
Koon
and
J. E.
Marsden
, “
Optimal control for holonomic and nonholonomic mechanical systems with symmetry and Lagrangian reduction
,”
SIAM J. Control Optim.
35
,
901
–
929
(
1997
).
A. M.
Bloch
and
J. E.
Marsden
, “
Controlling homoclinic orbits
,”
Theor. Comput. Fluid Dyn.
1
,
179
–
190
(
1989
).
A. M.
Bloch
,
N.
Leonard
, and
J. E.
Marsden
, “
Stabilization of mechanical systems using controlled Lagrangians
,”
Proc. CDC
36
,
2356
–
2361
(
1997
).
J. M.
Wendlandt
and
J. E.
Marsden
, “
Mechanical integrators derived from a discrete variational principle
,”
Physica D
106
,
223
–
246
(
1997
).
C.
Kane
,
J. E.
Marsden
, and
M.
Ortiz
, “
Symplectic energy momentum integrators
,”
J. Math. Phys.
40
,
3353
–
3371
(
1999
).
C. Kane, J. E. Marsden, M. Ortiz, and M. West, “Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems, Int. J. Num. Math. Engin. (to appear).
J. E.
Marsden
and
J.
Scheurle
, “
Pattern evocation and geometric phases in mechanical systems with symmetry
,”
Dyn. Stab. Syst.
10
,
315
–
338
(
1995
).
J. E. Marsden, J. Scheurle, and J. Wendlandt, “Visualization of orbits and pattern evocation for the double spherical pendulum,” ICIAM 95: Mathematical Research, edited by K. Kirchgässner, O. Mahrenholtz, and R. Mennicken (Academic-Verlag, Berlin, 1996), Vol. 87, pp. 213–232.
S.
Dermott
et al., “
A circumsolar ring of asteroidal dust in resonant lock with the Earth
,”
Nature (London)
369
,
719
–
723
(
1994
).
E.
Belbruno
and
J.
Miller
, “
Sun-perturbed earth-to-moon transfers with ballistic capture
,”
J. Guid. Control Dyn.
16
,
770
–
775
(
1993
).
A. M.
Bloch
,
P. S.
Krishnaprasad
,
J. E.
Marsden
, and
G.
Sánchez de Alvarez
, “
Stabilization of rigid body dynamics by internal and external torques
,”
Automatica
28
,
745
–
756
(
1992
).
P. J.
Enright
and
B. A.
Conway
, “
Discrete approximations to optimal trajectories using direct transcription and nonlinear programming
,”
J. Guid. Control Dyn.
15
,
994
–
1002
(
1992
).
A. L.
Herman
and
B. A.
Conway
, “
Direct optimization using collocation based on high-order Gauss–Lobatto quadrature rules
,”
J. Guid. Control Dyn.
19
,
592
–
599
(
1996
).
L.
Hiday-Johnston
and
K.
Howell
, “
Transfers between libration-point orbits in the elliptic restricted problem
,”
Celest. Mech. Dyn. Astron.
58
,
317
–
337
(
1994
).
C.
Howell
,
B.
Barden
, and
M.
Lo
, “
Application of dynamical systems theory to trajectory design for a libration point mission
,”
J. Astronaut. Sci.
45
,
161
–
178
(
1997
).
J. E.
Marsden
,
O. M.
O’Reilly
,
F. J.
Wicklin
, and
B. W.
Zombro
, “
Symmetry, stability, geometric phases, and mechanical integrators
,”
Nonlinear Sci. Today
1
,
4
–
11
, (
1991
);
J. E.
Marsden
,
O. M.
O’Reilly
,
F. J.
Wicklin
, and
B. W.
Zombro
,
Nonlinear Sci. Today
1
,
14
–
21
, (
1991
).
C. K. McCord, K. R. Meyer, and Q. Wang, The Integral Manifolds of the Three Body Problem, Mem. Am. Math. Soc. (1998), Vol. 628.
R. P.
McGehee
, “
Triple collision in the collinear three-body problem
,”
Invent. Math.
27
,
191
–
227
(
1974
).
R. W.
Easton
and
R.
McGehee
, “
Homoclinic phenomena for orbits doubly asymptotic to an invariant three-sphere
,”
Indiana Univ. Math. J.
28
,
211
–
240
(
1979
).
J. E. Marsden, Lectures on Mechanics, London Mathematical Society Lecture Note Series (Cambridge University Press, Cambridge, 1992), Vol. 174.
R. P. McGehee, “Parabolic orbits in the three-body problem. Dynamical systems,” Proceedings Symposium, of the University Bahia, Salvador, 1971 (Academic, New York, 1973), pp. 249–251.
M.
Henon
, “
Chaotic scattering modelled by an inclined billiard
,”
Physica D
33
,
132
–
156
(
1988
).
This content is only available via PDF.
© 2000 American Institute of Physics.
2000
American Institute of Physics
You do not currently have access to this content.
Sign in
Sign In
You could not be signed in. Please check your credentials and make sure you have an active account and try again.
Username ?
Password
Pay-Per-View Access
$40.00