Abstract
We present a procedure for the normalization of perturbed Keplerian problems in n dimensions based on Moser regularization of the Kepler problem and the invariants associated to the reduction process. The approach allows us not only to circumvent the problems introduced by certain classical variables used in the normalization of this kind of problems, but also to do both the normalization and reduction in one step. The technique is introduced for any dimensions and is illustrated for n = 2, 3 by relating Moser coordinates with Delaunay-like variables. The theory is applied to the spatial circular restricted three-body problem for the study of the existence of periodic and quasi-periodic solutions of rectilinear type.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Bacry, H., Ruegg, H., and Souriau, J.-M., Dynamical Groups and Spherical Potentials in Classical Mechanics, Comm. Math. Phys., 1966, vol. 3, no. 5, pp. 323–333
Belbruno, E.A., A New Family of Periodic Orbits for the Restricted Problem, Celestial Mech., 1981, vol. 25, no. 2, pp. 195–217
Benevieri, P., Gavioli, A., and Villarini, M., Existence of Periodic Orbits for Vector Fields via Fuller Index and the Averaging Method, Electron. J. Differential Equations, 2004, vol. 2004, no. 128, 14 pp.
Besse, A. L., Manifolds All of Whose Geodesics Are Closed, Ergeb. Math. Grenzgeb., vol. 93, Berlin: Springer, 1978.
Brouwer, D. and Clemence, G. M., Methods of Celestial Mechanics, New York: Acad. Press, 1961.
Chenciner, A. and Llibre, J., A Note on the Existence of Invariant Punctured Tori in the Planar Circular Restricted Three-Body Problem, Ergodic Theory Dynam. Systems, 1988, vol. 8, pp. 63–72.
Cordani, B., The Kepler Problem: Group Theoretical Aspects, Regularization and Quantization, with Application to the Study of Perturbations, Progr. Math. Phys., vol. 29, Basel: Birkhäuser, 2003.
Cushman, R., Reduction, Brouwer’s Hamiltonian, and the Critical Inclination, Celestial Mech., 1983, no. 31, no. 4, pp. 401–429. (Erratum: Reduction, Brouwer’s Hamiltonian, and the Critical Inclination, Celestial Mech., 1984, vol. 33, no. 3, p. 297.)
Cushman, R., Normal Forms for Hamiltonian Vector Fields with Periodic Flow, in Differential Geometric Methods in Mathematical Physics, Sh. Sternberg (Ed.), Dordrecht: Reidel, 1984, pp. 125–144.
Cushman, R., A Survey of Normalization Techniques Applied to Perturbed Keplerian Systems, in Dynamics Reported, C. K. R. T. Jones, U. Kirchgraber, and H. O. Walther (Eds.), Berlin: Springer, 1992, pp. 54–112.
Cushman, R. H. and Sadovskií, D. A., Monodromy in the Hydrogen Atom in Crossed Fields, Phys. D, 2000, vol. 142, nos. 1–2, pp. 166–196.
Cushman, R. and Sanders, J. A., The Constrained Normal Form Algorithm, Celestial Mech., 1988/1989, vol. 45, nos. 1–3, pp. 181–187.
Deprit, A., Canonical Transformations Depending on a Small Parameter, Celestial Mech., 1969/1970, vol. 1, pp. 12–30
Deprit, A., The Elimination of the Parallax in Satellite Theory, Celestial Mech., 1981, vol. 24, no. 2, pp. 111–153
Dubrovin, B.A., Fomenko, A. T., and Novikov, S.P., Modern Geometry — Methods and Applications: Part 2. The Geometry and Topology of Manifolds, Grad. Texts in Math., vol. 104, New York: Springer, 1985.
Féjoz, J., Averaging the Planar Three-Body Problem in the Neighborhood of Double Inner Collisions, J. Differential Equations, 2001, vol. 175, no. 1, pp. 175–187
Ferrer, S. and Lara, M., Families of Canonical Transformations by Hamilton–Jacobi–Poincaré Equation: Application to Rotational and Orbital Motion, J. Geom. Mech., 2010, vol. 2, no. 3, pp. 223–241
Fock, V., Zur Theorie des Wasserstoffatoms, Z. Phys., 1935, vol. 98, nos. 3–4, pp. 145–154.
Goursat, M. E., Les transformations isogonales en Mécanique, Comptes rendus hebdomadaires des séances de l’Académie des sciences, 1889, vol. 108, pp. 446–448
Györgyi, G., Kepler’s Equation, Fock Variables, Bacry’s Generators and Dirac Brackets, Il Nuovo Cimento A, 1968, vol. 53, no. 3, pp. 717–736
Han, Y., Li, Y., and Yi, Y., Invariant Tori in Hamiltonian Systems with High Order Proper Degeneracy, Ann. Henri Poincaré, 2010, vol. 10, no. 8, pp. 1419–1436
Han, M. Y. and Stehle, P., SU2 As a Classical Invariance Group, Il Nuovo Cimento A (10), 1967, vol. 48, pp. 180–187
Heckman, G. and de Laat, T., On the Regularization of the Kepler problem, J. Symplectic Geom., 2012, vol. 10, no. 3, pp. 463–473
Henrard, J., Virtual Singularities in the Artificial Satellite Theory, Celestial Mech., 1974, vol. 10, pp. 437–449.
Howison, R. C. and Meyer, K. R., Doubly-Symmetric Periodic Solutions of the Spatial Restricted Three-Body Problem, J. Differential Equations, 2000, vol. 163, no. 1, pp. 174–197
Kustaanheimo, P. and Stiefel, E., Perturbation Theory of Kepler Motion Based on Spinor Regularization, J. Reine Angew. Math., 1965, vol. 218, pp. 204–219
Levi-Civita, T., Sur la régularisation du problème des trois corps, Acta Math., 1920, vol. 42, no. 1, pp. 99–144.
Ligon, T. and Schaaf, M., On the Global Symmetry of the Classical Kepler Problem, Rep. Mathematical Phys., 1976, vol. 9, no. 3, pp. 281–300
Marchal, C., Collisions of Stars by Oscillating Orbits of the Second Kind, Acta Astronaut., 1978, vol. 5, no. 10, pp. 745–764
Marsden, J. and Weinstein, A., Reduction of Symplectic Manifolds with Symmetry, Rep. Mathematical Phys., 1974, vol. 5, no. 1, pp. 121–130
Meyer, K. R., Symmetries and Integrals in Mechanics, in Dynamical Systems: Proc. Sympos. (Univ. Bahia, Salvador, 1971), M. M. Peixoto (Ed.), New York: Acad. Press, 1973, pp. 259–272.
Meyer, K. R. and Offin, D., Introduction to Hamiltonian dynamical systems and the N-Body Problem, 3rd ed., Appl. Math. Sci., vol. 90, New York: Springer, 2017.
Meyer, K. R., Palacián, J. F., and Yanguas, P., Invariant Tori in the Lunar Problem, Publ. Mat., 2014, vol. 58, suppl., pp. 353–394.
Milnor, J., On the Geometry of the Kepler Problem, Amer. Math. Monthly, 1983, vol. 90, no. 6, pp. 353–365.
Milnor, J. and Stasheff, J. D., Characteristic Classes, Ann. Math. Stud., No. 76, Princeton: Princeton Univ. Press, 1974.
Moser, J., Regularization of Kepler’s Problem and the Averaging Method on a Manifold, Comm. Pure Appl. Math., 1970, vol. 23, pp. 609–636
Moser, J. and Zehnder, E., Notes on Dynamical Systems, Courant Lect. Notes Math., vol. 12, Providence, R.I., 2005.
Noether, E., Invariante Variationsprobleme, Nachr. v. d. Gesellsch. d. Wiss. zu Göttingen, Math.-Phys. Klasse, 1918, vol. 1918, pp. 235–257
Osácar, C. and Palacián, J., Decomposition of Functions for Elliptic Orbits, Celestial Mech. Dynam. Astronom., 1994, vol. 60, no. 2, pp. 207–223
Palacián, J. F., Normal Forms for Perturbed Keplerian Systems, J. Differential Equations, 2002, vol. 180, no. 2, pp. 471–519
Palacián, J. F., Sayas, F., and Yanguas, P., Regular and Singular Reductions in the Spatial Three-Body Problem, Qual. Theory Dyn. Syst., 2013, vol. 12, no. 1, pp. 143–182
Palacián, J. F., Sayas, F., and Yanguas, P., Flow Reconstruction and Invariant Tori in the Spatial Three-Body Problem, J. Differential Equations, 2015, vol. 258, no. 6, pp. 2114–2159
Palacián, J.F., Sayas, F., and Yanguas, P., Invariant Tori of the Spatial Three-Body Problem Related to Inner Rectilinear Motions, submitted for publication (2018).
Pauli, W., Jr., Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik, Z. Phys., 1926, vol. 36, no. 5, pp. 336–363
Reeb, G., Sur certaines propriétés topologiques des trajectoires des systèmes dynamiques, Acad. Roy. Sci. Lett. et Beaux-Arts de Belgique. Cl. des Sci. Mém., 1952, vol. 27, no. 9, 64 pp.
Sayas, F., Averaging, Reduction and Reconstruction in the Spatial Three-Body Problem, PhD Thesis, Universidad Pública de Navarra, 2015.
Scheifele, G. and Graf, O., Analytical Satellite Theories Based on a New Set of Canonical Elements, AIAA Paper, N. 74–838.
Sturmfels, B., Algorithms in Invariant Theory, 2nd ed., Wien: Springer, 2008.
Tremaine, S., Canonical Elements for Collision Orbits, Celestial Mech. Dynam. Astronom., 2001, vol. 79, no. 3, pp. 231–233
van der Meer, J.-C. and Cushman, R., Constrained Normalization of Hamiltonian Systems and Perturbed Keplerian Motion, Z. Angew. Math. Phys., 1986, vol. 37, no. 3, pp. 402–424
Yanguas, P., Palacián, J. F., Meyer, K. R., and Dumas, H. S., Periodic Solutions in Hamiltonian Systems, Averaging, and the Lunar Problem, SIAM J. Appl. Dyn. Syst., 2008, vol. 7, no. 2, pp. 311–340
Zhao, L., Quasi-Periodic Almost-Collision Orbits in the Spatial Three-Body Problem, Comm. Pure Appl. Math., 2015, vol. 68, no. 12, pp. 2144–2176
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Meyer, K.R., Palacián, J.F. & Yanguas, P. Normalization Through Invariants in n-dimensional Kepler Problems. Regul. Chaot. Dyn. 23, 389–417 (2018). https://doi.org/10.1134/S1560354718040032
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354718040032
Keywords
- Kepler Hamiltonian in n dimensions
- perturbed Keplerian problems
- Moser regularization
- Delaunay and Delaunay-like coordinates
- Keplerian invariants
- regular reduction
- periodic and quasi-periodic motions
- KAM theory for properly degenerate Hamiltonians