Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top
DOI:
https://doi.org/10.5890/DNC.2021.09.001Abstract
We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamiltonian system that preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$, when we add a perturbation the subalgebra $\mathbb{B}$ will no longer be preserved. We show how to transform the perturbed dynamical system to preserve $\mathbb{B}$ up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement.References
[1] Arnold, V.I. (1989), { Mathematical {Methods} of {Classical} {Mechanics}}. Graduate {Texts} in {Mathematics}, Springer-Verlag, 2 edn.
[2] Morrison, P.J. (1982), Poisson brackets for fluid and plasmas, { Mathematical {Methods} in {Hydrodynamics} and {Integrability} in {Dynamical} {Systems}}, New York, vol.88, p.36, American Institute of Physics.
[3] Sakurai, J.J. (2017), { Modern {Quantum} {Mechanics}}, Cambridge University Press, 2nd edn.
[4] Marsden, J.E., Morrison, P.J., and Weinstein, A.J. (1984), The {Hamiltonian} structure of the {BBGKY} hierarchy equations. Marsden, J.E. (ed.), { Fluids and plasmas: geometry and dynamics}, pp. 115-124, American Mathematical Society.
[5] Marsden, J.E., Montgomery, R., Morrison, P.J., and Thompson, W.B. (1986), Covariant poisson brackets for classical fields, { Annals of Physics}, {bf 169}, 29-47.
[6] Littlejohn, R.G. (1982), Hamiltonian perturbation theory in noncanonical coordinates. { Journal of Mathematical Physics}, {bf 23}, 742-747.
[7] Kolmogorov, A.N. (1954), On the preservation of conditionally periodic motions for a small change in {Hamilton}s function, { Dokl. Akad. Nauk, SSSR}, {bf 98}, 527-530.
[8] Arnold, V.I. (1963), Proof of a theorem by {A}. {N}. {Kolmogorov} on the persistence of quasi- periodic motions under small perturbations of the {Hamiltonian}, { Russian Mathematical Survey}, {bf 18}.
[9] Moser, J. (1973), { Stable and {Random} {Motions} in {Dynamical} {Systems}: {With} {Special} {Emphasis} on {Celestial} {Mechanics} ({AM}-77)}, Princeton University Press, rev - revised edn.
[10] Bost, J.B. (1986), Tores invariants des syst`{e}mes dynamiques hamiltoniens, { Ast{e}risque}, vol. 133-134, pp. 113-157.
[11] DeLaLlave, R., Gonz{a}lez, A., Jorba, A., and Villanueva, J. (2005), {KAM} theory without action-angle variables, { Nonlinearity}, {bf 18}, 855-895.
[12] Li, Y. and Yi, Y. (2002), Persistence of invariant tori in generalized {Hamiltonian} systems, { Ergod. Th. Dynam. Sys.}, {bf 22}.
[13] Alishah, H. and DeLaLlave, R. (2012) Tracing {KAM} {Tori} in {Presymplectic} {Dynamical} {Systems}, { Journal of Dynamics and Differential Equations}, {bf 24}, 685-711.
[14] Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M. (1984), A proof of {Kolmogorov}s theorem on invariant tori using canonical transformations defined by the {Lie} method. { Il Nuovo Cimento B (1971-1996)}, {bf 79}, 201-223.
[15] DeLaLlave, R. (2001), A tutorial on {KAM} theory, { Smooth ergodic {Theory} $&$ its applications}, Providence, vol.69, pp. 175-292, American Math Society.
[16] Broer, H.W. (2004), {KAM} theory: {The} legacy of {Kolmogorov}s 1954 paper. { Bulletin of the American Mathematical Society}, {bf 41}, 507-522.
[17] F{e}joz, J. (2016), Introduction to {KAM} theory with a view to celestial mechanics. Bergounioux, M., Peyr{e}, G., Schn"{o}rr, C., Caillau, J.-B., and Haberkorn, T. (eds.), { Variational {Methods}}, De Gruyter.
[18] Vittot, M. (2004), Perturbation {Theory} and {Control} in {Classical} or {Quantum} {Mechanics} by an {Inversion} {Formula}, { Journal of Physics A: Mathematical and General}, {bf 37}, 6337-6357, arXiv: math-ph/0303051.
[19] Marsden, J.E. and Ratiu, T.S. (2013), { Introduction to mechanics and symmetry: a basic exposition of classical mechanical systems}, vol.17. Springer Science & Business Media.
[20] Reed, M. and Simon, B. (1981), { Functional {Analysis}, {Volume} 1}. Academic Press.
[21] Giorgilli, A. (1995), Quantitative {Methods} in {Classical} {Perturbation} {Theory}, { From {Newton} to chaos: modern techniques for understanding and coping with chaos in {N}-body dynamical system}, pp. 21-38, Plenum Press, a.e. roy e b.d. steves edn.
[22] Deprit, A. (1967), Free {Rotation} of a {Rigid} {Body} {Studied} in the {Phase} {Plane}, { American Journal of Physics}, {bf 35}, 424-428.
[23] Gurfil, P., Elipe, A., Tangren, W., and Efroimsky, M. (2007), The {Serret}-{Andoyer} {Formalism} in {Rigid}-{Body} {Dynamics}: {I}. {Symmetries} and {Perturbations}, { Regular and Chaotic Dynamics}, {bf 12}, 389-425.
[24] Arnold, V.I. (1963) Small {Denominators} and problems of stability of motion in {Classical} and {Celestial} {Mechanics}, { Russ. Math. Surv.}, {bf 18}, 85.
[25] Pinzari, G. and Chierchia, L. (2010), Properly-degenerate {KAM} theory (following {V}. {I}. {Arnold}). { Discrete and Continuous Dynamical Systems - Series S}, {bf 3}, 545-578.
Article Metrics
Usage tracking begins September 1, 2026.