Summary
A relative equilibrium of a Hamiltonian system with symmetry is a point of phase space giving an evolution which is a one-parameter orbit of the action of the symmetry group of the system. The evolutions of sufficiently small perturbations of a formally stable relative equilibrium are arbitrarily confined to that relative equilibrium's orbit under the isotropy subgroup of its momentum. However, interesting evolution along that orbit, here called drift, does occur. In this article, linearizations of relative equilibria are used to construct a first order perturbation theory explaining drift, and also to determine when the set of relative equilibria near a given relative equilibrium is a smooth symplectic submanifold of phase space.
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References
R. Abraham and J. E. Marsden.Foundations of Mechanics, 2nd ed. Addision-Wesley, Reading, MA, 1978.
J. F. Adams.Lectures on Lie Groups. University of Chicago Press, Chicago, 1969.
J. M. Arms, J. E. Marsden, and V. Moncrief. Bifurcations of momentum mappings.Comm. Math. Phys., 78:455–478, 1981.
V. I. Arnold.Mathematical Methods of Classical Mechanics. Springer-Verlag, Berlin, 1978.
T. Bröcker and T. torn Dieck.Representations of Compact Lie Groups. Springer-Verlag, Berlin, 1985.
N. Burgoyne and R. Cushman. The decomposition of a linear mapping.Linear Algebra Appl., 8:515–519, 1974.
M. J. Field. Equivariant dynamical systems.Trans. Am. Math. Soc., 259:185–205, 1980.
R. Grossman, K. S. Krishnaprasad, and J. E. Marsden. The dynamics of two coupled rigid bodies. In M. Levi and F. Salam, editors,Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits, pages 373–378. SIAM, Philadelphia, 1988.
V. Guillemin and S. Sternberg.Symplectic Techniques in Physics. Cambridge University Press, Cambridge, 1984.
R. Hermann.Differential Geometry and the Calculus of Variations. Interdisiplinary Mathematics, vol. 17, 2nd ed., Math Sci Press, Brookline, MA, 1977.
M. W. Hirsh and S. Smale.Differential Equations, Dynamical Systems, and Linear Algebra. Academic Press, New York, 1974.
K. Hoffman and R. Kunze.Linear Algebra, 2nd ed. Prentice-Hall, 1971.
D. Holm, J. E. Marsden, T. Ratiu, and A. Weinstein. Stability of rigid body motion using the energy-Casimir method. In J. E. Marsden, editor,Fluids and Plasmas: Geometry and Dynamics, Contemporary Mathematics 28. AMS, Providence, RI, 1984.
M. Krupa. Bifurcations of relative equilibria.SIAM J. Math. Anal, 21:1453–1486, 1990.
J. B. Marion.Classical Dynamics of Particles and Systems, 2nd ed. Academic Press, New York, 1970.
J. E. Marsden.Lectures on Geometric Methods in Mathematical Physics. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 37. SIAM, Philadelphia, PA, 1981.
J. E. Marsden, R. Montgomery, and T. Ratiu. Reduction, symmetry, and phases in mechanics.Mem. Amer. Math. Soc., 88, 1990.
J. E. Marsden and T. S. Ratiu.Introduction to Mechanics and Symmetry. Springer-Verlag, Berlin, 1994.
J. E. Marsden, J. C. Simo, D. Lewis, and T. A. Posbergh. Block diagonalization and the energy-momentum method. In J. E. Marsden, P. S. Krishnaprasad, and J. C. Simo, editors,Dynamics and Control of Multibody Systems. Contemporary Mathematics, vol. 97, pages 297–313. AMS, Providence, RI, 1989.
J. E. Marsden and A. Weinstein. Reduction of symplectic manifolds with symmetry.Rep. Math. Phys., 5:121–130, 1974.
K. R. Meyer and G. R. Hall.Introduction to Hamiltonian Dynamical Systems and the n-Body Problem. Applied Mathematical Sciences, Vol. 90. Springer-Verlag, Berlin, 1991.
J. A. Montaldi, R. M. Roberts, and I. N. Stewart. Periodic solutions near equilibria of symmetric Hamiltonian systems.Philos. Trans. Roy. Soc. London Ser. A, 325:237–293, 1988.
J. Moser. On the theory of quasiperiodic motions.SLAM Rev., 8:145–172, 1966.
J. Moser. Convergent series expansions for quasi-periodic motions.Math. Ann., 169:136–176, 1967.
G. W. Patrick.Two axially symmetric coupled rigid bodies: relative equilibria, stability, bifurcations, and a momentum preserving symplectic integrator. Ph.D. thesis, University of California at Berkeley, 1991.
G. W. Patrick. Relative equilibria in hamiltonian systems: The dynamic interpretation of nonlinear stability on the reduced phase space.J. Geom. Phys., 9:111–119, 1992.
G. W. Patrick. Dynamics near stable relative equilibria at non-generic momenta: a numerical investigation. Fields Institute Communications, 1993. Also Technical Report TRE-94-8, Department of Mathematics and Statistics, University of Saskatchewan. Available by remote login to math.usask.ca using user-id info.
J. Simo, D. Lewis, and J. E. Marsden, Stability of relative equilibria. Part I: The reduced energy momentum method.Arch. Rational Mech. Anal, 115:15–59, 1991.
J. Simo, T. Posberg, and J. E. Marsden. Stability of coupled rigid body and geometrically exact rods: Block diagonalization and the energy-momentum method.Phys. Rep., 193:280–360, 1990.
S. Smale. Topology and mechanics. II. The planar n-body problem.Invent. Math., 11:45–64, 1970.
A. Weinstein.Lectures on Symplectic Manifolds, Regional Conference Series in Mathematics, AMS, Vol. 29. AMS, Providence, RI, 1977.
R. W. Wolverton, editor.Flight Performance Handbook for Orbital Operations, 2nd ed. Wiley, New York, 1963.
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Patrick, G.W. Relative equilibria of Hamiltonian systems with symmetry: Linearization, smoothness, and drift. J Nonlinear Sci 5, 373–418 (1995). https://doi.org/10.1007/BF01212907
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DOI: https://doi.org/10.1007/BF01212907