Relative Critical Points

Debra Lewis

Symmetry, integrability and geometry: methods and applications, Tome 9 (2013)

Résumé

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures — symplectic, Poisson, or variational — generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems — the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids — and generalizations of these systems.

Détail

Keywords: relative equilibria; symmetries; conservative systems; Riemann ellipsoids.

Bibliographie

[1] Abraham R., Marsden J. E., Foundations of mechanics, Advanced Book Program, 2nd ed., Benjamin/Cummings Publishing Co. Inc., Reading, Mass., 1978 | MR | Zbl

[2] Arms J. M., Cushman R. H., Gotay M. J., “A universal reduction procedure for {H}amiltonian group actions”, The Geometry of {H}amiltonian Systems (Berkeley, {CA}, 1989), Math. Sci. Res. Inst. Publ., 22, Springer, New York, 1991, 33–51 | DOI | MR

[3] Arms J. M., Gotay M. J., Jennings G., “Geometric and algebraic reduction for singular momentum maps”, Adv. Math., 79 (1990), 43–103 | DOI | MR | Zbl

[4] Arms J. M., Marsden J. E., Moncrief V., “Symmetry and bifurcations of momentum mappings”, Comm. Math. Phys., 78 (1981), 455–478 | DOI | MR | Zbl

[5] Arnold V. I., Mathematical methods of classical mechanics, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1978 | MR | Zbl

[6] Chandrasekhar S., Ellipsoidal figures of equilibrium, Dover, New York, 1987

[7] Chossat P., Lewis D., Ortega J. P., Ratiu T. S., “Bifurcation of relative equilibria in mechanical systems with symmetry”, Adv. in Appl. Math., 31 (2003), 10–45, arXiv: math.DG/9912232 | DOI | MR | Zbl

[8] Cushman R., Sjamaar R., “On singular reduction of {H}amiltonian spaces”, Symplectic Geometry and Mathematical Physics ({A}ix-en-{P}rovence, 1990), Progr. Math., 99, Birkhäuser Boston, Boston, MA, 1991, 114–128 | MR

[9] Dedekind R., “Zusatz zu der vorstehenden {A}bhandlung”, J. Reine Angew. Math., 58 (1861), 217–228 | DOI | Zbl

[10] Dirichlet G. L., “Untersuchungen über ein Problem der Hydrodynamik”, J. Reine Angew. Math., 58 (1861), 181–216 | DOI | Zbl

[11] Fassò F., Lewis D., “Stability properties of the {R}iemann ellipsoids”, Arch. Ration. Mech. Anal., 158 (2001), 259–292, arXiv: math.DG/0002011 | DOI | MR | Zbl

[12] Field M. J., “Equivariant dynamical systems”, Trans. Amer. Math. Soc., 259 (1980), 185–205 | DOI | MR | Zbl

[13] Golubitsky M., Stewart I., “Generic bifurcation of {H}amiltonian systems with symmetry”, Phys. D, 24 (1987), 391–405 | DOI | MR | Zbl

[14] Golubitsky M., Stewart I., Schaeffer D. G., Singularities and groups in bifurcation theory, v. II, Applied Mathematical Sciences, 69, Springer-Verlag, New York, 1988 | DOI | MR | Zbl

[15] Guillemin V., Sternberg S., Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984 | MR

[16] Hazeltine R. D., Holm D. D., Marsden J. E., Morrison P. J., “Generalized Poisson brackets and nonlinear Liapunov stability-application to reduced MHD”, International Conference on Plasma Physics Proceedings (Lausanne), v. 1, eds. M. Q. Tran, M. L. Sawley, Ecole Polytechnique Federale de Lausanne, Lausanne, 1984, 204–206

[17] Holm D. D., Marsden J. E., Ratiu T., Weinstein A., “Nonlinear stability of fluid and plasma equilibria”, Phys. Rep., 123 (1985), 1–116 | DOI | MR | Zbl

[18] Kamran N., Milson R., Olver P. J., “Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems”, Adv. Math., 156 (2000), 286–319, arXiv: solv-int/9904014 | DOI | MR | Zbl

[19] Kogan I. A., Olver P. J., “Invariant {E}uler–{L}agrange equations and the invariant variational bicomplex”, Acta Appl. Math., 76 (2003), 137–193 | DOI | MR | Zbl

[20] Krupa M., “Bifurcations of relative equilibria”, SIAM J. Math. Anal., 21 (1990), 1453–1486 | DOI | MR | Zbl

[21] Lewis D., “Bifurcation of liquid drops”, Nonlinearity, 6 (1993), 491–522 | DOI | MR | Zbl

[22] Lewis D., “Lagrangian block diagonalization”, J. Dynam. Differential Equations, 4 (1992), 1–41 | DOI | MR | Zbl

[23] Lewis D., Marsden J. E., Ratiu T. S., Simo J. C., “Normalizing connections and the energy-momentum method”, Hamiltonian Systems, Transformation Groups and Spectral Transform Methods (Montreal, {PQ}, 1989), Univ. Montréal, Montreal, QC, 1990, 207–227 | MR

[24] Lewis D., Ratiu T., Simo J. C., Marsden J. E., “The heavy top: a geometric treatment”, Nonlinearity, 5 (1992), 1–48 | DOI | MR | Zbl

[25] Lewis D., Simo J. C., “Energy methods in the stability analysis of relative equilibria of {H}amiltonian systems”, The Proceedings of the Sixth Symposium on Continuum Models and Discrete Systems (Dijon, June 1989), ed. G. A. Maugin, Longman, London, 1989, 162–183 | MR

[26] Lewis D., Simo J. C., “Nonlinear stability of rotating pseudo-rigid bodies”, Proc. Roy. Soc. London Ser. A, 427 (1990), 281–319 | DOI | MR | Zbl

[27] Marsden J. E., Ratiu T. S., Introduction to mechanics and symmetry. A basic exposition of classical mechanical systems, Texts in Applied Mathematics, 17, Springer-Verlag, New York, 1994 | DOI | MR | Zbl

[28] Marsden J. E., Ratiu T., “Reduction of {P}oisson manifolds”, Lett. Math. Phys., 11 (1986), 161–169 | DOI | MR | Zbl

[29] Marsden J. E., Scheurle J., “Lagrangian reduction and the double spherical pendulum”, Z. Angew. Math. Phys., 44 (1993), 17–43 | DOI | MR | Zbl

[30] Marsden J. E., Simo J. C., Lewis D., Posbergh T. A., “Block diagonalization and the energy-momentum method”, Dynamics and Control of Multibody Systems (Brunswick, ME, 1988), Contemp. Math., 97, Amer. Math. Soc., Providence, RI, 1989, 297–313 | DOI | MR

[31] Marsden J. E., Weinstein A., “Reduction of symplectic manifolds with symmetry”, Rep. Math. Phys., 5 (1974), 121–130 | DOI | MR | Zbl

[32] Meyer K. R., “Symmetries and integrals in mechanics”, Dynamical Systems, Proc. Sympos. (Univ. {B}ahia, {S}alvador, 1971), ed. M. Peixoto, Academic Press, New York, 1973, 259–272 | MR

[33] Montaldi J., “Persistence and stability of relative equilibria”, Nonlinearity, 10 (1997), 449–466 | DOI | MR | Zbl

[34] Olver P. J., Applications of {L}ie groups to differential equations, Graduate Texts in Mathematics, 107, Springer-Verlag, New York, 1986 | DOI | MR | Zbl

[35] Olver P. J., “Differential invariants”, Acta Appl. Math., 41 (1995), 271–284 | DOI | MR | Zbl

[36] Olver P. J., “Differential invariants and invariant differential equations”, Lie Groups Appl., 1 (1994), 177–192 | MR | Zbl

[37] Olver P. J., “Invariant variational problems and invariant flows via moving frames”, Variations, Geometry and Physics, eds. O. Krupková, D. Saunders, Nova Sci. Publ., New York, 2009, 209–235 | MR | Zbl

[38] Ortega J. P., Ratiu T. S., “The stratified spaces of a symplectic {L}ie group action”, Rep. Math. Phys., 58 (2006), 51–75 | DOI | MR | Zbl

[39] Patrick G. W., Roberts M., Wulff C., “Stability of {P}oisson equilibria and {H}amiltonian relative equilibria by energy methods”, Arch. Ration. Mech. Anal., 174 (2004), 301–344, arXiv: math.DS/0201239 | DOI | MR | Zbl

[40] Patrick G. W., Roberts M., Wulff C., “Stability transitions for axisymmetric relative equilibria of {E}uclidean symmetric {H}amiltonian systems”, Nonlinearity, 21 (2008), 325–352, arXiv: 0705.1552 | DOI | MR | Zbl

[41] Riemann B., “Ein {B}eitrag zu den {U}ntersuchungen über die {B}ewegung eines flüssigen gleichartigen {E}llipsoides”, Abh. d. Königl. Gesell. der Wis. zu Göttingen, 9 (1861), 3–36

[42] Rodríguez-Olmos M., “Stability of relative equilibria with singular momentum values in simple mechanical systems”, Nonlinearity, 19 (2006), 853–877, arXiv: math.DS/0506280 | DOI | MR | Zbl

[43] Rodríguez-Olmos M., Sousa-Dias M. E., “Nonlinear stability of {R}iemann ellipsoids with symmetric configurations”, J. Nonlinear Sci., 19 (2009), 179–219, arXiv: 0712.3081 | DOI | MR | Zbl

[44] Rosensteel G., “Gauge theory of {R}iemann ellipsoids”, J. Phys. A: Math. Gen., 34 (2001), L169–L178 | DOI | MR | Zbl

[45] Rosensteel G., “Geometric quantization of {R}iemann ellipsoids”, Group Theoretical Methods in Physics (Varna, 1987), Lecture Notes in Phys., 313, Springer, Berlin, 1988, 253–260 | DOI | MR

[46] Rosensteel G., “Rapidly rotating nuclei as {R}iemann ellipsoids”, Ann. Physics, 186 (1988), 230–291 | DOI | MR | Zbl

[47] Routh E. J., Advanced rigid dynamics, McMillan and Co., London, 1884

[48] Simo J. C., Lewis D., Marsden J. E., “Stability of relative equilibria. I: The reduced energy-momentum method”, Arch. Rational Mech. Anal., 115 (1991), 15–59 | DOI | MR | Zbl

[49] Simo J. C., Posbergh T. A., Marsden J. E., “Stability of coupled rigid body and geometrically exact rods: block diagonalization and the energy-momentum method”, Phys. Rep., 193 (1990), 279–360 | DOI | MR

[50] Sjamaar R., Lerman E., “Stratified symplectic spaces and reduction”, Ann. of Math. (2), 134 (1991), 375–422 | DOI | MR | Zbl

[51] Smale S., “Topology and mechanics, I”, Invent. Math., 10 (1970), 305–331 | DOI | MR | Zbl

[52] Smale S., “Topology and mechanics. II: The planar {$n$}-body problem”, Invent. Math., 11 (1970), 45–64 | DOI | MR | Zbl

[53] Zombro B., Holmes P., “Reduction, stability, instability and bifurcation in rotationally symmetric {H}amiltonian systems”, Dynam. Stability Systems, 8 (1993), 41–71 | DOI | MR | Zbl