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Motion in a Symmetric Potential on the Hyperbolic Plane

Published online by Cambridge University Press:  20 November 2018

Manuele Santoprete ,
Jürgen Scheurle  and
Sebastian Walcher
Manuele Santoprete
Affiliation:
Department of Mathematics, Wilfrid Laurier University, 75 University Avenue West, Waterloo, Ontario, CanadaN2L 3C5. e-mail: msantopr@wlu.ca
Jürgen Scheurle
Affiliation:
Zentrum Mathematik, TU München, Boltzmannstr. 3, 85747 Garching, Germany. e-mail: scheurle@ma.tum.de
Sebastian Walcher
Affiliation:
Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany. e-mail: walcher@matha.rwth-aachen.de
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Abstract

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We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.

Keywords

37J1570H3370F9937C8034C1420G20Hamiltonian systems with symmetrysymmetriesnon–compact symmetry groupssingular reduction
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 67 , Issue 2 , April 2015 , pp. 450 - 480
Copyright
Copyright © Canadian Mathematical Society 2015

References

[1] Bates, L., A smooth invariant not a smooth function of the invariant polynomials. Proc. Amer. Math. Soc. 135(2007), 3039–3040. http://dx.doi.org/10.1090/S0002-9939-07-08797-7 Google Scholar
[2] Birkes, D., Orbits of linear algebraic groups. Ann. Math. 93(1971), 459–475.http://dx.doi.org/10.2307/1970884 Google Scholar
[3] Cushman, R. and Bates, L., Global Aspects of Classical Integrable Systems. Birkhäuser, Basel, 1997.Google Scholar
[4] Dillen, F. and Kühnel, W., Ruled Weingarten surfaces in Minkowski 3-space. Manuscripta Math. 98(1999), 307–320. http://dx.doi.org/10.1007/s002290050142 Google Scholar
[5] Greuel, G.-M. and Pfister, G., A Singular 3-1-0 library for computing the ring of invariants of the additive groups. ainvar.lib, 2009.Google Scholar
[6] Greuel, G.-M., Pfister, G., and Schönemann, H., Singular 3–1–0—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de, 2009.Google Scholar
[7] Gröbner, W. and Knapp, H. Contributions to the method of Lie series. Bibliographisches Institut, Mannheim, 1967.Google Scholar
[8] Grosshans, F. D., The invariants of unipotent radicals of parabolic subgroups. Invent. Math. 73(1983), 1–9. http://dx.doi.org/10.1007/BF01393822 Google Scholar
[9] Grosshans, F. D., Scheurle, J., and Walcher, S., Invariant sets forced by symmetry. J. Geom. Mechanics 4(2012), 271–296. http://dx.doi.org/10.3934/jgm.2012.4.271 Google Scholar
[10] Grunewald, F. and Margulis, G., Transitive and quasitransitive actions of affine groups preserving a generalized Lorentz-structure. J. Geom. Phys. 5(1988), 493–531.http://dx.doi.org/10.1016/0393-0440(88)90017-4 Google Scholar
[11] Haboush, W. J., Reductive groups are geometrically reductive. Ann. of Math. (2) 102(1975), 67–83.http://dx.doi.org/10.2307/1970974 Google Scholar
[12] Hadžiev, D., Certain questions of the theory of vector invariants. (Russian) Mat. Sb. (N.S.) 72 (114)(1967), 420–435.Google Scholar
[13] Humphreys, J. E., Linear Algebraic Groups. Springer, New York, 1975.Google Scholar
[14] Kunz, E., Introduction to commutative algebra and algebraic geometry. Birkhäuser, Boston, MA, 1985.Google Scholar
[15] Luna, D., Fonctions differentiables invariantes sous l’operation d’un groupe reductif. Ann. Inst. Fourier (Grenoble) 26(1976), 33–49.Google Scholar
[16] Luna, D., Sur les orbites fermées des groupes algébriques reductifs. Invent. Math. 16(1972), 1–5.http://dx.doi.org/10.1007/BF01391210 Google Scholar
[17] Panyushev, D. I., On covariants of reductive algebraic groups. Indag. Math., N.S. 13(2002), 125–129.http://dx.doi.org/10.1016/S0019-3577(02)90010-8 Google Scholar
[18] Procesi, C. and Schwarz, G., Inequalities defining orbit spaces. Invent. Math. 81(1985), 539–554.http://dx.doi.org/10.1007/BF01388587 Google Scholar
[19] Hano, J. and Nomizu, K., On isometric immersions of the hyperbolic plane into the Lorentz–Minkovski space and the Monge–Ampère equation of certain type. Math. Ann. 262(1983), 245–253.http://dx.doi.org/10.1007/BF01455315 Google Scholar
[20] Schwarz, G. W., Smooth functions invariant under the action of a compact Lie group. Topology 14(1975), 63–68. http://dx.doi.org/10.1016/0040-9383(75)90036-1 Google Scholar
[21] Springer, T. A., Invariant theory. Lecture Notes in Math. 585, Springer–Verlag, Berlin–New York, 1977.Google Scholar
[22] Vinberg, E. B. and Popov, V. L., A certain class of quasihomogeneous affine varieties. (Russian) Izv. Akad. Nauk. SSSR Ser. Mat. 36(1972), 749–764.Google Scholar
[23] Walcher, S., On differential equations in normal form. Math. Ann. 291(1991), 293–314.http://dx.doi.org/10.1007/BF01445209 Google Scholar