On the Equivalence Between Type I Liouville Dynamical Systems in the Plane and the Sphere

  1. León, Miguel A. González 1
  2. Guilarte, Juan Mateos
  3. Mayado, Marina de la Torre 1
  1. 1 Universidad de Salamanca
    info

    Universidad de Salamanca

    Salamanca, España

    ROR https://ror.org/02f40zc51

Libro:
Integrability, Supersymmetry and Coherent States

ISBN: 978-3-030-20087-9,  978-3-030-20086-2

Año de publicación: 2019

Páginas: 359-373

Tipo: Capítulo de Libro

DOI: 10.1007/978-3-030-20087-9_16 GOOGLE SCHOLAR

Resumen

Separable Hamiltonian systems either in sphero-conical coordinates on an S2 sphere or in elliptic coordinates on a R2 plane are described in a unified way. A back and forth route connecting these Liouville Type I separable systems is unveiled. It is shown how the gnomonic projection and its inverse map allow us to pass from a Liouville Type I separable system with a spherical configuration space to its Liouville Type I partners where the configuration space is a plane and back. Several selected spherical separable systems and their planar cousins are discussed in a classical context.

Referencias bibliográficas

  • J. Liouville, Mémoire sur l’integration des équations différentielles du mouvement d’un nombre quelconque de points matériels. J. Math. Pures et Appl. 14, 257–299 (1849)
  • G. Morera, Sulla separazione delle variabili nelle equazioni del moto di un punto materiale su una superficie Atti. Sci. di Torino 16, 276–295 (1881)
  • A.M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras (Birkhäuser, Basel 1990)
  • M.A. Gonzalez Leon, J. Mateos Guilarte, M. de la Torre Mayado, Orbits in the problem of two fixed centers on the sphere. Regul. Chaotic Dyn. 22, 520–542 (2017)
  • H.W. Killing, Die Mechanik in den nicht-euklidischen Raumformen. J. Reine Angew. Math. 98(1), 1–48 (1885)
  • V.V. Kozlov, A.O. Harin, Kepler’s problem in constant curvature spaces. Celest. Mech. Dyn. Astr. 54(4), 393–399 (1992)
  • A.V. Borisov, I.S. Mamaev, Relations between integrable systems in plane and curved spaces. Celest. Mech. Dyn. Astr. 99(4), 253–260 (2007)
  • A. Albouy, The underlying geometry of the fixed centers problems, in Topological Methods, Variational Methods and Their Applications, ed. by H. Brezis, K.C. Chang, S.J. Li, P. Rabinowitz (World Scientific, Singapore, 2003), pp. 11–21
  • A. Albouy, T. Stuchi, Generalizing the classical fixed-centres problem in a non-Hamiltonian way. J. Phys. A 37, 9109–9123 (2004)
  • P. Appell, De l’homographie en mécanique. Am. J. Math. 12(1), 103–114 (1890)
  • P. Appell, Sur les lois de forces centrales faisant décrire à leur point d’application une conique quelles que soient les conditions initiales. Am. J. Math. 13(2), 153–158 (1891)
  • P.W. Higgs, Dynamical symmetry in a spherical geometry I. J. Phys. A: Math. Gen. 12(3), 309–323 (1979). H.I. Leemon, Dynamical symmetry in a spherical geometry II. J. Phys. A: Math. Gen. 12(4), 489–501 (1979)
  • A. Albouy, Projective dynamics and classical gravitation. Regul. Chaotic Dyn. 13(6), 525–542 (2008)
  • A. Albouy, There is a projective dynamics. Eur. Math. Soc. Newsl. 89, 37–43 (2013)
  • A. Albouy, Projective dynamics and first integrals. Regul. Chaotic Dyn. 20(3), 247–276 (2015)
  • A.V. Borisov, I.S. Mamaev, I.A. Bizyaev, The spatial problem of 2 bodies on a sphere. Reduction and stochasticity. Regul. Chaotic Dyn. 21(5), 556–580 (2016)
  • C. Neumann, De problemate quodam mechanico, quod ad primam integralium ultraellipticorum classem revocatur. J. Rein Angew. Math. 56, 46–63 (1859)
  • B.A. Dubrovine, Theta functions and non-linear equations. Russ. Math. Surv. 36(2), 11–92 (1981)
  • V.Z. Enolski, E. Hackmann, V. Kagramanova, J. Kung, C. L a ̈ $$\ddot {\mathrm {a}}$$ merzahl, Inversion of hyperelliptic integrals of arbitrary genus with applications to particle motion in general relativity. J. Geom. Phys. 61, 899–921 (2011)
  • R. Garnier, Sur une classe de systems differentielles abelians deduits de la theorie de equations lineaires. Rend. Circ. Mat. Palermo 43, 155–191 (1919)
  • O. Babelon, M. Talon, Separation of variables for the classical and quantum Neumann model. Nucl. Phys. B 379, 321–339 (1992)
  • M. Gadella, J. Negro, L.M. Nieto, G.P. Pronko, Two charged particles in the plane under a constant perpendicular magnetic field. Int. J. Theor. Phys. 50, 2019–2028 (2011)
  • M.A. Gonzalez Leon, J. Mateos Guilarte, M. de la Torre Mayado, Elementary solutions of the quantum planar two center problem. Europhys. Lett. 114, 30007 (2016)
  • A. Alonso Izquierdo, M.A. Gonzalez Leon, J. Mateos Guilarte, Kinks in a non-linear massive sigma model. Phys. Rev. Lett. 101, 131602 (2008). BPS and non BPS kinks in a massive non-linear S
  • 2-sigma model. Phys. Rev. D 79, 125003 (2009)
  • A. Alonso Izquierdo, M.A. Gonzalez Leon, J. Mateos Guilarte, M. de la Torre Mayado, On domain walls in a Ginzburg-Landau non-linear S2-sigma model. J. High Energy Phys. 2010(8), 1–29 (2010)
  • A. Alonso Izquierdo, A. Balseyro Sebastian, M.A. Gonzalez Leon, Domain walls in a non-linear 𝕊 2 $$\mathbb {S}^2$$ -sigma model with homogeneous quartic polynomial potential. J. High Energy Phys. 2018(11), 1–23 (2018)