Geometrodynamical Analysis to Characterize the Dynamics and Stability of a Molecular System through the Boundary of the Hill’s Region

Alberto Vergel¹, Rosa M. Benito¹, Juan C. Losada¹, Florentino Borondo²

¹Grupo de Sistemas Complejos, Departamento de Física y Mecánica, Escuela Técnica Superior de Ingenieros Agrónomos, Universidad Politécnica de Madrid, Madrid, Spain.
²Instituto de Ciencias Matemáticas (ICMAT), Madrid, Spain.
³Departamento de Química, Universidad Autónoma de Madrid, Madrid, Spain.

DOI: 10.4236/am.2014.516251                

Abstract

In this paper we study the dynamics and stability of a two-dimensional model for the vibrations of the LiCN molecule making use of the Riemannian geometry via the Jacobi-Levi-Civita equations applied to the Jacobi metric. The Stability Geometrical Indicator for short times is calculated to locate regular and chaotic trajectories as the relative extrema of this indicator. Only trajectories with initial conditions at the boundary of the Hill’s region are considered to characterize the dynamics of the system. The importance of the curvature of this boundary for the stability of trajectories bouncing on it is also discussed.

Keywords

Geometrodynamical Analysis, Molecular Nonlinear Dynamics, Stability Geometrical Indicator, Riemannian Geometry, Hamiltonian Systems

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Borondo, F. and Benito, R.M. (1995) Dynamics and Spectroscopy of Highly Excited Molecules in Frontiers of Chemical Physics. NATO ASI Series C. Kluwer, Dordrecht.
[2]	Skokos, Ch. (2010) The Lyapunov Characteristic Exponents and Their Computation. Lecture Notes in Physics, 790, 63. http://dx.doi.org/10.1007/978-3-642-04458-8_2
[3]	Lega, E., Guzzo, M. and Froeschlé, C. (2003) Detection of Arnold Diffusion in Hamiltonian Systems. Physica D, 182, 179-187. http://dx.doi.org/10.1016/S0167-2789(03)00121-0
[4]	Barrio, R. (2005) Sensitivity Tools vs. Poincaré Sections. Chaos, Solitons and Fractals, 25, 711. http://dx.doi.org/10.1016/j.chaos.2004.11.092
[5]	Dumas, H.S. and Laskar, J. (1993) Global Dynamics and Long-Time Stability in Hamiltonian Systems via Numerical Frequency Analysis. Physical Review Letters, 70, 2975. http://dx.doi.org/10.1103/PhysRevLett.70.2975
[6]	Skokos, Ch., Antonopoulos, Ch., Bountis, T.C. and Vrahatis, M.N. (2004) Detecting Order and Chaos in Hamiltonian Systems by the SALI Method. Journal of Physics A, 37, 6269. http://dx.doi.org/10.1088/0305-4470/37/24/006
[7]	Benitez, P., Losada, J.C., Benito, R.M. and Borondo, F. (2013) Analysis of the Full Vibrational Dynamics of the LiNC/ LiCN Molecular System. Progress and Challenges in Dynamical Systems, 54, 77-88. http://dx.doi.org/10.1007/978-3-642-38830-9_6
[8]	Vergel, A., Benito, R.M., Losada, J.C. and Borondo, F. (2014) Geometrical Analysis of the LiCN Vibrational Dynamics. A Stability Geometrical Indicator. Physical Review E, 89, Article ID: 022901. http://dx.doi.org/10.1103/PhysRevE.89.022901
[9]	Arranz, F.J., Borondo, F. and Benito, R.M. (2000) Transition from Order to Chaos in a Floppy Molecule: LiNC/LiCN. Chemical Physics Letters, 317, 451. http://dx.doi.org/10.1016/S0009-2614(99)01426-8
[10]	Borondo, F., Zembekov, A.A. and Benito, R.M. (1995) Quantum Manifestations of Saddle-Node Bifurcations. Chemical Physics Letters, 246, 421. http://dx.doi.org/10.1016/0009-2614(95)01147-X
[11]	Borondo, F., Zembekov, A.A. and Benito, R.M. (1996) Saddle-Node Bifurcations in the LiNC/LiCN Molecular System: Classical Aspects and Quantum Manifestations. Journal of Chemical Physics, 105, 5068. http://dx.doi.org/10.1063/1.472351
[12]	Borondo, F., Zembekov, A.A. and Benito, R.M. (1997) Semiclassical Quantization of Fragmented Tori: Application to Saddle-Node States of LiNC/LiCN. Journal of Chemical Physics, 107, 7934. http://dx.doi.org/10.1063/1.475147
[13]	Losada, J.C., Estebaranz, J.M., Benito, R.M. and Borondo, F. (1998) Local Frequency Analysis and the Structure of Classical Phase Space of the LiNC/LiCN Molecular System. Journal of Chemical Physics, 108, 63. http://dx.doi.org/10.1063/1.475365
[14]	Borondo, F., Losada, J.C. and Benito, R.M. (2001) Scars in Molecular Vibrations and Spectra of LiCN. Foundations of Physics, 31, 147-163. http://dx.doi.org/10.1023/A:1004160006269
[15]	Losada, J.C., Benito, R.M., Arranz, F.J. and Borondo, F. (2002) Frequency Map Analysis and Scars in Molecular Vibrations. International Journal of Quantum Chemistry, 86, 167-174. http://dx.doi.org/10.1002/qua.1620
[16]	Eisenhart, L.P. (1929) Dynamical Trajectories and Geodesics. Annals of Mathematics, 30, 591. http://dx.doi.org/10.2307/1968307
[17]	Cipriani, P. and Bari, M. (1998) Finsler Geometric Local Indicator of Chaos for Single Orbits in the Hénon-Heiles Hamiltonian. Physical Review Letters, 81, 5532. http://dx.doi.org/10.1103/PhysRevLett.81.5532
[18]	Horwitz, L., Zion, Y.B., Lewkowics, M., Schiffer, M. and Levitan, J. (2007) Geometry of Hamiltonian Chaos. Physical Review Letters, 98, Article ID: 234301. http://dx.doi.org/10.1103/PhysRevLett.98.234301
[19]	Zion, Y.B. and Horwitz, L. (2007) Detecting Order and Chaos in Three-Dimensional Hamiltonian Systems by Geometrical Methods. Physical Review E, 76, Article ID: 046220. http://dx.doi.org/10.1103/PhysRevE.76.046220
[20]	Zion, Y.B. and Horwitz, L. (2008) Applications of Geometrical Criteria for Transition to Hamiltonian Chaos. Physical Review E, 78, Article ID: 036209.
[21]	Zion, Y.B. and Horwitz, L. (2010) Controlling Effect of Geometrically Defined Local Structural Changes on Chaotic Hamiltonian Systems. Physical Review E, 81, Article ID: 046217.
[22]	Pettini, M. (1993) Geometrical Hints for a Nonperturbative Approach to Hamiltonian dynamics. Physical Review E, 47, 828. http://dx.doi.org/10.1103/PhysRevE.47.828
[23]	Casetti, L., Livi, R. and Pettini, M. (1995) Gaussian Model for Chaotic Instability of Hamiltonian Flows. Physical Review Letters, 74, 375. http://dx.doi.org/10.1103/PhysRevLett.74.375
[24]	Casetti, L. and Pettini, M. (1993) Analytic Computation of the Strong Stochasticity Threshold in Hamiltonian Dynamics Using Riemannian Geometry. Physical Review E, 48, 4320. http://dx.doi.org/10.1103/PhysRevE.48.4320
[25]	Casetti, L., Clementi, C. and Pettini, M. (1996) Riemannian Theory of Hamiltonian Chaos and Lyapunov Exponents. Physical Review E, 54, 5969. http://dx.doi.org/10.1103/PhysRevE.54.5969
[26]	Bari, M. and Cipriani, P. (1998) Geometry and Chaos on Riemann and Finsler Manifolds. Planetary and Space Science, 46, 1543. http://dx.doi.org/10.1016/S0032-0633(98)00059-2
[27]	Casetti, L., Pettini, M. and Cohen, E.G.D. (2000) Geometric Approach to Hamiltonian Dynamics and Statistical Mechanics. Physics Reports, 337, 237-241. http://dx.doi.org/10.1016/S0370-1573(00)00069-7
[28]	Cerruti-Sola, M. and Pettini, M. (1995) Geometric Description of Chaos in Self-Gravitating Systems. Physical Review E, 51, 53. http://dx.doi.org/10.1103/PhysRevE.51.53
[29]	Cerruti-Sola, M. and Pettini, M. (1996) Geometric Description of Chaos in Two-Degrees-of-Freedom Hamiltonian Systems. Physical Review E, 53, 179. http://dx.doi.org/10.1103/PhysRevE.53.179
[30]	Carmo, M.P. (1992) Riemannian Geometry. Birkh?user, Boston.
[31]	Carmo, M.P. (1976) Differential Geometry of Curves and Surfaces. Prentice Hall, Englewood Cliffs.
[32]	Saa, A. (2004) On the Viability of Local Criteria for Chaos. Annals of Physics, 314, 508-516. http://dx.doi.org/10.1016/j.aop.2004.08.008
[33]	Esser, R., Tennyson, J. and Wormer, P.E.S. (1982) An SCF Potential Energy Surface for Lithium Cyanide. Chemical Physics Letters, 89, 223-227. http://dx.doi.org/10.1016/0009-2614(82)80046-8
[34]	Benito, R.M., Borondo, F., Kim, J.H., Sumpter, B.G. and Ezra, G.S. (1989) Comparison of Classical and Quantum Phase Space Structure of Nonrigid Molecules, LiCN. Chemical Physics Letters, 161, 60-66. http://dx.doi.org/10.1016/S0009-2614(89)87032-0