Article Contents
Article Contents

# Dihedral molecular configurations interacting by Lennard-Jones and Coulomb forces

• * Corresponding author: Wieslaw Krawcewicz

The first and second authors are supported by the Department of Mathematical Sciences University of Texas at Dallas. The third author is supported by the Center for Applied Mathematics at Guangzhou University, Guangzhou China and the Department of Mathematical Sciences University of Texas at Dallas

• In this paper, we investigate nonlinear periodic vibrations of a group of particles with a planar dihedral configuration governed by the Lennard-Jones and Coulomb forces. Using the gradient equivariant degree, we provide a full topological classification of the periodic solutions with both temporal and spatial symmetries. In the process, we provide general formulae for the spectrum of the linearized system of equations describing the above configuration, which allows us to obtain the critical frequencies of the particles' motions. The obtained frequencies represent the set of all critical periods for small amplitude periodic solutions emerging from a given stationary symmetric orbit of solutions.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

 Citation:

• Figure 1.  Stationary solution to equation (7) with dihedral symmetries

Figure 2.  Relative motions of all 6 particles with $\lambda_0^2 = \frac{l^2}{\mu}$, $l = 1$ and $\mu$ near the eigenvalue $\mu_0 = 10.10496819$ of $\nabla^2 V(u^o)$

Figure 3.  Relative motions of all particles with $\lambda_0^2 = \frac{l^2}{\mu}$, $l = 1$ and $\mu$ near the eigenvalue $\mu = 6.442637681$ of $\nabla^2 V(u^o)$

Figure 4.  Relative motions of all particles with $\lambda_0^2 = \frac{l^2}{\mu}$, $l = 1$ and $\mu$ near the eigenvalue $\mu = 8.469351217$ of $\nabla^2 V(u^o)$

Figure 5.  Relative motions of all particles with Relative motions of all particles with $\lambda_0^2=\frac{l^2}{\mu}$, $l=1$ and $\mu$ near the eigenvalue $\mu=3.854423919$ of $\nabla^2 V(u^o)$

Table 1.  The values $\lambda_{j, l}$ in the critical set $\Lambda$

 $j$ $\mu_j$ $\lambda_{j, 1}$ $\lambda_{j, 2}$ $\lambda_{j, 3}$ $\lambda_{j, 4}$ 0 10.10496819 0.31458103 0.62916205 0.94374308 1.25832410 1 8.469351217 0.34361723 0.68723445 1.03085168 1.37446891 3 3.854423919 0.50935463 1.01870927 1.52806390 2.03741854 $2^+$ 6.442637681 0.62767390 1.25534781 1.88302171 2.51069561 $2^-$ 0.007288929 11.7130006 23.4260011 35.1390017 46.8520023

Table 2.  Maximal orbit types in $\mathscr W_{j,l}$

 $\mathscr W_{j,l}$, $l\ge 1$ maximal orbit types $\mathscr W_{0,l}$ $(D_6\times D_l)$ $\mathscr W_{1l}$ $({D_6}^{{\mathbb{Z}_1}}{ \times _{{D_6}}}{D_{6l}}) - ({D_2}^{{D_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2}^{{{\tilde D}_1}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$ $\mathscr W_{2,l}$ $({D_6}^{{\mathbb{Z}_2}}{ \times _{{D_3}}}{D_{3l}}) - ({D_2}^{{\mathbb{Z}_2}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}}) - ({D_2} \times {D_l})$ $\mathscr W_{3,l}$ $({D_6}^{{{\tilde D}_3}}{ \times _{{\mathbb{Z}_2}}}{D_{2l}})$
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