Global dynamics of the Buckingham problem [ Back ]

Date:
19.03.18   
Times:
15:30 to 16:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Jaume Llibre
University:
Universitat Autònoma de Barcelona

Abstract:

The Buckingham potential describes the Pauli repulsion energy and van der Waals energy for the interaction of two atoms as a function of the interatomic distance. The Hamiltonian which governs its dynamics is $$
H = \frac12 (p_x^2 + _py^2) + A \exp(-B \sqrt{x^2 + y^2}) -
\frac{M}{(x^2 + y^2)^3},
$$
where the constants $A$, $B$ and $M$ are positive.

The corresponding Hamiltonian systems with two degrees of freedom is integrable with the two independent first integrals given by the Hamiltonian $H$ and the angular momentum $C$.

Let $I_h$ (respectiveley $I_c$) be the set of points of the phase space on which $H$ (respectively $C$) takes the value $h$ (respectively $c$). Since $H$ and $C$ are first integrals, the sets $I_h$, $I_c$ and $I_{hc}= I_h \cap I_c$ are invariant under the flow of the Buckingham systems. We characterize the global flow of these systems when the parameters $A$, $B$ and $M$ vary. Thus we describe the foliation of the phase space by the invariant sets $I_h$ and the foliation of $I_h$ by the invariant sets $I_{hc}$.