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The quasihomogeneous ¢elds (featured by sums of homogeneous potentials) model a lot of concrete ¢elds met in problems of nonlinear particle dynamics belonging mainly to physics and astronomy, but not only. The particularly important aspect of singularities of the n-body problem in such ¢elds is being tackled. A Painleve¤ -type criterion for the existence of singularities and a necessary condition for collision singularities to occur are proved. An extension of the Lagrange^Jacobi relation to this case is proved, too. The main result of the paper proves that for the three-body problem all singularities are due to collisions. Some supplementary results are added: the impossibility of the simultaneous total collapse of the n bodies in in¢nite time, as well as the full understanding of the local behaviour of collision solutions within the framework of the two-body problem associated to quasihomogeneous ¢elds.Taking again into account Theorem 3.7, one sees thatAll that being established, we can state Theorem 6.5. No solution of the motion equations leads to a simultaneous total collapse in in¢nite time.Proof. Suppose that there exists a solution of Eqs (4) that leads to simultaneous total collision for t ! À1. According to Lemma 6.4, we shall have j € J Jj ! þ1 as t ! À1, or € J J ¼h h (constant), 8t 0.If € J J ! þ1 for t ! À1, then there exist an instant t 1 0 and a constant K 1 > 0 such that € J J > K 1 , 8t t 1 . Integrating this inequality twice between t 1 and t, we get ; 8t t 1 ;where ðJ s ; _ J J s Þ ¼ ðJ; _ J JÞðqðt s ÞÞ, s ¼ 0; 2, t 0 ¼ 0 (see also below). It follows that J ! þ1 for t ! À1, which contradicts Proposition 6.3.If € J J ! À1 for t ! À1, then there exist an instant t 2 0 and a constant K 2 < 0 such that € J J < K 2 , 8t t 2 . Integrating this inequality twice between t 2 and t, we getIt follows that J ! À1 for t ! À1, which contradicts again Proposition 6.3. Finally, if € J J ¼h h, 8t 0, we integrate this equality twice between 0 and t, obtainingIfh h 6 ¼ 0, the above formula shows that jJj ! þ1 for t ! À1, a contradiction. Ifh h ¼ 0 and _ J J 0 6 ¼ 0, the same formula gives jJj ! þ1 for t ! À1, a contradiction. Finally, ifh h ¼ 0 and _ J J 0 ¼ 0, by the same equality, J ¼ J 0 ¼ constant. But, by (), J 0 > 0, hence J cannot tend to zero as t ! À1, a contradiction. This completes the proof.

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We use helium atom scattering to investigate the structures formed by Li adsorption onto Cu(001) in the 0-2 ML regime. Submonolayer growth at 180 K proceeds through a sequence of ordered overlayers, including a c(2 × 2) structure at 0.5 ML and a series of 'ladder' superlattices around 0.6 ML. Beyond 1 ML, incommensurate, three-dimensional Li islands develop. A quantum close-coupled scattering analysis is performed to study the empirical He-surface potential of the structurally heterogeneous ladder structures. Good agreement with the measured distribution of diffracted intensity is obtained by describing the He-ladder interaction potential as the summation of only six one-dimensional Fourier components. The fitted potential indicates a remarkably flat surface that is punctuated by substantial, striped protrusions in the electron density. The result is consistent with the formation of one-dimensional Li wires, indicating an inhomogeneous metallization process.