Exact, E=0, Solutions for General Power-Law Potentials. I. Classical Orbits

For zero energy, $E=0$, we derive exact, classical solutions for {\em all} power-law potentials, $V(r)=-\gamma/r^\nu$, with $\gamma>0$ and $-\infty <\nu<\infty$. When the angular momentum is non-zero, these solutions lead to the orbits $\r(t)= [\cos \mu (\th(t)-\th_0(t))]^{1/\mu}$, for all $\mu \equiv \nu/2-1 \ne 0$. When $\nu>2$, the orbits are bound and go through the origin. This leads to discrete discontinuities in the functional dependence of $\th(t)$ and $\th_0(t)$, as functions of $t$, as the orbits pass through the origin. We describe a procedure to connect different analytic solutions for successive orbits at the origin. We calculate the periods and precessions of these bound orbits, and graph a number of specific examples. Also, we explain why they all must violate the virial theorem. The unbound orbits are also discussed in detail. This includes the unusual orbits which have finite travel times to infinity and also the special $\nu = 2$ case.Comment: LaTeX, 27 pages with 12 figures available from the authors or can be generated from Mathematica instructions at end of the fil

P2 receptors in atherosclerosis and postangioplasty restenosis

Atherosclerosis is an immunoinflammatory process that involves complex interactions between the vessel wall and blood components and is thought to be initiated by endothelial dysfunction [Ross (Nature 362:801–09, 1993); Fuster et al. (N Engl J Med 326:242–50, 1992); Davies and Woolf (Br Heart J 69:S3–S11, 1993)]. Extracellular nucleotides that are released from a variety of arterial and blood cells [Di Virgilio and Solini (Br J Pharmacol 135:831–42, 2002)] can bind to P2 receptors and modulate proliferation and migration of smooth muscle cells (SMC), which are known to be involved in intimal hyperplasia that accompanies atherosclerosis and postangioplasty restenosis [Lafont et al. (Circ Res 76:996–002, 1995)]. In addition, P2 receptors mediate many other functions including platelet aggregation, leukocyte adherence, and arterial vasomotricity. A direct pathological role of P2 receptors is reinforced by recent evidence showing that upregulation and activation of P2Y2 receptors in rabbit arteries mediates intimal hyperplasia [Seye et al. (Circulation 106:2720–726, 2002)]. In addition, upregulation of functional P2Y receptors also has been demonstrated in the basilar artery of the rat double-hemorrhage model [Carpenter et al. (Stroke 32:516–22, 2001)] and in coronary artery of diabetic dyslipidemic pigs [Hill et al. (J Vasc Res 38:432–43, 2001)]. It has been proposed that upregulation of P2Y receptors may be a potential diagnostic indicator for the early stages of atherosclerosis [Elmaleh et al. (Proc Natl Acad Sci U S A 95:691–95, 1998)]. Therefore, particular effort must be made to understand the consequences of nucleotide release from cells in the cardiovascular system and the subsequent effects of P2 nucleotide receptor activation in blood vessels, which may reveal novel therapeutic strategies for atherosclerosis and restenosis after angioplasty