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Slow neutron captures are responsible for the production of about 50% of elements heavier than iron, mainly, occurring during the asymptotic giant branch phase of low-mass stars (1 M/M ⊙ 3), where the main neutron source is the 13 C(α,n) 16 O reaction. This last is activated from locally-produced 13 C, formed by partial mixing of hydrogen into the He-rich layers. We present here the first attempt at describing a physical mechanism for the formation of the 13 C reservoir, studying the mass circulation induced by magnetic buoyancy and without adding new free parameters to those already involved in stellar modelling.Our approach represents the application, to the stellar layers relevant for s-processing, of recent exact, analytical 2D and 3D models for magneto-hydrodynamic processes at the base of convective envelopes in evolved stars in order to promote downflows of envelope material for mass conservation, during the occurrence of a dredge-up phenomenon. We find that the proton penetration is characterized by small concentrations, but extended over a large fractional mass of the He-layers, thus producing 13 C reservoirs of several 10 −3 M ⊙ .The ensuing 13 C-enriched zone has an almost flat profile, while only a limited production of 14 N occurs. In order to verify the effects of our new findings we show how the abundances of the main s-component nuclei can be accounted for in solar proportions and how our large 13 C-reservoir allows us to solve a few so far unexplained features in the abundance distribution of post-AGB objects.

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It is shown that the complete symmetry group for the Kepler problem, as introduced by Krause, can be derived by Lie group analysis. The same result is true for any autonomous system.

Searching for a Lagrangian may seem either a trivial endeavour or an impossible task. In this paper we show that the Jacobi last multiplier associated with the Lie symmetries admitted by simple models of classical mechanics produces (too?) many Lagrangians in a simple way. We exemplify the method by such a classic as the simple harmonic oscillator, the harmonic oscillator in disguise [H Goldstein, Classical Mechanics, 2nd edition (Addison-Wesley, Reading, 1980)] and the damped harmonic oscillator. This is the first paper in a series dedicated to this subject.

After giving a brief account of the Jacobi last multiplier for ordinary differential equations and its known relationship with Lie symmetries, we present a novel application which exploits the Jacobi last multiplier to the purpose of finding Lie symmetries of first-order systems. Several illustrative examples are given.

No abstract

Superintegrable systems on the two-dimensional sphere S 2 and the hyperbolic plane H 2