On the variational approach to the periodic n-body problem

This expository paper gathers some of the results obtained by the author in recent works in collaboration with Davide Ferrario and Vivina Barutello, focusing on the periodic n-body problem from the perspective of the calculus of variations and minimax theory. These researches were aimed at developing a systematic variational approach to the equivariant periodic n-body problem in the two and three-dimensional space. The purpose of this paper is to expose the main problems and achievements of this approach. The material here was exposed in the talk that given at the Meeting CELMEC IV promoted by SIMCA (Società italiana di Meccanica Celeste).     

Bulk organic δ13C and C/N ratios as palaeosalinity indicators within a Scottish isolation basin

Microfossils in isolation basin sediments are frequently used to reconstruct sea‐level change, but preservation problems and non‐analogue situations can limit their usefulness. Here we investigate the potential of stable carbon isotopes (δ¹³C) and C/N ratios from bulk organic matter, as an alternative proxy of salinity within isolation basin sediments from a basin in northwest Scotland. Within the Holocene sediment δ¹³C and C/N are determined largely by the mean weighted values of the predominant source of the organic material. Analysis of modern materials and comparison with the diatom record shows that the marine parts of the sequence are dominated by high δ¹³C and variable C/N. In the fresh water sequences the organic material is a mixture of both freshwater aquatic and terrestrial plant input that have relatively low δ¹³C and high C/N. The application of δ¹³C and C/N ratios in the studied basin in general follow the environmental change recorded by the diatoms and shows the potential of bulk organic matter in the investigation of salinity change in isolation basins. Copyright © 2005 John Wiley & Sons, Ltd.     

On Motions Near the Lagrange Equilibrium Point L 4 in the Case of Routh's Critical Mass Ratio

We deal with the planar restricted circular problem of three bodies. We study trajectories in a small neighborhood of the Lagrange equilibrium point L ₄ when mass ratio is close to Routh's value. In particular, we show that the case of proper degeneracy takes place and for most initial conditions trajectories are conditionally-periodic. We obtain an approximate representation of families of periodic solution emanating from the equilibrium point L ₄. We also show that in the case of instability of L ₄ the trajectories starting in a vicinity of L ₄ remain in a finite domain forever. We give an upper bound of this domain. To carry out our investigation, we analyze the dynamics of a general Hamiltonian system with two degrees of freedom in the case of 1 : 1 resonance in detail.This revised version was published online in October 2005 with corrections to the Cover Date.     

Optical Spectroscopy of Comet C/2000 WM1 (LINEAR) at the Guillermo Harro Astrophysical Observatory in Mexico

We present a preliminary analysis of medium resolution optical spectra of comet C/2000 WM1 (LINEAR) obtained on 22 November 2001. Theemission lines of the molecules C₂, C₃, CN, NH₂,H₂O⁺ and presumably CO (Asundi and triplet bands) and C₂ ^-were identified in these spectra. By analysing the brightnessdistributions of the C₂, C₃, CN emission lines along theslit of the spectrograph we determined some physical parameters of theseneutrals, such as their lifetimes and expansion velocities inthe coma. The Franck–Condon factors for the CO Asundi bands and C₂ ^- bands were calculated using a Morse potential model.     

The influence of diurnal temperatures on the hydrochemistry of a tufa‐depositing stream

At‐a‐station diurnal variations in carbonate hydrochemistry were measured during four observation periods at Davys Creek, a tufa‐depositing stream in central NSW, Australia. Major ion concentrations and continuously logged measurements of specific conductivity, pH and temperature showed that changes in the amount of CaCO₃ deposited upstream of the study reach were directly related to changes in diurnal water temperatures, which control the rate of CO₂ efflux to the atmosphere. The greatest upstream losses occurred during the mid‐afternoon water temperature peak, whereas the lowest upstream losses occurred at sunrise, when water temperatures were at their lowest. Cloudy days at all times of the year produced small diurnal water temperatures ranges (c. 2–5°C) and, consequently, relatively small changes in upstream CaCO₃ loss (23–50 mg L^?1) through the day. Clear sunny days, especially during summer months, produced large diurnal water temperature changes (up to c. 11°C), which in turn triggered diurnal changes in upstream CaCO₃ loss of up to 100 mg L^?1. By implication, the active reach of tufa deposition must advance downstream and increase in length during the evening and vice versa during the day. Given that the temperature of Davys Creek waters are a function of insolation, changes in the reach of tufa deposition under baseflow conditions are a direct function of the prevailing weather. This has implications for the palaeoclimatic interpretation of fossil tufa deposits. Copyright © 2003 John Wiley & Sons, Ltd.