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We propose a relation which predicts the low-frequency thermal conductivity of a one-dimensional (1D) system from the thermal conductivity and bulk viscosity at higher frequency. Our theory is based on the assumption that "ballistic" transport by sound waves dominates the heat transport. For a system with equal heat capacities (c(p) = c(v)) this relation is particularly simple. We test the prediction by simulating a chain of particles with quartic interparticle potentials under zero pressure conditions. As the frequency omega --> 0 the theory predicts that the energy current power spectrum diverges as omega(-1/2), not seen in previous simulations. Because we simulate very long chains to long times we do observe the crossover into this regime. The bulk viscosity of a 1D chain has been determined via simulation. It is found to be finite for our system, in contrast to the thermal conductivity which is infinite.

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We combine molecular dynamics (MD) simulations and experiment, both small-angle neutron (SANS) and small-angle X-ray scattering (SAXS), to determine the precise structure of bilayers composed of 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylglycerol (POPG), a lipid commonly encountered in bacterial membranes. Experiment and simulation are used to develop a one-dimensional scattering density profile (SDP) model suitable for the analysis of experimental data. The joint refinement of such data (i.e., SANS and SAXS) results in the area per lipid that is then used in the fixed-area simulations. In the final step, the direct comparison of simulated-to-experimental data gives rise to the detailed structure of POPG bilayers. From these studies we conclude that POPG's molecular area is 66.0 ( 1.3 Å 2 , its overall bilayer thickness is 36.7 ( 0.7 Å, and its hydrocarbon region thickness is 27.9 ( 0.6 Å, assuming a simulated value of 1203 Å 3 for the total lipid volume.

We examine the nature of the stationary character of the Hamilton action S for a space-time trajectory ͑worldline͒ x͑t͒ of a single particle moving in one dimension with a general time-dependent potential energy function U͑x , t͒. We show that the action is a local minimum for sufficiently short worldlines for all potentials and for worldlines of any length in some potentials. For long enough worldlines in most time-independent potentials U͑x͒, the action is a saddle point, that is, a minimum with respect to some nearby alternative curves and a maximum with respect to others. The action is never a true maximum, that is, it is never greater along the actual worldline than along every nearby alternative curve. We illustrate these results for the harmonic oscillator, two different nonlinear oscillators, and a scattering system. We also briefly discuss two-dimensional examples, the Maupertuis action, and newer action principles.

Magnetic fields play a crucial role at all stages of the formation of low-mass stars and planetary systems. In the final stages, in particular, they control the kinematics of in-falling gas from circumstellar discs, and the launching and collimation of spectacular outflows. The magnetic coupling with the disc is thought to influence the rotational evolution of the star, while magnetized stellar winds control the braking of more evolved stars and may influence the migration of planets. Magnetic reconnection events trigger energetic flares which irradiate circumstellar discs with high energy particles that influence the disc chemistry and set the initial conditions for planet formation. However, it is only in the past few years that the current generation of optical spectropolarimeters has allowed the magnetic fields of forming solar-like stars to be probed in unprecedented detail. In order to do justice to the recent extensive observational programs new theoretical models are being developed that incorporate magnetic fields with an observed degree of complexity. In this review we draw together disparate results from the classical electromagnetism, molecular physics/chemistry and the geophysics literature, and demonstrate how they can be adapted to construct models of the large scale magnetospheres of stars and planets. We conclude by examining how the incorporation of multipolar magnetic fields into new theoretical models will drive future progress in the field through the elucidation of several observational conundrums.(Some figures in this article are in colour only in the electronic version) 3.4. Difference between a spherical and Cartesian tensor approach 13 4. Magnetospheric accretion models with multipolar magnetic fields 14 4.1. Development of PFSS models and comparison with MHD field extrapolations 14 4.2. Potential field models of T Tauri magnetospheres with complex fields 16 4.3. 3D MHD models of T Tauri magnetospheres with non-dipolar fields 18 5. Summary and applications to outstanding problems 19 Acknowledgments 21 Appendix A. Relations between the equatorial and polar field strength for a multipole of arbitrary order l 21 Appendix A.1. Odd-order multipoles 21 Appendix A.2. Even-order multipoles 22 Appendix B. Electrostatic expansion using Cartesian tensors 22 Appendix B.1. The dipole term 23 Appendix B.2. The quadrupole term 23 References 24

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We review the development and practical uses of a generalized Maupertuis least action principle in classical mechanics, in which the action is varied under the constraint of fixed mean energy for the trial trajectory. The original Maupertuis (Euler-Lagrange) principle constrains the energy at every point along the trajectory. The generalized Maupertuis principle is equivalent to Hamilton's principle. Reciprocal principles are also derived for both the generalized Maupertuis and the Hamilton principles. The Reciprocal Maupertuis Principle is the classical limit of Schrödinger's variational principle of wave mechanics, and is also very useful to solve practical problems in both classical and semiclassical mechanics, in complete analogy with the quantum Rayleigh-Ritz method. Classical, semiclassical and quantum variational calculations are carried out for a number of systems, and the results are compared. Pedagogical as well as research problems are used as examples, which include nonconservative as well as relativistic systems.