Bipolar coordinates

Abstract: A new, general formula that connects the derivatives of the free energy along the selected, generalized coordinates of the system with the instantaneous force acting on these coordinates is derived. The instantaneous force is defined as the force acting on the coordinate of interest so that when it is subtracted from the equations of motion the acceleration along this coordinate is zero. The formula applies to simulations in which the selected coordinates are either unconstrained or constrained to fixed values. It is shown that in the latter case the formula reduces to the expression previously derived by den Otter and Briels. If simulations are carried out without constraining the coordinates of interest, the formula leads to a new method for calculating the free energy changes along these coordinates. This method is tested in two examples - rotation around the C-C bond of 1,2-dichloroethane immersed in water and transfer of fluoromethane across the water-hexane interface. The calculated free energies are compared with those obtained by two commonly used methods. One of them relies on determining the probability density function of finding the system at different values of the selected coordinate and the other requires calculating the average force at discrete locations along this coordinate in a series of constrained simulations. The free energies calculated by these three methods are in excellent agreement. The relative advantages of each method are discussed.

Abstract: A manifold is called a complex manifold if it can be covered by coordinate patches with complex coordinates in which the coordinates in overlapping patches are related by complex analytic transformations. On such a manifold scalar multiplication by i in the tangent space has an invariant meaning. An even dimensional 2n real manifold is called almost complex if there is a linear transformation J defined on the tangent space at every point (and varying differentiably with respect to local coordinates) whose square is minus the identity, i.e. if there is a real tensor field h' satisfying

Abstract: The equations of ionospheric electrodynamics are developed for a geomagnetic field of general configuration, with specific application to coordinate systems based on Magnetic Apex Coordinates. Two related coordinate systems are proposed: Modified Apex Coordinates, appropriate for calculations involving electric fields and magnetic-field-aligned currents; and Quasi-Dipole Coordinates, appropriate for calculations involving height-integrated ionospheric currents. Distortions of the geomagnetic field from a dipole cause modifications to the equations of electrodynamics, with distortion factors exceeding 50% at some geographical locations. Under the assumption of equipotential geomagnetic-field lines, it is shown how the field-line-integrated electrodynamic equations can be expressed in two dimensions in magnetic latitude and longitude, and how the height-integrated and field-aligned current densities can be calculated. Expressions are derived for the simplified calculation of magnetic perturbations above and below the ionosphere associated with the three-dimensional current system. It is shown how the base vectors for the Modified Apex coordinate system can be applied to map electric fields, plasma-drift velocities, magnetic perturbations, and Poynting fluxes along the geomagnetic field to other altitudes, automatically taking into account changes in magnitude and direction of these vector quantities along the field line. Similarly, it is shown how Quasi-Dipole coordinates are useful for expressing horizontal ionospheric currents, equivalent currents, and ground-level magnetic perturbations. A computer code is made available for efficient calculation of the various coordinates, base vectors, and related quantities described in this article.