Geometric continuum models for fluid lipid membranes are considered using classical field theory, within a covariant variational approach. The approach is cast as a higher-derivative Lagrangian formulation of continuum classical field theory, and it can be seen as a covariant version of the field theoretical variational approach that uses the height representation. This novel Lagrangian formulation is presented first for a generic reparametrization invariant geometric model, deriving its equilibrium equation, or shape equation, and its linear and angular stress tensors, using the classical Canham-Helfrich elastic bending energy for illustration. The robustness of the formulation is established by extending it to the presence of external forces, and to the case of heterogenous lipid membranes, breaking reparametrization invariance. In addition, a useful and compact general expression for the second variation of the free energy is obtained within the Lagrangian formulation, as a first step towards the study of the stability of membrane configurations. The simple structure of the expressions derived for the basic entities that appear in the mechanics of a lipid membrane is a direct consequence of the well established power of a Lagrangian variational approach. The paper is self-contained, and it is meant to provide, besides a new framework, also a convenient introduction to the mechanics of lipid membranes.

In this paper we consider twice-dimensionally reduced, generalized Seiberg-Witten (S-W) equations, defined on a compact Riemann surface. A novel feature of the reduction technique is that the resulting equations produce an extra “Higgs field”. Under suitable regularity assumptions, we show that the moduli space of gauge-equivalent classes of solutions to the reduced equations, is a smooth Kähler manifold and construct a pre-quantum line bundle over the moduli space of solutions.

A general model is formulated for a universal fleet of all unmanned vehicles, including Aerial Vehicles (UAVs), Ground Vehicles (UGVs), Sea Vehicles (USVs) and Underwater Vehicles (UUVs), as a geometric Kähler dynamics and control system. Based on the Newton-Euler dynamics of each vehicle, a control system for the universal autonomous fleet is designed as a combined Lagrangian and Hamiltonian form. The associated continuous system representing a very large universal fleet is given in Appendix in the form of the Kähler-Ricci flow.

In this paper, we study rotational surfaces in the pseudo-Galilean three-space $\mathbb G_3^1$ with pseudo-Euclidean rotations and isotropic rotations. In particular, we investigate properties of geodesics on rotational surfaces in $\mathbb G_3^1$ and give some examples.

We pose a new problem of collinear central configurations in Newtonian $n$-body problem. It is known that the configuration of two bodies moving along the Newtonian force is always a collinear central configuration. Can we add new two bodies on the straight line of initial two bodies without changing the move of the initial two bodies and the configuration of the four bodies is central, too? We call it 2+2 Moulton configuration. We find three special solutions to this problem and find each mass of new two bodies is zero.