U(1)-invariant membranes: the geometric formulation, Abel and pendulum differential equations

TL;DR

This paper develops a geometric framework for U(1)-invariant membranes, linking their dynamics to Abel and pendulum differential equations, and provides exact solutions in five-dimensional Minkowski space.

Contribution

It introduces a geometric formulation of membrane dynamics involving Abel equations and classifies exact solutions in specific spacetime dimensions.

Findings

The dynamics reduce to Abel differential equations with variable coefficients.

Exact solutions for membranes in 5D Minkowski space are obtained and classified.

Radial membrane dynamics can be described by the pendulum equation when radial component depends only on time.

Abstract

The geometric approach to study the dynamics of U(1)-invariant membranes is developed. The approach reveals an important role of the Abel nonlinear differential equation of the first type with variable coefficients depending on time and one of the membrane extendedness parameters. The general solution of the Abel equation is constructed. Exact solutions of the whole system of membrane equations in the D=5 Minkowski space-time are found and classified. It is shown that if the radial component of the membrane world vector is only time dependent then the dynamics is described by the pendulum equation.