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Gauss's Law: Planar Symmetry

Gauss's Law: Planar Symmetry

A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...

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Self-assembly of Complex Two-dimensional Shapes from Single-stranded DNA Tiles

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G-Strands on symmetric spaces.

Alexis Arnaudon¹, Darryl D Holm¹, Rossen I Ivanov²

¹Department of Mathematics, Imperial College, London SW7 2AZ, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences

|April 18, 2017

Summary

This summary is machine-generated.

This study explores G-strand equations, extending chiral models in particle physics using symmetric spaces. New integrable models and Camassa-Holm equations are derived, revealing odd function solutions.

Keywords:

Camassa–Holm equation Lie groups chiral model integrability

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Area of Science:

Theoretical Physics
Mathematical Physics

Background:

Classical chiral models are fundamental in particle physics.
Symmetric spaces provide a framework for studying broken symmetries.

Purpose of the Study:

To extend classical chiral models using G-strand equations.
To derive and analyze integrable models on Lie groups and symmetric spaces.
To investigate G-strands on infinite-dimensional diffeomorphism groups.

Main Methods:

Derivation of G-strand equations on finite-dimensional Lie groups and symmetric spaces.
Analysis of specific examples: SU(2)/S¹ and SO(4)/SO(3).
Study of G-strands on the infinite-dimensional group of diffeomorphisms with Sobolev norm.

Main Results:

Complete integrability of several G-strand models on finite-dimensional Lie groups.
Identification of a new integrable nine-dimensional system from the SO(4)/SO(3) model.
Derivation of 1+2 Camassa-Holm equations from G-strands on diffeomorphism groups.
Characterization of odd function solutions for these equations.

Conclusions:

G-strand equations offer a versatile framework for integrable systems in theoretical physics.
The study introduces novel integrable models and connections to Camassa-Holm equations.
Odd functions play a crucial role in the solutions on specific spaces.