A Lie-Algebraic Formulation for Triply Orthogonal and General Coordinate Systems in Three-Dimensional Euclidean and Lorentz Spaces

B. A. Shipman; P. D. Shipman

doi:10.5802/jolt.789

B. A. Shipman; P. D. Shipman

Journal of Lie Theory, Volume 24 (2014) no. 2, pp. 397-419

Abstract

We give a Lie-algebraic formulation for the interacting geometries of orthogonal families of coordinate surfaces in 3-dimensional Euclidean- and Lorentz-orthogonal coordinate systems. A study of the Gauss-Lamé equations and their variational equations in this setting leads to formulas for constructing more general 3-dimensional coordinate transformations. To motivate the general constructions, we begin with special cases of orthogonal coordinate systems in 3-dimensional Lorentz space, built from orthogonal systems in the plane.

arXiv Zbl

DOI: 10.5802/jolt.789

Classification: 53C21, 53C12, 53A05, 53A35, 53Z05
Keywords: Orthogonal coordinate systems, Gauss-Lame equations, Lorentz space

@article{JOLT_2014_24_2_a4,
     author = {B. A. Shipman and P. D. Shipman},
     title = {A {Lie-Algebraic} {Formulation} for {Triply} {Orthogonal} and {General} {Coordinate} {Systems} in {Three-Dimensional} {Euclidean} and {Lorentz} {Spaces}},
     journal = {Journal of Lie Theory},
     pages = {397--419},
     year = {2014},
     volume = {24},
     number = {2},
     doi = {10.5802/jolt.789},
     zbl = {1300.53023},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.789/}
}

TY  - JOUR
AU  - B. A. Shipman
AU  - P. D. Shipman
TI  - A Lie-Algebraic Formulation for Triply Orthogonal and General Coordinate Systems in Three-Dimensional Euclidean and Lorentz Spaces
JO  - Journal of Lie Theory
PY  - 2014
SP  - 397
EP  - 419
VL  - 24
IS  - 2
UR  - https://jolt.centre-mersenne.org/articles/10.5802/jolt.789/
DO  - 10.5802/jolt.789
ID  - JOLT_2014_24_2_a4
ER  -

%0 Journal Article
%A B. A. Shipman
%A P. D. Shipman
%T A Lie-Algebraic Formulation for Triply Orthogonal and General Coordinate Systems in Three-Dimensional Euclidean and Lorentz Spaces
%J Journal of Lie Theory
%D 2014
%P 397-419
%V 24
%N 2
%U https://jolt.centre-mersenne.org/articles/10.5802/jolt.789/
%R 10.5802/jolt.789
%F JOLT_2014_24_2_a4

B. A. Shipman; P. D. Shipman. A Lie-Algebraic Formulation for Triply Orthogonal and General Coordinate Systems in Three-Dimensional Euclidean and Lorentz Spaces. Journal of Lie Theory, Volume 24 (2014) no. 2, pp. 397-419. doi: 10.5802/jolt.789

Cited by Sources: