Hamiltonian point of view of non-Euclidean geometry and elliptic functions

Authors
Citation
V. Jurdjevic, Hamiltonian point of view of non-Euclidean geometry and elliptic functions, SYST CONTR, 43(1), 2001, pp. 25-41
Citations number
14
Categorie Soggetti
AI Robotics and Automatic Control
Journal title
SYSTEMS & CONTROL LETTERS
ISSN journal
01676911 → ACNP
Volume
43
Issue
1
Year of publication
2001
Pages
25 - 41
Database
ISI
SICI code
0167-6911(20010515)43:1<25:HPOVON>2.0.ZU;2-L
Abstract
This paper offers a new way of looking at the classical geometries and the theory of elliptic functions through Hamiltonian systems on Lie groups. In particular, the paper shows that: (i) the classical models of non-Euclidean geometries are canonically induced by bi-invariant sub-Riemannian metrics on Lie groups which act by left-actions on the underlying space (ii) there is a class of canonical variational problems on Lie groups G whose projecti ons on homogeneous spaces G/K generalize Euler's elasticae and include all curves of constant curvature and all f-functions of Weierstrass; (iii) comp lex Lie groups unify non-Euclidean geometries and complex elasticae offer a distinctive look at the elliptic functions. (C) 2001 Elsevier Science B.V. All rights reserved.