THE CLASSICAL BERNOULLI-EULER ELASTIC CURVE IN A MANIFOLD
Abstract
In this study, we describe the classical Bernoulli-Euler elastic curve in a manifold by the property that the velocity vector field of the curve is harmonic. Then, a condition is obtained for the elastic curve in a manifold. Finally, we give an example which provides the condition mentioned in this paper and illustrate it with a figure.
Keywords
Full Text:
PDFReferences
A. Altın, On the Energy of Frenet Vectors Fields in R n . Cogent Mathematics, 2017.
P. Baird and J. C. Wood: Harmonic Morphisms Between Riemannian Manifold. Clarendos press, Oxford, 2003.
P. M. Chacón, A. M. Naveira, and J. M. Weston: On the Energy of Distributions, with Application to the Quarternionic Hopf Fibration. Monatshefte fûr Mathematik,
, 281-294. 2001.
P. A. Chacón and A. M. Naveira: Corrected Energy of Distributions on Riemannian Manifold. Osaka Journal Mathematics, Vol 41, 97-105, 2004.
L. Euler: Additamentum de curvis elasticis. In Methodus Inveniendi Lineas Curvas Maximi Minimive Probprietate Gaudentes. Lausanne, 1744.
H. A Goldstine: A History of the Calculus of Variations From the 17th Through the 19th Century. Springer, New York, 1980.
J. Guven, D. M.Valencia and J. Vazquez-Montejo: Environmental bias and elastic curves on surfaces. Phys. A: Math Theory. 47 355201,2014.
R. Huang: A Note on the p-elastica in a Constant Sectional Curvature Manifold. Journal of Geometry and Physics, Vol. 49, pp. 343-349, 2004.
B. O’Neill: Elementary Diffrential Geometry. Academic Press Inc., 1966.
T. Sakai, Riemannian Geometry. American Mathematical Society, 1996.
D. H. Steinberg: Elastic Curves in Hyperbolic Space. Doctoral Thesis, Case Western Reserve University, UMI Microform 9607925, 72p. 1995,
C. M. Wood: On the Energy of a Unit Vector Field. Geometrae Dedicata, 64, 319-330, 1997.
DOI: https://doi.org/10.22190/FUMI1903473A
Refbacks
- There are currently no refbacks.
ISSN 0352-9665 (Print)