Anholonomik Koordinatlar Boyunca Yönlü Enerji Fonksiyonelleri

Yıl 2022, , 46 - 60, 24.03.2022

Ridvan Cem Demırkol

https://doi.org/10.17798/bitlisfen.991769

Öz

Bu makalede 3-boyutlu Öklid uzayında bazı özel yönlü enerji fonksiyonelleri birim vektör alanlarının yönlü enerjileri ve pseudo-açıları hesaplanarak incelenmiştir. Bu yaklaşımdan hareketle hız vektörlerinin yönlü enerji fonksiyonellerince belirlenen kritik noktaları tanımlanmıştır. Daha sonra yönlü enerji fonksiyonellerinin ekstremum değerleri ve harmonik dönüşümleri için bazı şartlar tartışılmıştır. Sonuç olarak, vektör alanlarının total eğrilmesi veya enerjisi olarak bilinen kavramları genelleştirecek şekilde eğrilik vektör alanının yönlü harmonik ve biharmonik denklemleri tanımlanmıştır.

Anahtar Kelimeler

Yönlü enerji fonksiyoneli, yönlü enerji minimizesi, harmoniklik ve biharmoniklik

Directional Energy Functionals Through Anholonomic Coordinates

Öz

In this paper, a special case of directional energy functional is investigated by computing the directional energy and pseudoangle of unit vector fields in the ordinary three-dimensional space. This approach is also extended simultaneously to define the critical points of the directional energy functionals of the velocity fields. Then, the restriction of the harmonic maps and the extrema of the directional energy functionals is considered, Finally, we compute directional harmonic and biharmonic equations of the curvature vector fields to generalize total bending or energy of vector fields.

Anahtar Kelimeler

Directional energy functional, directional energy minimizer, harmonicity and biharmonicty

Kaynakça

• [1] Wiegmink G. 1995. Total bending of vector fields on Riemannian manifolds. Mathematische Annalen 303 (1): 325-344.
• [2] Wiegmink G. 1996. Total bending of vector fields on the sphere S³. Differential Geometry and its Applications, 6 (3): 219-236.
• [3] Gluck H., Ziller W. 1986. On the volume of a unit vector field on the three-sphere. Commentarii Mathematici Helvetici, 61 (1): 177-192.
• [4] Brito F.G. 2000. Total bending of flows with mean curvature correction. Differential Geometry And its Applications, 12 (2): 157-163.
• [5] Wood C.M. 1997. On the energy of a unit vector field. Geometriae Dedicata, 64 (3): 319-330.
• [6] Chacon P.M., Naveira A.M., Weston J.M. 2001. On the energy of distributions on Riemannian manifolds. Osaka Journal of Mathematics, 41 (1): 97-105.
• [7] Chacon P.M., Naveira, A.M., Weston, J.M. 2001. On the energy of distributions, with application to the quaternionic Hopf fibrations. Monatshefte für Mathematik, 133 (4): 281-294.
• [8] Altın A. 2011. On the energy and pseudoangle of Frenet vector fields in R_{v}ⁿ. Ukrainian Mathematical Journal, 63 (6): 969-976.
• [9] Altın A. 2017. Minimizing the energy of the velocity vector field of curve in R³. The Online Mathematical Journal, 7 (2): 91-94.
• [10] Altın A. 2015. The energy of a domain on the surface. Ukrainian Mathematical Journal, 67 (4): 641-647.
• [11] Körpınar T., Demirkol R.C. 2017. Energy on a timelike particle in dynamical and electrodynamical force fields in De-Sitter space. Revista Mexicana de Física, 63 (6): 560-568.
• [12] Körpınar T., Demirkol R.C. 2017. A new characterization on the energy of elastica with the energy of Bishop vector fields in Minkowski space. Journal of Advanced Physics, 6 (4): 562-569.
• [13] Körpınar T., Demirkol R.C. 2017. A new approach on the curvature dependent energy for elastic curves in a Lie group. Honam Mathematical Journal, 39 (4): 637-647.
• [14] Inoguchi J.I. 2004. Submanifolds with harmonic mean curvature vector field in contact 3- manifolds. Colloq. Math. 100 (2): 163-179.
• [15] Sasahara T. 2005. Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors. Publ. Math. Debrecen, 67 (3-4): 285-303.
• [16] Chen B.Y. 1996. A report on submanifolds of finite type. Soochow Journal of Mathematics, 22 (2): 117-337.
• [17] Chen B.Y. 2014. Total mean curvature and submanifolds of finite type, Vol: 27, World Sci. Publishing Company.
• [18] Chen B.Y. 1995. Submanifolds in de sitter space-time satisfying ΔH=λH. Israel Journal of Mathematics, 91 (1): 373-391.
• [19] Chen B.Y. 1994. Some classification theorems for submanifolds in Minkowski space-time. Archiv der Mathematik, 62 (2): 177-182.
• [20] Turhan E., Körpinar T. 2010. On characterization of time-like horizontal biharmonic curves in the Lorentzian Heisenberg group Heis3. Zeitschrift für Naturforschung A, 65 (8-9): 641-648.
• [21] Körpinar T. 2018. On T-magnetic biharmonic particles with energy and angle in the three dimensional Heisenberg group H3. Advances in Applied Clifford Algebras, 28 (1): 1-15.
• [22] Körpinar, T. 2014. New characterizations for minimizing energy of biharmonic particles in Heisenberg spacetime. International Journal of Theoretical Physics, 53 (9): 3208-3218.
• [23] Körpinar T. 2015. Bianchi type-I cosmological models for inextensible flows of biharmonic particles by using curvature tensor field in spacetime. International Journal of Theoretical Physics, 54 (6): 1762-1774.
• [24] Körpınar T., Turhan E. 2013. Biharmonic S-curves according to Sabban frame in Heisenberg group Heis³. Boletim da Sociedade Paranaense de Matemática, 31 (1): 205-211.
• [25] Turhan E., Körpinar T. 2011. On characterization canal surfaces around timelike horizontal biharmonic curves in Lorentzian Heisenberg group Heis3. Zeitschrift für Naturforschung A, 66 (6-7): 441-449.
• [26] Schief W.K., Rogers C. 1999. Binormal motion of curves of constant curvature and torsion Generation of soliton surfaces. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 455 (1988): 3163-3188.
• [27] Sakai T. 1996. Riemannian geometry, Vol: 149, American Mathematical Society.