Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
권호 표지
DOI QR코드

DOI QR Code

ENERGY ON A PARTICLE IN DYNAMICAL AND ELECTRODYNAMICAL FORCE FIELDS IN LIE GROUPS

  • Received : 2018.01.06
  • Accepted : 2018.02.26
  • Published : 2018.06.25
Download PDF
⟨ Previous Next ⟩

Abstract

In this study, we firstly define equations of motion based on the traditional model Newtonian mechanics in terms of the Frenet frame adapted to the trajectory of the moving particle in Lie groups. Then, we compute energy on the moving particle in resultant force field by using geometrical description of the curvature and torsion of the trajectory belonging to the particle. We also investigate the relation between energy on the moving particle in different force fields and energy on the particle in Frenet vector fields.

Keywords

References

  1. A. Altin, On the energy and Pseduoangle of Frenet Vector Fields in $R^n_v$, Ukrainian Mathematical J. 63(6) (2011), 969-975. https://doi.org/10.1007/s11253-011-0556-2
  2. V.I. Arnold, Sur la geometrie differentielle des groupes de Lie de dimension infinie et ses applications 'a l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966), 319-361.
  3. H. Bacry and J.M. Levy-Leblond, Possible Kinematics, J. Math. Phys. 9 (1968), 1605-1614. https://doi.org/10.1063/1.1664490
  4. E. Boeckx and L. Vanhecke, Harmonic and minimal vector fields on tangent and unit tangent bundles, Differential Geom. Appl. 13 (2000), 77-93. https://doi.org/10.1016/S0926-2245(00)00021-8
  5. P.M. Chacon and A.M. Naveira, Corrected Energy of Distribution on Riemannian Manifolds, Osaka J. Math. 41 (2004), 97-105.
  6. G.A. Garcia and J.S. Morales, Equations of motion of a relativistic charged particle with curvature dependent actions, Palestine Journal of Math. 3(2) (2014), 218-230.
  7. O. Gil-Medrano, Relationship between volume and energy of vector fields, Differential Geometry and its App. 15 (2001), 137-152. https://doi.org/10.1016/S0926-2245(01)00053-5
  8. H. Gluck and W. Ziller, On the volume of the unit vector fields on the three sphere, Comment Math. Helv. 61 (1986), 177-192. https://doi.org/10.1007/BF02621910
  9. H. Goldstein, C. Poole, and J. Safko, Classical Mechanics, Pearson, New Jersey (2002).
  10. F. Gonzales-Catoldo, G. Gutierrez and J.M. Yanez, Sliding down an arbitrary curve in the presence of friction, American Journal of Phys. 85(2) (2017), 108-114. https://doi.org/10.1119/1.4966628
  11. J.C. Gonzalez-Davila, and L. Vanhecke, Examples of minimal unit vector fields, Ann. Global Anal. Geom. 18 (2000), 385-404. https://doi.org/10.1023/A:1006788819180
  12. A. Gray, E. Abbena and S. Salamon, Modern Differential Geometry of curves and surfaces with Mathematica. CRC Press (1997).
  13. G. Hayward, Gravitational action for spacetimes with nonsmooth boundaries, Phys. Rev. D. 47(8) (1994).
  14. D.L. Johnson, Volume of flows, Proc. Amer. Math. Soc. 104 (1988), 923-932. https://doi.org/10.1090/S0002-9939-1988-0964875-4
  15. T. Korpinar, New Characterizations for Minimizing Energy of Biharmonic Particles in Heisenberg Spacetime, International Journal of Theoretical Phys. 53 (2014), 3208-3218. https://doi.org/10.1007/s10773-014-2118-5
  16. T. Korpinar, Constant Energy of Time Involute Particles of Biharmonic Particles in Bianchi Type-I Cosmological Model Spacetime, International Journal of Theoretical Phys. 54 (2015), 1654-1660. https://doi.org/10.1007/s10773-014-2366-4
  17. T. Korpinar, On T-magnetic Biharmonic Particles with Energy and Angle in the Three Dimensional Heisenberg Group H, Advances in Applied Clifford Algebras. 28 (2018).
  18. T. Korpinar and R.C. Demirkol, New Characterization on the Energy of Elastica with the Energy of Bishop Vector Fields in Minkowski Space, Journal of Advanced Phys. 6 (2017), 562-569. https://doi.org/10.1166/jap.2017.1375
  19. T. Korpinar and R.C. Demirkol, A New Approach on the Curvature Dependent Energy for Elastic Curves in a Lie Group, Honam Mathematical Journal. 39 (2017), 637-647.
  20. T. Korpinar and R.C. Demirkol, A New Approach on the Enegry of Elastica and Non-Elastica in Minkowski Space ${\mathbb{E}}^4_2$, Bull. Braz. Math. Soc., New Series. (2017), 1-19.
  21. M. Mansuripur, On the Electrodynamics of Moving Permanent Dipoles in External Electromagnetic Fields. Metamaterials, Fundamentals and App. (2014), 1-29.
  22. S.W. NcCuskey, Introduction to a Celestial Mechanics. Adison Wesley (1963).
  23. O.Z. Okuyucu, I. Gok and N. Ekmekci, Bertrand Curves in Three Dimensional Lie Groups. Miskolc Math. Notes, 17 (2017), 999-1010. https://doi.org/10.18514/MMN.2017.1314
  24. S. Pramanik, S. Ghosha and P. Palb, Electrodynamics of a generalized charged particle in doubly special relativity framework, Annals of Phys. 346 (2014), 113-128. https://doi.org/10.1016/j.aop.2014.04.009
  25. T. Roberts and S. Schleif, What is the experimental basis of Special Relativity? Usenet Physics FAQ. (2007).
  26. A. Romano, Classical Mechanics with Mathematica. Birkhauser, (2012).
  27. J.A. Santiagoa, G. Chacon-Acosta, O. Gonzalez-Gaxiola, and G. Torres-Vargas, Geometry of classical particles on curved surfaces, Revista Mexicana de Fis. 63 (2017), 26-31.
  28. C.M. Wood, On the Energy of a Unit Vector Field,Geom. Dedic. 64 (1997), 319-330. https://doi.org/10.1023/A:1017976425512
  29. G. Wiegmink, Total bending of vector fields on Riemannian manifolds, Math. Ann. 303 (1995), 325-344. https://doi.org/10.1007/BF01460993