Abstract
§1 shows that a particle sensible to a magnetic field obeys a new Lorentz-world-force law with respect to a special geometrical structure. Precisely, the energy of the magnetic field is incorporated into a Riemann-Jacobi metric and the Lorentz equation of charged particle motion results from this metric and from the rotor of the magnetic field (external electromagnetic field). Prom geometrical point of view the magnetic lines belong to a class of geodesics, and the magnetic surfaces appears like ruled surfaces generated by geodesics. §2 studies the stable manifold and the unstable manifold of an hyperbolic equilibrium point of a stationary magnetic dynamical system as ruled submanifolds (a surface, and a geodesic), and gives properties of a stationary magnetic field with heteroclinic structure. §3 analyses the morphology of an elementary magnetic field which admit two hyperbolic equilibrium points whose flow invariant manifolds intersect giving rise to a heteroclinic structure and a family of spatial polycyles.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R.G. Beil, Comparison of unified field theories, Tensor, N.S., 5-6, 2 (1995), 175–183.
F.M. Crampin, A linear connection associated with any second order differential equation field, In: Eds.L. Tamassy, J. Szenthe, New Developments in Geometry, 77–86, Kluwer Academic Publishers, 1996.
T. Kwaguchi, R. Miron, A Lagrangian model for gravitation and electromagnetism, Tensor, N.S., 48 (1989), 153–169.
E. Petrişor, The dynamics of a magnetic field around a heteroclinic structure, lucrare prezentatä 1a Workshop on Appl. Math., Inst, de Mat. al Academiei Bucureşti, 14-16 Sept. 1995; Seminarul de Mecanică, nr.48, Univ. de Vest din Timişoara.
Sabba S. Ştefanescu, C. Udrişte, Magnetic Field Lines around Filliform Electrical Circuits of Right Angle Type, Int. Conf. Diff. Geom. Appl., August 24-29, 1992, Bucharest, Scientific Bulletin, Applied Mathematics and Physics, Series A, 55(1993), 3-18, Politehnica University of Bucharest.
B. Sandstede, A. Scheel, Forced symmetry breaking of homoclimic cycles, Nonlinearity, The Institute of Physics and The London Mathematical Society, 8, 2(1995), 333–365.
C. Udrişte, M. Postolache, A. Udrişte, Energy of Magnetic Field Generated by Filliform Electrical Circuits of Right Angle Type, Proc. Int. Conf. Diff. Geom. Appl., August 24-29, 1992, Bucharest, Tensor N.S. 54 1993 (in printing).
C. Udrişte, M. Postolache, A. Udrişte, Numerical Simulation of Dynamic Magnetical System, Third Int.Symposium ‘Chaotic Dynamical Systems’, Utrecht, The Netherlands, June 14-17, 1992, Scientific Bulletin, Applied Mathematics and Physics, Politehnica University of Bucharest, 55 (1993), 51–64.
C. Udriçte, A. Udrişte, Properties of the magnetic lines and surfaces, Proceedings of 23 — rd Conference on Geometry and Topology, Sept 27-29, 1993, 203–208, Babes-Bolyai University, Cluj-Napoca.
C. Udrişte, A. Udrişte, C. Dumitrescu, T. Vasile, Geometry of magnetic flow, Proceedings of International Workshop on Diff. Geom. Appl., Bucharest, July 25-30, 1993; Scientific Bulletin, Applied Mathematics and Physics, Politehnica University of Bucharest, 55, 3-4 (1993), 279–283.
C. Udrişte, S. Udrişte, Biot-Savart-Laplace dynamical systems, Workshop on Global Analysis, Diff.Geom. and Lie Algebras, Aristotle University of Thessaloniki, Dec.16-18, 1993; Balkan Journal of Geometry and Its Applications, 1, 2(1996), 125–136.
C. Udrişte, A. Udrişte, V. Balan, M. Postolache, Magnetic Dynamical Systems, Proc. of the Colloquium on Diff. Geom, 25-30 July, 1994, Debrecen, Hungary; New Development in Differential Geometry, Edited by L. Tamassy, J. Szenthe, Kluwer Academic Publishers, 1996, 407–414; Analele Stiintifice ale Univ.Al.I.Cuza Iasi, IV, Informatica, (1995), 105-126.
C. Udrişte, A. Udrişte, V. Balan, M. Postolache, Phase Portraits and Critical Elements of Magnetic Fields Generated by Piecewise Rectilinear Electric Circuits, Int.Conf.on Lagrange and Finsler Geometry with Application to Diffusion in Physics and Biology, January 19-23, 1994, Braşov, Romania; In: Eds. P. Antonelli, R. Miron, Lagrange and Finsler Geometry, 177–187, Kluwer Academic Publishers, 1995.
C. Udrişte, A. Udrişte, V. Balan, M. Postolache, Magnetic field generated by two coplanar electrical circuits of fixed angle type and its field lines, Proc. of the 24-th National Conference of Geometry and Topology, Timişoara, July 5-9, 1994, 285–301.
A. Udrişte, C. Udrişte, Magnetic dynamics around electrical circuits, Workshop on Global Analysis, Differential Geometry and Lie Algebras, Aristotle University of Thessaloniki, December 16-18, 1995, Ed. Gr.Tsagas, BSG Proceedings 1(1997), 109–122, Geometry Balkan Press.
A. Udrişte, New geometrical model for particle dynamics, Workshop on Global Analysis, Differential Geometry and Lie Algebras, Aristotle University of Thessaloniki, October 21-25, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Udriste, A., Udriste, C. (1999). Dynamics Induced by a Magnetic Field. In: Szenthe, J. (eds) New Developments in Differential Geometry, Budapest 1996. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5276-1_31
Download citation
DOI: https://doi.org/10.1007/978-94-011-5276-1_31
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6220-6
Online ISBN: 978-94-011-5276-1
eBook Packages: Springer Book Archive