From Euler-Lagrange equations to canonical nonlinear connections

Mircea Neagu

Archivum Mathematicum (2006)

  • Volume: 042, Issue: 3, page 255-263
  • ISSN: 0044-8753

Abstract

top
The aim of this paper is to construct a canonical nonlinear connection Γ = ( M ( α ) β ( i ) , N ( α ) j ( i ) ) on the 1-jet space J 1 ( T , M ) from the Euler-Lagrange equations of the quadratic multi-time Lagrangian function L = h α β ( t ) g i j ( t , x ) x α i x β j + U ( i ) ( α ) ( t , x ) x α i + F ( t , x ) .

How to cite

top

Neagu, Mircea. "From Euler-Lagrange equations to canonical nonlinear connections." Archivum Mathematicum 042.3 (2006): 255-263. <http://eudml.org/doc/249806>.

@article{Neagu2006,
abstract = {The aim of this paper is to construct a canonical nonlinear connection $\Gamma =(M_\{(\alpha )\beta \}^\{(i)\}, N_\{(\alpha )j\}^\{(i)\})$ on the 1-jet space $J^\{1\}(T,M)$ from the Euler-Lagrange equations of the quadratic multi-time Lagrangian function \[ L=h^\{\alpha \beta \}(t)g\_\{ij\}(t,x)x\_\{\alpha \}^\{i\}x\_\{\beta \}^\{j\}+U\_\{(i)\}^\{(\alpha )\}(t,x)x\_\{\alpha \}^\{i\}+F(t,x)\,. \]},
author = {Neagu, Mircea},
journal = {Archivum Mathematicum},
keywords = {1-jet fibre bundles; nonlinear connections; quadratic Lagrangian functions; 1-jet fibre bundles; nonlinear connections; quadratic Lagrangian functions},
language = {eng},
number = {3},
pages = {255-263},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {From Euler-Lagrange equations to canonical nonlinear connections},
url = {http://eudml.org/doc/249806},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Neagu, Mircea
TI - From Euler-Lagrange equations to canonical nonlinear connections
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 3
SP - 255
EP - 263
AB - The aim of this paper is to construct a canonical nonlinear connection $\Gamma =(M_{(\alpha )\beta }^{(i)}, N_{(\alpha )j}^{(i)})$ on the 1-jet space $J^{1}(T,M)$ from the Euler-Lagrange equations of the quadratic multi-time Lagrangian function \[ L=h^{\alpha \beta }(t)g_{ij}(t,x)x_{\alpha }^{i}x_{\beta }^{j}+U_{(i)}^{(\alpha )}(t,x)x_{\alpha }^{i}+F(t,x)\,. \]
LA - eng
KW - 1-jet fibre bundles; nonlinear connections; quadratic Lagrangian functions; 1-jet fibre bundles; nonlinear connections; quadratic Lagrangian functions
UR - http://eudml.org/doc/249806
ER -

References

top
  1. Asanov G. S., Gauge-Covariant Stationary Curves on Finslerian and Jet Fibrations and Gauge Extension of Lorentz Force, Tensor (N.S.) 50 (1991), 122–137. (1991) MR1164848
  2. Asanov G. S., Jet Extension of Finslerian Gauge Approach, Fortschr. Phys. 38(8) (1990), 571–610. (1990) Zbl0744.53035MR1076500
  3. Cordero L. A., Dodson C. T. J., de Léon M., Differential Geometry of Frame Bundles, Kluwer Academic Publishers, 1989. (1989) Zbl0673.53001MR0980716
  4. Eells J., Lemaire L., A Report on Harmonic Maps, Bull. London Math. Soc. 10 (1978), 1–68. (1978) Zbl0401.58003MR0495450
  5. Giachetta G., Mangiarotti L., Sardanashvily G., Covariant Hamiltonian Field Theory, http://xxx.lanl.gov/hep-th/9904062, (1999). (1999) 
  6. Gotay M. J., Isenberg J., Marsden J. E., Momentum Maps and the Hamiltonian Structure of Classical Relativistic Fields, http://xxx.lanl.gov/hep/9801019, (1998). (1998) 
  7. Holm D. D., Marsden J. E., Raţiu T. S., The Euler-Poincaré Equations and Semidirect Products with Applications to Continuum Theories, http://xxx.lanl.gov/chao-dyn/9801015, (1998). (1998) Zbl0951.37020MR1627802
  8. Kamran N., Olver P. J., Le Probléme d’Equivalence à une Divergence prés dans le Calcul des Variations des Intégrales Multiple, C. R. Acad. Sci. Paris, 308, Série I, (1989), 249–252. (1989) MR1006072
  9. Michor P. W., Raţiu T. S., On the Geometry of the Virasoro-Bott Group, Helderman-Verlag, No. 8 (1998), 293–309. (1998) Zbl0945.58005MR1650358
  10. Miron R., Anastasiei M., The Geometry of Lagrange Spaces: Theory and Applications, Kluwer Academic Publishers, 1994. (1994) Zbl0831.53001MR1281613
  11. Miron R., Kirkovits M. S., Anastasiei M., A Geometrical Model for Variational Problems of Multiple Integrals, Proc. Conf. Diff. Geom. Appl., June 26-July 3, (1988), Dubrovnik, Yugoslavia. (1988) MR1040070
  12. Neagu M., Generalized Metrical Multi-Time Lagrange Geometry of Physical Fields, Forum Math. 15(1) (2003), 63–92. Zbl1027.53090MR1957279
  13. Neagu M., Ricci and Bianchi Identities for h -Normal Γ -Linear Connections on J 1 ( T , M ) , Hindawi Publishing Corporation, Int. J. Math. Math. Sci. 34 (2003), 2177–2192. MR1991962
  14. Neagu M., The Geometry of Relativistic Rheonomic Lagrange Spaces, Proceedings of Workshop on Diff. Geom., Global Analysis, Lie Algebras, No. 5, 142–169, (2001); Editor: Prof. Dr. Grigorios Tsagas, Aristotle University of Thessaloniki, Greece. Zbl0996.53049MR1896912
  15. Neagu M., Udrişte C., Oană A., Multi-Time Dependent Sprays and h-Traceless Maps, Balkan J. Geom. Appl. 10(2) (2005), 76–92. MR2235108
  16. Olver P. J., Applications of Lie Groups to Differential Equations, Springer-Verlag, 1986. (1986) Zbl0588.22001MR0836734
  17. Olver P. J., Canonical Elastic Moduli, J. Elasticity 19, 189–212, (1988). (1988) Zbl0658.73014MR0940120
  18. Saunders D., The Geometry of Jet Bundles, Cambridge University Press, New-York, London, 1989. (1989) Zbl0665.58002MR0989588
  19. Vondra A., Symmetries of Connections on Fibered Manifolds, Arch. Math. (Brno) 30 (1994), 97–115. (1994) Zbl0813.35006MR1292562

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.