Skip to main content
SpringerLink
Log in

Lagrangian Curves in a 4-Dimensional Affine Symplectic Space

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Lagrangian curves in \(\mathbb {R}^{4}\) entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in \(\mathbb {R}^{4}\) and determine Lagrangian geodesics.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. An exterior differential form is semi-basic if it annihilates the vertical vectors of the fibration.

  2. A smooth immersed curve \(\delta:I \to\varLambda_{+}^{2}\) is null if its tangent vectors are isotropic (null) with respect to the conformal structure of \(\varLambda_{+}^{2}\).

  3. Contrary to the moving frame construction, the invariantization does not restrict to locally free actions. See [19].

  4. We first write R above in terms of the Λ(X (i),X (j)) so has to visualize the Gram-Schmidt process. The expression of R −1 in terms of the symplectic curvatures κ i is given afterwards.

References

  1. Álvarez Paiva, J.C., Durán, C.E.: Geometric invariants of fanning curves. Adv. Appl. Math. 42(3), 290–312 (2009)

    Article  MATH  Google Scholar 

  2. Barbot, T., Charette, V., Drumm, T., Goldman, W.M., Melnick, K.: A primer on the (2+1) Einstein universe. In: Recent Developments in Pseudo-Riemannian Geometry. ESI Lect. Math. Phys., pp. 179–229. Eur. Math. Soc., Zurich (2008)

    Chapter  Google Scholar 

  3. Boulier, F., Hubert, E.: diffalg: Description, Help Pages and Examples of Use. Symbolic Computation Group. University of Waterloo, Waterloo (1998). http://www-sop.inria.fr/members/Evelyne.Hubert/diffalg/

    Google Scholar 

  4. Capovilla, R., Guven, J., Rojas, E.: Null Frenet-Serret dynamics. Gen. Relativ. Gravit. 38(4), 689–698 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cartan, E.: Sur le problème général de la déformation. In: C.R. Congrés de Strasbourg, pp. 397–406 (1920)

    Google Scholar 

  6. Chern, S.S., Wang, H.C.: Differential geometry in symplectic space. I. Sci. Rep. Nat. Tsing Hua Univ. 4, 453–477 (1947)

    MathSciNet  Google Scholar 

  7. Deconchy, V.: Hypersurfaces in symplectic affine geometry. Differ. Geom. Appl. 17(1), 1–13 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Fels, M., Olver, P.J.: Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55(2), 127–208 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Feoli, A., Nesterenko, V.V., Scarpetta, G.: Functionals linear in curvature and statistics of helical proteins. Nucl. Phys. B 705(3), 577–592 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Griffiths, P.: On Cartan’s method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry. Duke Math. J. 41, 775–814 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  11. Griffiths, P.A.: Exterior Differential Systems and the Calculus of Variations. Progress in Mathematics, vol. 25. Birkhäuser Boston, Cambridge (1983)

    Book  MATH  Google Scholar 

  12. Guillemin, V., Sternberg, S.: Variations on a Theme by Kepler. American Mathematical Society Colloquium Publications., vol. 42. Am. Math. Soc., Providence (1990)

    MATH  Google Scholar 

  13. Hubert, E.: diffalg: Extension to non Commuting Derivations. INRIA, Sophia Antipolis (2005). http://www.inria.fr/members/Evelyne.Hubert/diffalg

  14. Hubert, E.: Differential algebra for derivations with nontrivial commutation rules. J. Pure Appl. Algebra 200(1–2), 163–190 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  15. Hubert, E.: The maple package aida—Algebraic Invariants and Their Differential Algebras. INRIA (2007). http://www.inria.fr/members/Evelyne.Hubert/aida

  16. Hubert, E.: Differential invariants of a Lie group action: syzygies on a generating set. J. Symb. Comput. 44(3), 382–416 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hubert, E.: Algebraic and differential invariants. In: Cucker, F., Krick, T., Pinkus, A., Szanto, A. (eds.) Foundations of Computational Mathematics, Budapest 2011. London Mathematical Society Lecture Note Series, vol. 403. Cambridge University Press, Cambridge (2012)

    Google Scholar 

  18. Hubert, E.: Generation properties of Maurer-Cartan invariants. (2012, preprint). http://hal.inria.fr/inria-00194528

  19. Hubert, E., Kogan, I.A.: Smooth and algebraic invariants of a group action. Local and global constructions. Found. Comput. Math. 7(4), 345–383 (2007)

    Article  MathSciNet  Google Scholar 

  20. Hubert, E., Olver, P.J.: Differential invariants of conformal and projective surfaces. SIGMA 3(097) (2007)

  21. Inoguchi, J., Lee, S.: Null curves in Minkowski 3-space. Int. Electron. J. Geom. 1(2), 40–83 (2008)

    MATH  MathSciNet  Google Scholar 

  22. Ivey, T.A., Landsberg, J.M.: Cartan for Beginners: Differential Geometry Via Moving Frames and Exterior Differential Systems. Graduate Studies in Mathematics, vol. 61. Am. Math. Soc., Providence (2003)

    Google Scholar 

  23. Jensen, G.R.: Deformation of submanifolds of homogeneous spaces. J. Differ. Geom. 16(2), 213–246 (1981)

    MATH  Google Scholar 

  24. Kamran, N., Olver, P., Tenenblat, K.: Local symplectic invariants for curves. Commun. Contemp. Math. 11(2), 165–183 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Kogan, I.A., Olver, P.J.: Invariant Euler-Lagrange equations and the invariant variational bicomplex. Acta Appl. Math. 76(2), 137–193 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kuznetsov, Y., Plyushchay, M.: (2+1)-dimensional models or relativistic particles with curvature and torsion. J. Math. Phys. 35(6), 2772–2778 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mansfield, E.L.: A Practical Guide to the Invariant Calculus. Cambridge University Press, Cambridge (2010)

    Book  MATH  Google Scholar 

  28. Marí Beffa, G.: On completely integrable geometric evolutions of curves of Lagrangian planes. Proc. R. Soc. Edinb. A 137(1), 111–131 (2007)

    Article  MATH  Google Scholar 

  29. Marí Beffa, G.: Hamiltonian evolution of curves in classical affine geometries. Phys. D, Nonlinear Phenom. 238(1), 100–115 (2009)

    Article  MATH  Google Scholar 

  30. McKay, B.: Lagrangian submanifolds in affine symplectic geometry. Differ. Geom. Appl. 24(6), 670–689 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. Musso, E., Grant, J.D.E.: Coisotropic variational problems. J. Geom. Phys. 50(1–4), 303–338 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Musso, E., Nicolodi, L.: Closed trajectories of a particle model on null curves in anti-de Sitter 3-space. Class. Quantum Gravity 24(22), 5401–5411 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  33. Musso, L., Nicolodi, E.: Reduction for constrained variational problems on 3-dimensional null curves. SIAM J. Control Optim. 47(3), 1399–1414 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  34. Musso, E., Nicolodi, L.: Symplectic applicability of Lagrangian surfaces. SIGMA 5, Paper 067, 18 (2009)

    MathSciNet  Google Scholar 

  35. Musso, E., Nicolodi, L.: Hamiltonian flows on null curves. Nonlinearity 23(9), 2117–2129 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  36. Nersessian, A., Manvelyan, R., Müller-Kirsten, H.J.W.: Particle with torsion on 3d null-curves. Nucl. Phys. B, Proc. Suppl. 88, 381–384 (2000). Constrained dynamics and quantum gravity, 1999, (Villasimius)

    Article  MATH  Google Scholar 

  37. Nesterenko, V.V.: Curvature and torsion of the world curve in the action of the relativistic particle. J. Math. Phys. 32(12), 3315–3320 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  38. Nesterenko, V.V., Feoli, A., Scarpetta, G.: Dynamics of relativistic particles with Lagrangians dependent on acceleration. J. Math. Phys. 36(10), 5552–5564 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  39. Nesterenko, V.V., Feoli, A., Scarpetta, G.: Complete integrability for Lagrangians dependent on acceleration in a spacetime of constant curvature. Class. Quantum Gravity 13(5), 1201–1211 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  40. Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107. Springer, New York (1986)

    MATH  Google Scholar 

  41. Olver, P.J.: Equivalence, Invariants and Symmetry. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  42. Olver, P.J.: Generating differential invariants. J. Math. Anal. Appl. 333, 450–471 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  43. Sharpe, R.W.: Differential Geometry. Graduate Texts in Mathematics, vol. 166. Springer, New York (1997)

    MATH  Google Scholar 

  44. Valiquette, F.: Geometric affine symplectic curve flows in \(\Bbb{R}^{4}\). Differ. Geom. Appl. 30(6), 631–641 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departement of Mathematical Sciences, Politecnico di Torino, Turin, Italy

    Emilio Musso

  2. GALAAD, INRIA Méditerranée, Sophia Antipolis, France

    Evelyne Hubert

Authors
  1. Emilio Musso
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Evelyne Hubert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emilio Musso.

Additional information

This research is partially supproted by MIUR (Italy) under the PRIN project Varieta’ reali e complesse: geometria, topologia e analisi armonica. E. Musso partially supported by GNSAGA of INDAM.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Musso, E., Hubert, E. Lagrangian Curves in a 4-Dimensional Affine Symplectic Space. Acta Appl Math 134, 133–160 (2014). https://doi.org/10.1007/s10440-014-9874-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-014-9874-3

Keywords

Mathematics Subject Classification

Search

Navigation