March  2018, 10(1): 69-92. doi: 10.3934/jgm.2018003

The projective Cartan-Klein geometry of the Helmholtz conditions

Setor de Ciências Exatas, Centro Politécnico, Caixa Postal 019081, CEP 81531-990, Curitiba, PR, Brazil

* Corresponding author

Received  April 2016 Revised  August 2017 Published  December 2017

We show that the Helmholtz conditions characterizing differential equations arising from variational problems can be expressed in terms of invariants of curves in a suitable Grassmann manifold.

Citation: Carlos Durán, Diego Otero. The projective Cartan-Klein geometry of the Helmholtz conditions. Journal of Geometric Mechanics, 2018, 10 (1) : 69-92. doi: 10.3934/jgm.2018003
References:
[1]

A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach, preprint, arXiv: 1306. 5318v5. Google Scholar

[2]

J. C. Álvarez Paiva and C. E. Durán, Geometric invariants of fanning curves, Adv. in Appl. Math., 42 (2009), 290-312.  doi: 10.1016/j.aam.2006.07.008.  Google Scholar

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I. Anderson, Introduction to variational bicomplex, Contemp. Math., 132 (1992), 51-73.   Google Scholar

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M. de León and P. R. Rodrigues, The inverse problem of Lagrangian dynamics for higher-order differential equations -a geometrical approach, Inverse Prob., 8 (1992), 525-540.  doi: 10.1088/0266-5611/8/4/006.  Google Scholar

[5]

C. E. Durán and D. Otero, The projective symplectic geometry of higher order variational problems: Minimality conditions, J. Geom. Mech., 8 (2016), 305-322.  doi: 10.3934/jgm.2016009.  Google Scholar

[6]

C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible grassmannians, Diff. Geom. Appl., 49 (2016), 447-472.  doi: 10.1016/j.difgeo.2016.10.001.  Google Scholar

[7]

C. E. Durán and H. Vitório, Moving planes, Jacobi curves and the dynamical approach to Finsler geometry, European Journal of Mathematics, 3 (2017), 1245-1274.   Google Scholar

[8]

H. Flanders, The Schwarzian as a curvature, J. Differential Geometry, 4 (1970), 515-519.  doi: 10.4310/jdg/1214429647.  Google Scholar

[9]

S. R. Garcia, The eigenstructure of complex symmetric operators, Recent Advances in Operator Theory, Operator Theory: Advances and Applications, 179 (2008), 169-183.   Google Scholar

[10]

I. Gelfand and S. Fomin, Calculus of Variations, Silverman Prentice-Hall, Inc., Englewood Cliffs, N. J. 1963.  Google Scholar

[11]

P. Piccione and D. V. Tausk, A students' guide to symplectic spaces, Grassmannians and Maslov index, Publicações Matemáticas do IMPA, Rio de Janeiro, 2008.  Google Scholar

[12]

W. SarletE. Engels and L. Y. Bahar, Time-dependent linear systems derivable from a variational principle, Int. J. Eng. Sci., 20 (1982), 55-66.  doi: 10.1016/0020-7225(82)90072-6.  Google Scholar

[13]

P. Vassiliou and I. Lisle, Geometric Approaches to Differential Equations, Australian Mathematical Society Lecture Series, Cambridge University Press, 2000. Google Scholar

[14]

E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces, Chelsea Publishing Co., New York, 1962.  Google Scholar

show all references

References:
[1]

A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach, preprint, arXiv: 1306. 5318v5. Google Scholar

[2]

J. C. Álvarez Paiva and C. E. Durán, Geometric invariants of fanning curves, Adv. in Appl. Math., 42 (2009), 290-312.  doi: 10.1016/j.aam.2006.07.008.  Google Scholar

[3]

I. Anderson, Introduction to variational bicomplex, Contemp. Math., 132 (1992), 51-73.   Google Scholar

[4]

M. de León and P. R. Rodrigues, The inverse problem of Lagrangian dynamics for higher-order differential equations -a geometrical approach, Inverse Prob., 8 (1992), 525-540.  doi: 10.1088/0266-5611/8/4/006.  Google Scholar

[5]

C. E. Durán and D. Otero, The projective symplectic geometry of higher order variational problems: Minimality conditions, J. Geom. Mech., 8 (2016), 305-322.  doi: 10.3934/jgm.2016009.  Google Scholar

[6]

C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible grassmannians, Diff. Geom. Appl., 49 (2016), 447-472.  doi: 10.1016/j.difgeo.2016.10.001.  Google Scholar

[7]

C. E. Durán and H. Vitório, Moving planes, Jacobi curves and the dynamical approach to Finsler geometry, European Journal of Mathematics, 3 (2017), 1245-1274.   Google Scholar

[8]

H. Flanders, The Schwarzian as a curvature, J. Differential Geometry, 4 (1970), 515-519.  doi: 10.4310/jdg/1214429647.  Google Scholar

[9]

S. R. Garcia, The eigenstructure of complex symmetric operators, Recent Advances in Operator Theory, Operator Theory: Advances and Applications, 179 (2008), 169-183.   Google Scholar

[10]

I. Gelfand and S. Fomin, Calculus of Variations, Silverman Prentice-Hall, Inc., Englewood Cliffs, N. J. 1963.  Google Scholar

[11]

P. Piccione and D. V. Tausk, A students' guide to symplectic spaces, Grassmannians and Maslov index, Publicações Matemáticas do IMPA, Rio de Janeiro, 2008.  Google Scholar

[12]

W. SarletE. Engels and L. Y. Bahar, Time-dependent linear systems derivable from a variational principle, Int. J. Eng. Sci., 20 (1982), 55-66.  doi: 10.1016/0020-7225(82)90072-6.  Google Scholar

[13]

P. Vassiliou and I. Lisle, Geometric Approaches to Differential Equations, Australian Mathematical Society Lecture Series, Cambridge University Press, 2000. Google Scholar

[14]

E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces, Chelsea Publishing Co., New York, 1962.  Google Scholar

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